Complexity Bounds for Relational Algebra over Document Spanners

TL;DR

This paper studies the computational complexity of evaluating relational algebra queries over document spanners, identifying intractability in generalized cases and proposing syntactic constraints for efficient evaluation.

Contribution

It introduces new complexity bounds for relational algebra over regex-based document spanners, especially under schemaless semantics and with difference operators, and proposes automaton-based evaluation techniques.

Findings

Intractability of join and difference operations under schemaless semantics.

Polynomial delay evaluation achievable with syntactic constraints.

Automaton-based compilation method for efficient query evaluation.

Abstract

We investigate the complexity of evaluating queries in Relational Algebra (RA) over the relations extracted by regex formulas (i.e., regular expressions with capture variables) over text documents. Such queries, also known as the regular document spanners, were shown to have an evaluation with polynomial delay for every positive RA expression (i.e., consisting of only natural joins, projections and unions); here, the RA expression is fixed and the input consists of both the regex formulas and the document. In this work, we explore the implication of two fundamental generalizations. The first is adopting the "schemaless" semantics for spanners, as proposed and studied by Maturana et al. The second is going beyond the positive RA to allowing the difference operator. We show that each of the two generalizations introduces computational hardness: it is intractable to compute the natural join of two regex formulas under the schemaless semantics, and the difference between two regex formulas under both the ordinary and schemaless semantics. Nevertheless, we propose and analyze syntactic constraints, on the RA expression and the regex formulas at hand, such that the expressive power is fully preserved and, yet, evaluation can be done with polynomial delay. Unlike the previous work on RA over regex formulas, our technique is not (and provably cannot be) based on the static compilation of regex formulas, but rather on an ad-hoc compilation into an automaton that incorporates both the query and the document. This approach also allows us to include black-box extractors in the RA expression.

Equations

$$\alpha :=\varnothing \mid ϵ\mid \sigma \mid (\alpha \vee \alpha )\mid (\alpha \cdot \alpha )\mid {\alpha }^{\ast }\mid x\{\alpha \}$$

$$\text{llceil}\alpha \text{rrceil}(\mathbf{d})=\{\mu \mid ([1,|\mathbf{d}|+1⟩,\mu )\in [\alpha ](\mathbf{d})\}$$

$${\alpha }_{\mathsf{mail}}:=x_{\text{mail}}\{\gamma @\gamma .\gamma \}$$

$${\alpha }_{\mathsf{name}}:=(x_{\text{first}}\{\delta \}\text{textvisiblespace}{x}_{\text{last}}\{\delta \})\vee (x_{\text{last}}\{\delta \})$$

$${\alpha }_{\mathsf{phone}}:=x_{\text{phone}}\{{\beta }^{\ast }\}$$

$$\alpha_{\mathsf{info}}:=\Gamma^{*}\cdot(\epsilon\vee\hookleftarrow)\cdot\alpha_{\mathsf{name}}\cdot\text{\textvisiblespace}\cdot\Big{(}(\alpha_{\mathsf{phone}}\cdot\text{\textvisiblespace}\vee\epsilon)\cdot\alpha_{\mathsf{mail}}\Big{)}\cdot\hookleftarrow\cdot\Gamma^{*}$$

$$(q_0,i_0)\stackrel{o_1}{\to }\cdots (q_{m-1},i_{m-1})\stackrel{o_m}{\to }(q_m,i_m)$$

$$\alpha_{\mathsf{UKm}}:=\Big{(}\epsilon\vee(\Gamma^{*}\cdot\hookleftarrow)\Big{)}\cdot\Gamma^{*}\cdot\text{\textvisiblespace}x_{\textit{mail}}\{\gamma\mathtt{@}\gamma\mathtt{.}\mathsf{uk}\}\cdot\hookleftarrow\cdot\Gamma^{*}$$

$${\gamma }_1:={\gamma }_{x_1}\cdots {\gamma }_{x_n}\cdot \mathsf{a}$$

$${\gamma }_{x_i}:=(x_i^{1,\mathsf{t}}\{ϵ\}\cdots x_i^{m,\mathsf{t}}\{ϵ\})\vee (x_i^{1,\mathsf{f}}\{ϵ\}\cdots x_i^{m,\mathsf{f}}\{ϵ\}).$$

$${\gamma }_2:=\mathsf{a}\cdot ({\delta }_1\cdots {\delta }_m)$$

$$\phi :=(x\vee y\vee z)\wedge (\neg x\vee y\vee \neg z).$$

$${\delta }_1=x^{1,\mathsf{t}}\{ϵ\}\vee y^{1,\mathsf{t}}\{ϵ\}\vee z^{1,\mathsf{t}}\{ϵ\}$$

$${\delta }_2=x^{2,\mathsf{f}}\{ϵ\}\vee y^{2,\mathsf{t}}\{ϵ\}\vee z^{2,\mathsf{f}}\{ϵ\}$$

$${\gamma }_2:=\mathsf{a}\cdot (x^{1,\mathsf{t}}\{ϵ\}\vee y^{1,\mathsf{t}}\{ϵ\}\vee z^{1,\mathsf{t})}\{ϵ\})\cdot$$

$$(x^{2,\mathsf{f}}\{ϵ\}\vee y^{2,\mathsf{t}}\{ϵ\}\vee z^{2,\mathsf{f}}\{ϵ\}).$$

$${\gamma }_1:=$$

$$\displaystyle\big{(}{y}^{1,\mathsf{t}}\{\epsilon\}{y}^{2,\mathsf{t}}\{\epsilon\}\lor{y}^{1,\mathsf{f}}\{\epsilon\}{y}^{2,\mathsf{f}}\{\epsilon\}\big{)}\cdot$$

$$\displaystyle\big{(}{z}^{1,\mathsf{t}}\{\epsilon\}{z}^{2,\mathsf{t}}\{\epsilon\}\lor{z}^{1,\mathsf{f}}\{\epsilon\}{z}^{2,\mathsf{f}}\{\epsilon\}\big{)}\cdot\mathsf{a}\,.$$

$$\mathsf{dom}({\mu }_1)=\{x^{1,\mathsf{t}},x^{2,\mathsf{t}},y^{1,\mathsf{f}},y^{2,\mathsf{f}},z^{1,\mathsf{f}},z^{2,\mathsf{f}}\}$$

$$\mathsf{dom}({\mu }_2)=\{x^{1,\mathsf{f}},x^{2,\mathsf{f}},y^{1,\mathsf{t}},y^{2,\mathsf{t}},z^{1,\mathsf{t}},z^{2,\mathsf{t}}\}$$

$${\rho }_1:=(q_0,1)\stackrel{x⊢}{\to }(q_1,1)\stackrel{\mathtt{a}}{\to }(q_1,2)\stackrel{⊣x}{\to }(q_2,2)$$

$${\rho }_2:=(q_0,1)\stackrel{\mathtt{a}}{\to }(q_2,2)$$

$$\mathsf{funcRGX}⊊\mathsf{dfuncRGX}⊊\mathsf{seqRGX}$$

$$\text{llceil}\mathsf{funcRGX}\text{rrceil}⊊\text{llceil}\mathsf{dfuncRGX}\text{rrceil}=\text{llceil}\mathsf{seqRGX}\text{rrceil}$$

$$(x_1\{{\Sigma }^{\ast }\}\vee y_1\{{\Sigma }^{\ast }\})\cdots (x_n\{{\Sigma }^{\ast }\}\vee y_n\{{\Sigma }^{\ast }\})$$

$${\beta }_i:=((x_i\{ϵ\}\cdot \mathtt{a})\vee x_i\{\mathtt{a}\})$$

$${\gamma }_1:={\beta }_1\cdots {\beta }_n$$

$${\gamma }_2^i=$$

$${\beta }_{i_2+1}\cdots {\beta }_{i_3-1}\cdot {\delta }_{i_3}\cdot {\beta }_{i_3+1}\cdots {\beta }_n$$

$${\gamma }_2:=\underset{1\le i\le m}{\bigvee }{\gamma }_2^i.$$

$$\phi =(x\vee y\vee z)\wedge (\neg x\vee y\vee \neg z)$$

Full text

We show that each of the two generalizations introduces computational hardness: it is intractable to compute the natural join of two regex formulas under the schemaless semantics, and the difference between two regex formulas under both the ordinary and schemaless semantics. Nevertheless, we propose and analyze syntactic constraints, on the RA expression and the regex formulas at hand, such that the expressive power is fully preserved and, yet, evaluation can be done with polynomial delay. Unlike the previous work on RA over regex formulas, our technique is not (and provably cannot be) based on the static compilation of regex formulas, but rather on an ad-hoc compilation into an automaton that incorporates both the query and the document. This approach also allows us to include black-box extractors in the RA expression.

1. Introduction

The abundance and availability of valuable textual resources position text analytics as a standard component in data-driven workflows. To facilitate the integration with textual content, a core operation is Information Extraction (IE)—the extraction of structured data from text. IE arises in a large variety of domains, including biology and biomedical analysis, social media analysis, cyber security,111See, e.g., the TA-COS workshop at http://www.ta-cos.org/. system and network log analysis, and business intelligence, to name a few (DBLP:journals/ftdb/Sarawagi08, ; chakraborty2014text, ). Rules for IE are used in commercial systems and academic prototypes for text analytics, either as a standalone extraction language or within machine-learning models. A common paradigm for rule programming is the one supported by IBM’s SystemT (DBLP:conf/acl/LiRC11, ; DBLP:conf/naacl/ChiticariuDLRZ18, ), which exposes a collection of atomic (sometimes called “primitive”) extractors of relations from text (e.g., tokenizer, dictionary lookup, part-of-speech tagger and regular-expression matcher), together with a relational algebra for manipulating these relations. In Xlog (DBLP:conf/vldb/ShenDNR07, ), user-defined functions provide the atomic extractors, and Datalog is used for relational manipulation. In DeepDive (DBLP:journals/sigmod/SaRR0WWZ16, ), rules are used for generating features that are translated into the factors of a statistical model with machine-learned parameters. Feature declaration combines atomic extractors alongside relational operators thereof.

Document spanners

In this work, we explore complexity aspects of IE within the framework of document spanners (or just spanners for short) (DBLP:journals/jacm/FaginKRV15, ). In this framework, a document is a string over a fixed finite alphabet, and a spanner extracts from every input document a relation of intervals within the document. An interval, called span, is represented by its starting and ending indices in the document. An example of a spanner is a regex formula, which is a regular expression with capture variables that correspond to the relational attributes. The most studied language for specifying spanners is that of the regular spanners: the closure of regex formulas under the classic relational algebra: projection, natural join, union, and difference (DBLP:journals/jacm/FaginKRV15, ). Equally expressive formalisms include non-recursive Datalog over regex formulas (DBLP:journals/tods/FaginKRV16, ) and the variable-set automaton (vset-automaton for short), which is a nondeterministic finite-state automaton (NFA) that can open and close variables while running.

Since the framing of the spanner framework, there has been a considerable effort to delineate the computational complexity of spanner evaluation, with a special focus on the regular representations (regex formulas and vset-automata) of the atomic extractors. Florenzano et al. (DBLP:conf/pods/FlorenzanoRUVV18, ) studied the data complexity (where the spanner is fixed and the input consists of only the document), and so did Fagin et al. (DBLP:journals/corr/abs-1712-08198, ) who showed that the closure of regex formulas under Datalog characterizes the class of polynomial-time spanners. Freydenberger et al. (DBLP:journals/mst/FreydenbergerH18, ; DBLP:conf/pods/FreydenbergerKP18, ) studied the combined complexity (where the input consists of both the query and the document) for conjunctive queries, and unions of conjunctive queries, over spanners. More recently, Amarilli et al. (DBLP:journals/corr/AmarilliBMN18, ) presented an evaluation algorithm with tractability properties under both data and combined complexity; we further discuss this algorithm later on.

For complexity analysis, there are important advantages to yardsticks that take the atomic extractors (e.g., regex formulas or vset-automata) as input, rather than regarding them small or fixed. First, the size of these extractors can be quite large in practice. Taking examples from RegExLib.com, each of the regexes for recognizing the RFC 2822 mailbox format (regexp id 711) and date format (regexp id 969) uses more than 350 ASCII symbols, and a regex for identifying US addresses (regexp id 1564) uses more than 2,000 ASCII symbols. Furthermore, automata may be constructed by automatic (machine-learning) processes that achieve accuracy through the granularity of the automaton. The paradigm of Artificial Neural Networks (ANNs) in natural-language processing has motivated the conversion of ANN models such as recurrent neural networks and convolutional neural networks into automata (DBLP:journals/corr/abs-1808-09357, ; DBLP:journals/nn/OmlinG96, ; DBLP:conf/icml/WeissGY18, ), where the number of states may reach tens of thousands to match the expressiveness of the numeric parameters (DBLP:conf/icml/WeissGY18, ). Another advantage of regarding the atomic extractors as input is more technical: polynomial-time combined complexity allows to incorporate vset-automata of which size may depend on the input document. This approach allows to establish tractability even if we join with schemaless spanners that cannot be represented as RA expressions over regular spanners, such as string equality (DBLP:conf/pods/FreydenbergerKP18, ).

Schema-based functionality vs schemaless sequentiality

As defined by Fagin et al. (DBLP:journals/jacm/FaginKRV15, ), the spanners are schema-based in the sense that every spanner is associated with a fixed and finite set $X$ of variables, playing the roles of attributes in relational databases, so that every tuple they extract from a document assigns a value to each variable of $X$. The regex formulas conform to this property in the sense that every parse tree contains exactly one occurrence of each variable; such regex formulas are said to be functional. Freydenberger (DBLP:conf/icdt/Freydenberger17, ) applied the property of functionality to vset-automata: a vset-automaton is functional if every accepting path properly opens and closes every variable exactly once.

The functionality property can be tested in polynomial time for both regex formulas (DBLP:journals/jacm/FaginKRV15, ) and vset-automata (DBLP:journals/mst/FreydenbergerH18, ). Moreover, functional vset-automata generalize functional regex formulas in the sense that every instance of the former can be transformed in linear time into an instance of the latter (but not necessarily the other way around). Beyond that, functional vset-automata (and regex formulas) possess various desired tractability features (DBLP:conf/pods/FreydenbergerKP18, ). First, they can be evaluated with polynomial delay under combined complexity. Second, the natural join of two functional vset-automata can be compiled in polynomial time into one functional vset-automaton, and so can the union of two vset-automata and the projection of a vset-automaton to a subset of its variables. Consequently, every combination of functional vset-automata can be evaluated with polynomial delay, as long as this combination is via the positive operators of the relational algebra.

More recently, Maturana et al. (DBLP:conf/pods/MaturanaRV18, ) introduced a schemaless version of spanners that allows for incomplete extraction from documents, in the spirit of the SPARQL model (DBLP:journals/tods/PerezAG09, ). There, two extracted tuples may assign spans to different sets of variables. The analog of functionality is sequentiality: a regex formula is sequential is every parse tree includes at most one occurrence of every variable, and a vset-automaton is sequential if every accepting path properly opens and closes every variable at most once. Again, in polynomial time we can test for sequentiality and transform a sequential regex formula into a sequential vset-automaton; moreover, sequential vset-automata can be evaluated with polynomial delay under combined complexity (DBLP:conf/pods/MaturanaRV18, ). In fact, the aforementioned algorithm of Amarilli et al. (DBLP:journals/corr/AmarilliBMN18, ) enumerates with polynomial delay under combined complexity, and, under data complexity, with constant delay following a linear pre-processing of the document.222This is the spanner analog of a recent line of work on the enumeration complexity of database and string queries (DBLP:conf/csl/BaganDG07, ; DBLP:conf/icdt/CarmeliK18, ; DBLP:conf/pods/NiewerthS18, ; DBLP:conf/icdt/Segoufin13, ). Since functional vset-automata are also sequential, this algorithm also applies to the schema-based spanners, and improves upon (and, in fact, generalizes the applicability of) the constant-delay algorithm of Florenzano et al. (DBLP:conf/pods/FlorenzanoRUVV18, ).

Contribution

The state of affairs leaves open two fundamental questions regarding the combined complexity of query evaluation.

• •

Does the tractability for the positive relational algebra generalize from the schema-based case to the schemaless case?

• •

Does the tractability extend beyond the positive operators (in either the schema-based or schemaless case)? In particular, can we enumerate with polynomial delay the difference between two functional vset-automata?

We prove that the answers to both questions are negative. More specifically, it is NP-complete to determine whether the natural join of two sequential regex formulas is nonempty (Theorem 3.1), and it is NP-complete to determine whether the difference between two given functional regex formulas is nonempty (Theorem 4.1).

We formulate various syntactic restrictions that allow to avoid hardness. In particular, we show that polynomial delay is retained if we bound the number of common variables between the two operands of the natural join and difference. For the natural join, we also present a normal form for schemaless regex formulas and vset-automata, namely disjunctive functional, that are more restricted than, yet as expressive as, their sequential counterparts; yet, the natural join of two disjunctive-functional vset-automata can be compiled into a disjunctive-functional vset-automaton in polynomial time (hence, evaluated with polynomial delay).

In contrast to the natural join, the tractability of the difference between vset-automata with a bounded number of common variables cannot be established via compilation into a single vset-automaton. This is due to the simple reason that, in the case of Boolean spanners, the problem is the same as the difference between two NFAs, where the compilation necessitates an exponential blowup (DBLP:journals/tcs/Jiraskova05, ). Nevertheless, we establish the tractability by transforming the difference into a natural join with a special vset-automaton that is built ad-hoc for the input document.

In summary, our complexity upper bounds are established in two main approaches. The first is based on a document-independent compilation of the input vset-automata (or regex formulas) into a new vset-automaton. The second is based on a compilation of both the input vset-automata and the input document into a new, ad-hoc vset-automaton. We refer to first approach as static compilation and to the second as ad-hoc compilation.

We compose our tractability results into more general queries by proposing a new complexity measure that is specialized to spanners. Recall that the evaluation problem has three components: the document, the atomic spanners (e.g., regex formulas), and the relational algebra that combines the atomic spanners, which we refer to as the RA tree. Under combined complexity, all three are given as input; under data complexity, the document is given as input and the rest are fixed; there is also the expression complexity (DBLP:conf/stoc/Vardi82, ) where the document is fixed and the rest are given as input. We propose the extraction complexity, where the RA tree is fixed, and the input consists of the document and the atomic spanners (mapped to their corresponding positions in the RA tree). We present and discuss conditions that cast the extraction complexity tractable (polynomial-delay evaluation) and intractable (NP-hard nonemptiness). Interestingly, since the tractability of an RA tree is based on ad-hoc compilation, we can incorporate there any polynomial-time spanner, as long as its dimension is bounded by a constant.

Organization

The rest of the paper is organized as follows. In Section 2, we present the basic terminology and concepts. We investigate the complexity of the natural-join operator in Section 3 and the difference operator in Section 4. We extend our development to the extraction complexity in Section 5, and conclude in Section 6. To meet space constraints, some of the proof are given in the Appendix.

2. Preliminaries

We first introduce the main definitions and terminology, mainly from the literature on document spanners (DBLP:journals/jacm/FaginKRV15, ; DBLP:conf/pods/MaturanaRV18, ).

2.1. Document Spanners

Documents and spans

We fix a finite alphabet $\Sigma$ of symbols. By a document or string we refer to a finite sequence $\mathbf{d}=\sigma_{1}\cdots\sigma_{n}$ over $\Sigma$ (that is, each $\sigma_{i}$ is in $\Sigma$), that is, a member of $\Sigma^{*}$. The length $n$ of the document $\mathbf{d}=\sigma_{1}\cdots\sigma_{n}$ is denoted by $|\mathbf{d}|$. A span is a pair $[i,j\rangle$ of indices $1\leq i\leq j\leq n+1$ that marks a substring of $\mathbf{d}$. The term $\mathbf{d}_{[i,j\rangle}$ denotes the substring $\sigma_{i}\cdots\sigma_{j-1}$. Note that $\mathbf{d}_{[i,i\rangle}$ is the empty string, and that $\mathbf{d}_{[1,n+1\rangle}$ is $\mathbf{d}$. Note also that the spans $[i,i\rangle$ and $[j,j\rangle$, where $i\neq j$, are different objects, even though the substrings $\mathbf{d}_{[i,i\rangle}$ and $\mathbf{d}_{[j,j\rangle}$ are equal. We denote by $\mathsf{spans}$ the set of all spans of all strings, that is, all expressions $[i,j\rangle$ where $1\leq i\leq j$. By $\mathsf{spans}(\mathbf{d})$ we denote the set of all spans of $\mathbf{d}$.

Schemaless spanners

We assume a countably infinite set $\mathsf{Vars}$ of variables, and assume that $\mathsf{Vars}$ is disjoint from $\Sigma$ and $\Sigma^{*}$. A schemaless (document) spanner is a function that maps each document into a finite collection of tuples (referred to as mappings) that assign spans to variables. More formally, a mapping to $\mathbf{d}$ is a function $\mu$ from a finite set of variables, called the domain of $\mu$ and denoted $\mathsf{dom}(\mu)$, into $\mathsf{spans}(\mathbf{d})$. A schemaless spanner is a function $P$ that maps every document $\mathbf{d}$ into a finite set $P(\mathbf{d})$ of mappings.

For a schemaless spanner $P$ and a document $\mathbf{d}$, different mappings in $P(\mathbf{d})$ may have different domains. This stands in contrast to the (schema based) spanners of Fagin et al. (DBLP:journals/jacm/FaginKRV15, ), where $P$ is such that there exists a set $V_{P}$ of variables where every document $\mathbf{d}$ and mapping $\mu\in P(\mathbf{d})$ satisfy $\mathsf{dom}(\mu)=V_{P}$; in this case, we may refer to $P$ as a schema-based spanner.

Example 2.1.

Let $\Gamma$ be the alphabet consists of lowercase and uppercase English letters: $a,\cdots,z,A\cdots,Z$; digits: $0,\cdots,9$; and symbols: ␣ that stands for whitespace, ‘.’ and ‘$\mathsf{@}$’. Let $\Delta=\{\hookleftarrow\}$ where $\hookleftarrow$ stands for end of line. The input document $\mathbf{d}_{\mathsf{Students}}$ over $\Gamma\cup\Delta$ given in Figure 1 holds personal information on students. (Some of the positions are marked underneath for convenience.) Each line in the document describes information on a student in the following format: first name (if applicable), last name, phone number (if applicable) and email address. There are spaces in between these elements. The schemaless document spanner $P_{\textit{StudInfo}}$ extracts from the input document $\mathbf{d}_{\mathsf{Students}}$ the following set of mappings, given in a table for convenience.

Note that the empty cells in the table stand for undefined. That is, we can conclude, for example, that $x_{\textit{last}}\notin\mathsf{dom}(\mu_{2})$.

In the next sections, we discuss different representation languages for schemaless spanners. Whenever a schemaless spanner is represented by a description $q$, we denote by $\mathord{\llceil{q}\rrceil}$ the actual schemaless spanner that $q$ represents. We are using the notation $\mathord{\llceil{\cdot}\rrceil}$ in order to clearly distinguish the schemaless semantics from the schema based semantics of Fagin et al. (DBLP:journals/jacm/FaginKRV15, ) who use $\mathord{\llbracket{\cdot}\rrbracket}$. This distinction is critical in the case of the vset-automata that we define later on.

2.2. Regex Formulas

One way of representing a schemaless spanner is by means of a regex formula, which is a regular expression with capture variables, as allowed by the grammar

[TABLE]

where $\sigma\in\Sigma$ and $x\in\mathsf{Vars}$. For convenience, we sometimes put regex formulas in parentheses and also omit parentheses, as long as the meaning remains clear. We denote by $\mathsf{Vars}(\alpha)$ the set of variables that appear in $\alpha$. By $\mathsf{RGX}$ we denote the class of regex formulas.

Following Maturana et al. (DBLP:conf/pods/MaturanaRV18, ), we interpret regex formulas as schemaless spanners in the following manner. The following grammar defines the application of a regex formula $\alpha$ on a document $\mathbf{d}=\sigma_{1}\cdots\sigma_{n}$, where the result is a pair $(s,\mu)$ where $s$ is a span of $\mathbf{d}$ and $\mu$ is a mapping to $\mathbf{d}$.

• •

$[{\emptyset}](\mathbf{d}):=\emptyset$;

• •

$[{\epsilon}](\mathbf{d}):=\{([i,i\rangle,\emptyset)\mid i=1,\dots,n\}$;

• •

$[{\sigma}](\mathbf{d}):=\{([i,i+1\rangle,\emptyset)\mid\sigma_{i}=\sigma\}$;

• •

$[{x\{\alpha\}}](\mathbf{d}):=\{([i,j\rangle,\mu\cup\{x\mapsto[i,j\rangle\})\mid([i,j\rangle,\mu)\in[{\alpha}](\mathbf{d})$ and $x\not\in\mathsf{dom}(\mu)\}$;

• •

$[{\alpha_{1}\vee\alpha_{2}}](\mathbf{d}):=[{\alpha_{1}}](\mathbf{d})\cup[{\alpha_{2}}](\mathbf{d})$;

• •

$[{\alpha_{1}\cdot\alpha_{2}}](\mathbf{d}):=\{([i,j\rangle,\mu_{1}\cup\mu_{2})\mid\exists i^{\prime}\mbox{ s.t. }([i,i^{\prime}\rangle,\mu_{1})\in[{\alpha_{1}}](\mathbf{d})$, $([i^{\prime},j\rangle,\mu_{2})\in[{\alpha_{2}}](\mathbf{d})$, and $\mathsf{dom}(\mu_{1})\cap\mathsf{dom}(\mu_{2})=\emptyset\}$;

• •

$[{\alpha^{*}}](\mathbf{d}):=\bigcup_{i=0}^{\infty}[{\alpha^{i}}](\mathbf{d})$ where $\alpha^{i}$ stands for the concatenation of $i$ copies of $\alpha$.

The result of applying $\alpha$ to $\mathbf{d}$ is then defined as follows.

[TABLE]

We denote by $\mathord{\llceil{\mathsf{RGX}}\rrceil}$ the class of schemaless spanners that can be expressed using the regex formulas. Similarly, for every subclass $\mathsf{R}\subseteq\mathsf{RGX}$, we denote by $\mathord{\llceil{\mathsf{R}}\rrceil}$ the class of spanners expressible by an expression in $\mathsf{R}$.

Syntactic restrictions

Fagin et al. (DBLP:journals/jacm/FaginKRV15, ) introduced the class of regex formulas that are interpreted as schema-based spanners, namely the functional regex formulas. To define functional regex formulas, we first use the following inductive definition. A regex formula $\alpha$ is functional for a set $V\subseteq\mathsf{Vars}$ of variables if:

• •

$\alpha\in\Sigma^{*}$ and $V=\emptyset$;

• •

$\alpha=\alpha_{1}\vee\alpha_{2}$ and each $\alpha_{i}$ is functional for $V$;

• •

$\alpha=\alpha_{1}\cdot\alpha_{2}$ and there exists $V_{1}\subseteq V$ such that $\alpha_{1}$ is functional for $V_{1}$ and $\alpha_{2}$ is functional for $V\setminus V_{1}$;

• •

$\alpha=\alpha_{0}^{*}$ and $\alpha_{0}$ is functional for $\emptyset$;

• •

$\alpha=x\{\alpha_{0}\}$ and $\alpha_{0}$ is functional for $V\setminus\{x\}$.

Finally, a regex formula $\alpha$ is functional if it is functional for the set $\mathsf{Vars}(\alpha)$ of its variables.

Maturana et al. (DBLP:conf/pods/MaturanaRV18, ) pointed at a wider fragment of regex formulas, namely the sequential regex formula, that has some desirable properties, as will be discussed later. A regex formula $\alpha$ is sequential if the following conditions hold:

• •

Every sub-formula of the form $\alpha_{1}\cdot\alpha_{2}$ satisfies $\mathsf{Vars}(\alpha_{1})\cap\mathsf{Vars}(\alpha_{2})=\emptyset$;

• •

Every sub-formula of the form $\alpha^{*}$ satisfies $\mathsf{Vars}(\alpha)=\emptyset$;

• •

Every sub-formula of the form $x\{\alpha\}$ satisfies $x\not\in\mathsf{Vars}(\alpha)$.333We added this restriction to the original definition (DBLP:conf/pods/MaturanaRV18, ) since it was mistakenly omitted, as the authors confirmed.

We denote by $\mathsf{funcRGX}$ and $\mathsf{seqRGX}$ the classes of functional and sequential regex formulas, respectively. Maturana et al. (DBLP:conf/pods/MaturanaRV18, ) showed that $\mathsf{funcRGX}\subsetneq\mathsf{seqRGX}$, that is, every functional regex formula is sequential, but some sequential regex formulas are not functional, as the next example illustrates.

Example 2.2.

Let us define the following regex formulas over the alphabet $\Gamma\cup\Delta$ from Example 2.1:

[TABLE]

[TABLE]

[TABLE]

where $\gamma:=(a\vee\cdots\vee z)^{*}$, $\delta:=(A\vee\cdots\vee Z)(a\vee\cdots\vee z)^{*}$,and $\beta:=(0\vee\cdots\vee 9)^{*}$. Based on the previous regex formulas, we define the regex formula that represents the schemaless spanner $P_{\textit{StudInfo}}$ from Example 2.1:

[TABLE]

Note that this is regex formula is sequential but not functional since the variables $x_{\textit{first}}$ and $x_{\textit{phone}}$ are optional.

2.3. Vset-Automata

In addition to regex formulas, we use the variable-set automata (abbreviated vset-automata) for representing schemaless spanners, as defined by Maturana et al. (DBLP:conf/pods/MaturanaRV18, ) as a schemaless adaptation of the vset-automata of Fagin et al. (DBLP:journals/jacm/FaginKRV15, ).

A vset-automaton, VA for short, is a tuple $(Q,q_{0},F,\delta)$, where $Q$ is set of states, $q_{0}\in Q$ is the initial state, $F\subseteq Q$ is the set of accepting states, 444The original definition by Fagin et al. (DBLP:journals/jacm/FaginKRV15, ) used a single accepting state. We can extend this definition to multiple accepting states without changing the expressive power by simulating a single accepting state with epsilon transitions. and $\delta$ is a transition relation consisting of epsilon transitions of the form $(q,\epsilon,p)$, letter transitions of the form $(q,\sigma,p)$ and variable transitions of the form $(q,\mathop{v{\vdash}},p)$ or $(q,\mathbin{{\dashv}v},p)$ where $q,p\in Q$, $\sigma\in\Sigma$, and $v\in\mathsf{Vars}$. The symbols $\mathop{v{\vdash}}$ and $\mathbin{{\dashv}v}$ are special symbols to denote the opening or closing of a variable $v$. We define the set $\mathsf{Vars}(A)$ as the set of all variables $v$ that are mentioned in some transition of $A$. For every finite set $V\subseteq\mathsf{Vars}$ we define the set $\Gamma_{V}:=\{\mathop{v{\vdash}},\mathbin{{\dashv}v}:v\in V\}$ of variable operations. A run $\rho$ over a document $\mathbf{d}:=\sigma_{1}\cdots\sigma_{n}$ is a sequence of the form

[TABLE]

where:

• •

the $i_{j}$ are indexes in $\mathord{\{1,\dots,n+1\}}$ such that $i_{0}=1$ and $i_{m}=n+1$;

• •

each $o_{j}$ is in $\Sigma\cup\{\epsilon\}\cup\Gamma_{\mathsf{Vars}(A)}$;

• •

$i_{j+1}=i_{j}$ whenever $o_{j}\in\Gamma_{\mathsf{Vars}(A)}$, and $i_{j+1}=i_{j}+1$ otherwise;

• •

for all $j>0$ we have $(q_{j-1},o_{j},q_{j})\in\delta$.

A run $\rho$ is called valid if for every variable $v$ the following hold:

• •

$v$ is opened (or closed) at most once;

• •

if $v$ is opened at some position $i$ then it is closed at some position $j$ with $i\leq j$;

• •

if $v$ is closed at some position $j$ then it is opened at some position $i$ with $i\leq j$.

A run is called accepting if its last state is an accepting state, i.e., $q_{m}\in F$. For an accepting and valid run $\rho$, we define $\mu_{\rho}$ to be the mapping that maps the variable $v$ to the span $[i_{j},i_{j^{\prime}}\rangle$ where $o_{i_{j}}=\mathop{v{\vdash}}$ and $o_{i_{j^{\prime}}}=\mathbin{{\dashv}v}$. Finally, the result $\mathord{\llceil{A}\rrceil}(\mathbf{d})$ of applying the schemaless spanner represented by $A$ on a document $\mathbf{d}$ is defined as the set of all assignments $\mu_{\rho}$ for all valid and accepting runs $\rho$ of $A$ on $\mathbf{d}$. A VA is called sequential if all of its accepting runs are valid, and it is called functional if each such run also include all of its variables $\mathsf{Vars}(A)$. Note that sequential VAs corresponds with schemaless spanners, whereas functional with complete.

Example 2.3.

Let $A$ be the following sequential VA:

q_{0}$$q_{1}$$q_{2}$$\Sigma$$\mathop{x{\vdash}}$$\Sigma$$\mathbin{{\dashv}x}$$\Sigma$$\Sigma

Omitting the transition from $q_{0}$ to $q_{2}$ results in a functional VA. The same schemaless spanner as that represented by $A$ is given by the sequential regex formula $\alpha:=(\Sigma^{*}x\{\Sigma^{*}\}\Sigma^{*})\vee(\Sigma^{+})$ where $\Sigma^{+}$ stands for $\Sigma\Sigma^{*}$.

2.4. Algebraic Operators

Before we define the algebra over schemaless spanners, we present some basic definitions. Two mappings $\mu_{1}$ and $\mu_{2}$ are compatible if they agree on every common variable, that is, $\mu_{1}(x)=\mu_{2}(x)$ for all $x\in\mathsf{dom}(\mu_{1})\cap\mathsf{dom}(\mu_{2})$. In this case, we define $\mu:=\mu_{1}\cup\mu_{2}$ as the mapping with $\mathsf{dom}(\mu)=\mathsf{dom}(\mu_{1})\cup\mathsf{dom}(\mu_{2})$ such that $\mu(x)=\mu_{1}(x)$ for all $x\in\mathsf{dom}(\mu_{1})$ and $\mu(x)=\mu_{2}(x)$ for $x\in\mathsf{dom}(\mu_{2})$.

The correspondents of the relational-algebra operators are defined similarly to the SPARQL formalism (DBLP:journals/tods/PerezAG09, ). In particular, the operators union, projection, natural join, and difference are defined as follows for all schemaless spanners $P_{1}$ and $P_{2}$ and documents $\mathbf{d}$.

• •

Union:   The union $P:=P_{1}\cup P_{2}$ is defined by $P(\mathbf{d}):=P_{1}(\mathbf{d})\cup P_{2}(\mathbf{d})$.

• •

Projection:   The projection $P:=\pi_{Y}P_{1}$ is defined by P(\mathbf{d})=\mathord{\{\mu\scalebox{1.0}[0.85]{\raisebox{8.99994pt}{\mbox{\rotatebox{270.0}{\leftharpoonup}}} }Y\mid\mu\in P_{1}(\mathbf{d})\}} where

$\leftharpoonup$

stands for the restriction of $\mu$ to the variables in $\mathsf{dom}(\mu)\cap Y$.

• •

Natural join:   The (natural) join $P:=P_{1}\bowtie P_{2}$ is defined to be such that $P(\mathbf{d})$ consists of all mappings $\mu_{1}\cup\mu_{2}$ such that $\mu_{1}\in P_{1}(\mathbf{d})$, $\mu_{2}\in P_{2}(\mathbf{d})$ and $\mu_{1}$ and $\mu_{2}$ are compatible.

• •

Difference:   The difference $P:=P_{1}\setminus P_{2}$ is defined to be such that $P(\mathbf{d})$ consists of all mappings $\mu_{1}\in P_{1}(\mathbf{d})$ such that no $\mu_{2}\in P_{2}(\mathbf{d})$ is compatible with $\mu_{1}$.

We allow the use of these operators for spanners represented by regex formulas or VAs and also for more complex spanner representations, e.g., $\mathord{\llceil{A_{1}}\rrceil}\bowtie\mathord{\llceil{A_{2}}\rrceil}$. In this case, we use an the abbreviated notation $\mathord{\llceil{A_{1}\bowtie A_{2}}\rrceil}$ instead of $\mathord{\llceil{A_{1}}\rrceil}\bowtie\mathord{\llceil{A_{2}}\rrceil}$. We make the clear note that when the above operators are applied on schema-based spanners, they are the same as those of Fagin et al. (DBLP:journals/jacm/FaginKRV15, ).

Example 2.4.

Let us consider our input document $\mathbf{d}_{\mathsf{Students}}$ from Figure 1. Assume one wants to filter out from the results obtained by applying the spanner $P_{\textit{StudInfo}}$ from Example 2.2 on $\mathbf{d}_{\mathsf{Students}}$ the mappings that correspond with students from universities within the UK. It is given that students study in the UK if and only if their email addresses end with the letters ‘$\mathsf{uk}$’. We phrase the following regex formula that extracts such email addresses:

[TABLE]

where $\gamma$ is as defined in Example 2.2. In this case, the desired output is given by $\mathord{\llceil{\alpha_{\mathsf{info}}\setminus\alpha_{\mathsf{UKm}}}\rrceil}(\mathbf{d}_{\mathsf{Students}})$ who consists of the mappings $\mu_{1}$ and $\mu_{2}$ from Example 2.1.

2.5. Complexity

Let $\mathcal{L}$ be a representation language for schemaless spanners (e.g., the class of regex formulas or the class of VAs). Given $q\in\mathcal{L}$ and a document $\mathbf{d}$, we are interested in the decision problem that checks whether $\mathord{\llceil{q}\rrceil}(\mathbf{d})$ is not empty. In that case, we are also interested in evaluating $\mathord{\llceil{q}\rrceil}(\mathbf{d})$. Note that we study the combined complexity of these problems, as both $q$ and $\mathbf{d}$ are regarded as input.

Under the combined complexity, “polynomial time” is not a proper yardstick of efficiency for evaluating $\mathord{\llceil{q}\rrceil}(\mathbf{d})$, since this set can contain exponentially many mappings. We thus use efficiency yardsticks of enumeration (DBLP:journals/ipl/JohnsonP88, ). In particular, our evaluation algorithm takes $q$ and $\mathbf{d}$ as input, and it outputs all the mappings of $\mathord{\llceil{q}\rrceil}(\mathbf{d})$, one by one, without duplicates. The algorithm runs in polynomial total time if its execution time is polynomial in the combined size of $q$, $\mathbf{d}$ and $\mathord{\llceil{q}\rrceil}(\mathbf{d})$. The delay of the evaluation algorithm refers to the maximal time that passes between every two consecutive mappings. A well-known observation is that polynomial delay implies polynomial total time (but not necessarily vice versa), and that NP-hardness of the nonemptiness problem implies that no evaluation algorithm runs in polynomial total time, or else $\text{P}=\text{NP}$.

While deciding whether $\mathord{\llceil{q}\rrceil}(\mathbf{d})\neq\emptyset$ is $\mathrm{NP}$-hard whenever $q$ is given as a VA (DBLP:conf/icdt/Freydenberger17, ), this is not the case for sequential (and hence functional) VA:

Theorem 2.5.

(DBLP:journals/corr/AmarilliBMN18, )*   Given a sequential VA $A$ and a document $\mathbf{d}$, one can enumerate $\mathord{\llceil{A}\rrceil}(\mathbf{d})$ with polynomial delay.*

We call two schemaless spanner representations $q_{1}$ and $q_{2}$ equivalent if $\mathord{\llceil{q_{1}}\rrceil}\equiv\mathord{\llceil{q_{2}}\rrceil}$, that is, $\mathord{\llceil{q_{1}}\rrceil}$ and $\mathord{\llceil{q_{2}}\rrceil}$ are identical. Note that the translation of functional and sequential regex formulas to equivalent functional and sequential VAs, respectively, can be done in linear time (DBLP:conf/pods/FreydenbergerKP18, ; DBLP:conf/pods/MaturanaRV18, ). Hence, our lower bounds are usually shown for the nonemptiness of regex formulas and our upper bounds for the evaluation of VAs.

3. The Natural-Join Operator

To establish complexity upper bounds on the evaluation of schema-based spanners, Freydenberger et al. (DBLP:conf/pods/FreydenbergerKP18, ) used static compilation to compile the query (where the operands are regex formulas or VAs) into a single VA. In particular, they showed that two functional VAs can be compiled in polynomial time into a single equivalent VA that is also functional. Consequently, we can enumerate with polynomial delay the mappings of $\mathord{\llceil{A_{1}\bowtie A_{2}}\rrceil}(\mathbf{d})$, given functional VAs $A_{1}$ and $A_{2}$. The question is whether it generalizes to schemaless spanners: can we efficiently enumerate the mappings of $\mathord{\llceil{A_{1}\bowtie A_{2}}\rrceil}(\mathbf{d})$, given sequential (but not necessarily functional) $A_{1}$ and $A_{2}$? This is no longer the case, as the next theorem implies, even under the yardstick of expression complexity (DBLP:conf/stoc/Vardi82, ) in which the document is regarded as fixed. (Recall that a sequential regex formula can be translated in polynomial time into an equivalent VA (DBLP:conf/pods/MaturanaRV18, ).)

Theorem 3.1.

The following decision problem is $\mathrm{NP}$-complete. Given two sequential regex formulas $\gamma_{1}$ and $\gamma_{2}$ and an input document $\mathbf{d}$, is $\mathord{\llceil{\gamma_{1}\bowtie\gamma_{2}}\rrceil}(\mathbf{d})$ nonempty? The problem remains $\mathrm{NP}$-hard even if $\mathbf{d}$ is assumed to be of length one.

Proof.

Membership in $\mathrm{NP}$ is straightforward, so we focus on NP-hardness. We show a reduction from 3-CNF-satisfiability which is also known as 3SAT (Garey:1990:CIG:574848, ). The input for 3SAT is a formula $\varphi$ with the free variables $x_{1},\ldots,x_{n}$ such that $\varphi$ has the form $C_{1}\wedge\cdots\wedge C_{m}$, where each $C_{j}$ is a clause. In turn, each clause is a disjunction of three literals, where a literal has the form $x_{i}$ or $\neg x_{i}$ for $i=1,\dots,n$. The goal is to determine whether there is an assignment $\tau:\{x_{1},\ldots,x_{n}\}\rightarrow\{0,1\}$ that satisfies $\varphi$. Given a 3CNF formula $\varphi$, we construct two sequential regex formulas $\gamma_{1}$ and $\gamma_{2}$ such that there is a satisfying assignment for $\varphi$ if and only if $\mathord{\llceil{\gamma_{1}\bowtie\gamma_{2}}\rrceil}(\mathbf{d})\neq\emptyset$, where $\mathbf{d}$ is the document that consists of a single letter $\mathsf{a}$.

To construct $\gamma_{1}$ and $\gamma_{2}$, we associate every variable $x_{i}$ with $2m$ corresponding capture variables $x_{i}^{j,\ell}$ for $1\leq j\leq m$ and $\ell\in\mathord{\{\mathsf{t},\mathsf{f}\}}$. We then define

[TABLE]

where

[TABLE]

Intuitively, $\gamma_{x_{i}}$ verifies that the assignment to $x_{i}$ is consistent in all of the clauses. We then define

[TABLE]

where $\delta_{j}$ is the disjunction of regex formulas $\beta$ such that $\beta=x_{i}^{j,\mathsf{f}}\{\epsilon\}$ if $\neg{x_{i}}$ appears in $C_{j}$, and $\beta=x_{i}^{j,\mathsf{t}}\{\epsilon\}$ if $x_{i}$ appears in $C_{j}$. Intuitively, $\gamma_{2}$ verifies that at least one disjunct in each clause is evaluated true.

Let us consider the following example where

[TABLE]

In this case, we have

[TABLE]

[TABLE]

and, therefore,

[TABLE]

We also have

[TABLE]

It follows directly from the definition that both $\gamma_{1}$ and $\gamma_{2}$ are sequential. Moreover, $\mathord{\llceil{\gamma_{1}\bowtie\gamma_{2}}\rrceil}(\mathbf{d})$ is nonempty if and only if there are compatible mappings $\mu_{1}\in\mathord{\llceil{\gamma_{1}}\rrceil}(\mathbf{d})$ and $\mu_{2}\in\mathord{\llceil{\gamma_{2}}\rrceil}(\mathbf{d})$. Since $\gamma_{1}$ ends with the letter $\mathsf{a}$ whereas $\gamma_{2}$ starts with the letter $\mathsf{a}$, it holds that $\mu_{1}\in\mathord{\llceil{\gamma_{1}}\rrceil}(\mathbf{d})$ and $\mu_{2}\in\mathord{\llceil{\gamma_{2}}\rrceil}(\mathbf{d})$ are compatible if and only if $\mathsf{dom}(\mu_{1})\cap\mathsf{dom}(\mu_{2})=\emptyset$. We will show that $\mathord{\llceil{\gamma_{1}\bowtie\gamma_{2}}\rrceil}(\mathbf{d})$ is nonempty if and only if there is a satisfying assignment to $\varphi$.

The “only if” direction

Suppose that $\mathord{\llceil{\gamma_{1}\bowtie\gamma_{2}}\rrceil}(\mathbf{d})$ is nonempty. In this case, a satisfying assignment $\tau$ to $\varphi$ is encoded by the domain of $\gamma_{2}$ in the following way: if $x_{i}^{j,\ell}\in\mathsf{dom}(\mu_{2})$ then $\tau(x_{i})=\ell$. Observe that $\tau$ is well defined, due to the definition of $\gamma_{2}$.

In our example, the mapping $\mu_{1}\in\mathord{\llceil{\gamma_{1}}\rrceil}(\mathsf{a})$ with

[TABLE]

and the mapping $\mu_{2}\in\mathord{\llceil{\gamma_{2}}\rrceil}(\mathsf{a})$ with

[TABLE]

are compatible, and the satisfying assignment $\tau$ is encoded by $\mathsf{dom}(\mu_{2})$ and is given by $\tau(x)=\mathsf{f}$, $\tau(y)=\mathsf{t}$ and $\tau(z)=\mathsf{t}$.

The “if” direction

If there is a satisfying assignment $\tau$ to $\varphi$, then define the mappings $\mu_{1}\in\mathord{\llceil{\gamma_{1}}\rrceil}(\mathbf{d})$ and $\mu_{2}\in\mathord{\llceil{\gamma_{2}}\rrceil}(\mathbf{d})$ are defined by $x_{i}^{j,\ell}\in\mathsf{dom}(\mu_{2})$ whenever $j=\tau(x_{i})$ and $x_{i}^{j,\ell}\in\mathsf{dom}(\mu_{2})$ whenever $j\neq\tau(x_{i})$. These mapping are compatible, since $\mathsf{dom}(\mu_{1})\cap\mathsf{dom}(\mu_{2})=\emptyset$. We conclude that $\mathord{\llceil{\gamma_{1}\bowtie\gamma_{2}}\rrceil}(\mathbf{d})$ is nonempty.

We conclude the $\mathrm{NP}$-hardness of the problem of determining whether $\mathord{\llceil{\gamma_{1}\bowtie\gamma_{2}}\rrceil}(\mathbf{d})$ is nonempty, as claimed.

∎

In what follows, we suggest two different approaches to deal with this hardness.

3.1. Bounded Number of Shared Variables

We now consider the task of computing $\mathord{\llceil{A_{1}\bowtie A_{2}}\rrceil}(\mathbf{d})$, given sequential VAs $A_{1}$ and $A_{2}$ and a document $\mathbf{d}$. Next, we show that compiling the join into a new sequential VA is Fixed Parameter Tractable (FPT) when the parameter is the number of common variables.

Lemma 3.2.

The following problem is FPT when parametrized by $|\mathsf{Vars}({A_{1}})\cap\mathsf{Vars}({A_{2}})|$. Given two sequential VAs $A_{1}$ and $A_{2}$, construct a sequential VA that is equivalent to ${A_{1}\bowtie A_{2}}$.

Since we have a polynomial delay algorithm for the evaluation of sequential VAs (Theorem 2.5) and the size of the resulting VA is FPT in $|\mathsf{Vars}({A_{1}})\cap\mathsf{Vars}({A_{2}})|$, we have the following immediate conclusion.

Theorem 3.3.

Given two sequential VAs $A_{1}$ and $A_{2}$ and a document $\mathbf{d}$, one can evaluate $\mathord{\llceil{A_{1}\bowtie A_{2}}\rrceil}(\mathbf{d})$ with FPT delay parameterized by $|\mathsf{Vars}({A_{1}})\cap\mathsf{Vars}({A_{2}})|$.

In the rest of this section, we discuss the proof of Lemma 3.2. As was shown by Freydenberger et al. (DBLP:conf/pods/FreydenbergerKP18, ) if $A$ is a functional VA then for every state $q$ of $A$ and every variable $v\in\mathsf{Vars}(A)$, all of the possible runs from the initial state $q_{0}$ to $q$ include the same variable operations. Formally, for every state $q$ there is a function ${c}_{q}$, namely the variable configuration function, that assigns a label from $\{\mathsf{o},\mathsf{c},\mathsf{w}\}$, standing for “open,” “close,” and “wait,” to every variable in $\mathsf{Vars}(A)$, as follows. First, ${c}_{q}(x)=\mathsf{o}$ if every run from $q_{0}$ to $q$ opens $x$ but does not close it. Second, ${c}_{q}(x)=\mathsf{c}$ if every run from $q_{0}$ to $q$ opens and closes $x$. Third, ${c}_{q}(x)=\mathsf{w}$ if no run from $q_{0}$ to $q$ opens or closes variable $x$.

In sequential VAs, however, not all of the accepting runs open and close all of the variables and therefore it makes more sense to replace the label $\mathsf{w}$ with the label $\mathsf{u}$ that stands for “unseen”. In addition, in sequential VAs as opposed to functional, there might be a state $q$ for which there are two (different) runs from $q_{0}$ to $q$ such that the first opens and closes the variable $x$ whereas the second does not even open $x$. For this case, we add to the set of labels the label $\mathsf{d}$ that stands for “done” meaning that variable $x$ cannot be seen after reaching state $q$. Hence, “done” can also be understood as “unseen or closed, depending on what happened before”. We formalize these notions right after the next example.

Example 3.4.

Let us examine the following two accepting runs of the sequential VA $A$ from Example 2.3 on the input document $\mathbf{d}:=\mathtt{a}$:

[TABLE]

[TABLE]

The run $\rho_{1}$ gets to state $q_{2}$ after opening and closing $x$ while $\rho_{2}$ gets to $q_{2}$ without opening $x$. Thus, in state $q_{2}$ the variable configuration of $x$ is $\mathsf{d}$.

This “nondeterministic” behavior of sequential VAs is reflected in an extended variable configuration function $\tilde{c}_{q}$ for every state $q$ whose co-domain is the set $\{\mathsf{u},\mathsf{o},\mathsf{c},\mathsf{d}\}$. Since all of the accepting runs of a sequential VA are valid, given a state $q$, exactly one of the following holds:

• •

all runs from $q_{0}$ to $q$ open $x$; in this case $\tilde{c}_{q}(x)=\mathsf{o}$;

• •

all runs from $q_{0}$ to $q$ (open and) close $x$; in this case $\tilde{c}_{q}(x)=\mathsf{c}$;

• •

all runs from $q_{0}$ to $q$ do not open $x$; in this case $\tilde{c}_{q}(x)=\mathsf{u}$;

• •

at least one run from $q_{0}$ to $q$ (opens and) closes $x$ and at least one does not open $x$; in this case $\tilde{c}_{q}(x)=\mathsf{d}$.

A sequential VA $A$ is semi-functional for $x$, if for every state $q$ it holds that $\tilde{c}_{q}(x)\in\{\mathsf{o},\mathsf{c},\mathsf{u}\}$. We say that $A$ is semi-functional for $X$ if it is semi-functional for every $x\in X$.

Example 3.5.

The sequential VA $A$ from Example 2.3 is not semi-functional for $x$ because $\tilde{c}_{q_{2}}(x)=\mathsf{d}$, as reflected from the runs $\rho_{1}$ and $\rho_{2}$ presented in the previous example. However, the following equivalent sequential VA $A^{\prime}$ is semi-functional for $x$:

q_{0}$$q_{1}$$q^{\mathsf{c}}_{2}$$q^{\mathsf{u}}_{2}$$\Sigma$$\mathop{x{\vdash}}$$\Sigma$$\mathbin{{\dashv}x}$$\Sigma$$\Sigma$$\Sigma

Observe that the ambiguity we had in state $q_{2}$ of $A$ is resolved since it is replaced with two states, each corresponding to a unique configuration.

We show that for every sequential VA $A$, every state $q$ of $A$ and every variable $v$, we can compute $\tilde{c}_{q}(v)$ efficiently, and based on that we can translate $A$ into an equivalent sequential VA that is semi-functional for $X$. We show that the total runtime is FPT parameterized by $|X|$.

Lemma 3.6.

Given a sequential VA $A$ and $X\subseteq\mathsf{Vars}(A)$, one can construct in $O(2^{|X|}(n+m))$ time a sequential VA $A^{\prime}$ that is equivalent to $A$ and semi-functional for $X$ where $n$ is the number of states of $A$ and $m$ is the number of its transitions.

Example 3.7.

The sequential VA $A^{\prime}$ from Example 3.5 can be obtained from the automaton $A$ from Example 2.3 by replacing $q_{2}$ with two states $q^{\mathsf{u}}_{2}$ and $q^{\mathsf{c}}_{2}$ such that $q^{\mathsf{u}}_{2}$ corresponds with the paths in from $q_{0}$ to $q_{2}$ in which variable $x$ was unseen and $q^{\mathsf{c}}_{2}$ corresponds with the paths in from $q_{0}$ to $q_{2}$ in which variable $x$ was closed, and by changing the transitions accordingly. The algorithm from the previous Lemma generalizes this idea.

We refer the reader to Footnote 4 in the definition of a VA and note that, as in the previous example, there are cases where, to be semi-functional, a VA must have more than a single accepting state.

If two sequential VAs are semi-functional for their common variables, their join can be computed efficiently:

Lemma 3.8.

Given two sequential VAs $A_{1}$ and $A_{2}$ that are semi-functional for $\mathsf{Vars}(A_{1})\cap\mathsf{Vars}(A_{2})$ one can construct in polynomial time a sequential VA $A$ that is semi-functional for $\mathsf{Vars}(A_{1})\cap\mathsf{Vars}(A_{2})$ and equivalent to ${A_{1}\bowtie A_{2}}$.

The proof of this Lemma uses the same product construction as that for functional VAs presented by Freydenberger et al. (DBLP:conf/pods/FreydenbergerKP18, , Lemma 3.10). What allow us to use the same construction is (a) the fact it ignores the non-common variables and (b) the fact we can treat both $A_{1}$ and $A_{2}$ as functional VAs over $\mathsf{Vars}(A_{1})\cap\mathsf{Vars}(A_{2})$.

We can now move to compose the proof of Lemma 3.2: Given two sequential VAs $A_{1}$ and $A_{2}$, we invoke the algorithm from Lemma 3.6 and obtain two equivalent sequential VAs $\tilde{A}_{1}$ and $\tilde{A}_{2}$, respectively, such that each $\tilde{A}_{i}$ is semi-functional for $\mathsf{Vars}(A_{1})\cap\mathsf{Vars}(A_{2})$. Then, we use Lemma 3.8 to join $\tilde{A}_{1}$ and $\tilde{A}_{2}$. Note that the runtime is indeed FPT parametrized by $\mathsf{Vars}(A_{1})\cap\mathsf{Vars}(A_{2})$.

3.2. Restricting to Disjunctive Functional

Another approach to obtain a tractable evaluation of the join is by restricting the syntax of the regex formulas while preserving expressiveness. A regex formula $\gamma$ is said to be disjunctive functional if it is a finite disjunction of functional regex formula $\gamma_{1},\ldots,\gamma_{n}$. We denote the class of disjunctive functional regex formulas as $\mathsf{dfuncRGX}$.

Note that every disjunctive functional regex formula is also sequential. However, the regex formula $z\{\Sigma^{*}\}\cdot(x\{\Sigma^{*}\}\vee y\{\Sigma^{*}\})$ is sequential, yet it is not disjunctive functional. It also holds that every functional regex formula is disjunctive functional regex formula with a single disjunct. We can therefore conclude that we have the following:

[TABLE]

Note that here we treat the regex formulas as syntactic objects.

Equivalently, a disjunctive functional VA $A$ is the sequential VA whose states are the disjoint union of the states of a finite number $n$ of functional VAs $A_{1},\ldots,A_{n}$ and whose transitions are those of $A_{1},\ldots,A_{n}$, with the addition of a new initial state $q_{0}$ that is connected with epsilon transitions to each of the initial states of the $A_{i}$’s. Notice that being disjunctive functional is only a syntactic restriction and not semantic as can be concluded from the following proposition.

Proposition 3.9.

The following hold:

1. (1)

For every sequential regex formula there exists an equivalent disjunctive functional regex formula. 2. (2)

For every sequential VA there exists an equivalent disjunctive functional VA.

Since $\mathsf{funcRGX}$ corresponds with schema-based spanners whereas $\mathsf{seqRGX}$ with schemaless and due to the previous proposition we can conclude the following:

[TABLE]

Note that here we refer to the schemaless spanners represented by the regex formulas.

Example 3.10.

Consider the following sequential regex formula:

[TABLE]

Note that if we want to translate it into an equivalent disjunctive functional regex formula then we need at least one disjunct for each possible combination $z_{1}\{\Sigma^{*}\}\cdots z_{n}\{\Sigma^{*}\}$ where $z_{i}\in\{x_{i},y_{i}\}$. This implies a lower bound on the length of the shortest equivalent disjunctive functional regex formula. Similarly, let us consider the following sequential VA:

q_{0}$$q_{1}$$\cdots$$\small{q_{n-1}}$$q_{f}$$\mathop{x_{1}{\vdash}}$$\mathop{y_{1}{\vdash}}$$\Sigma$$\mathbin{{\dashv}x_{1}}$$\mathbin{{\dashv}y_{1}}$$\Sigma$$\mathop{x_{n}{\vdash}}$$\mathop{y_{n}{\vdash}}$$\Sigma$$\mathbin{{\dashv}x_{n}}$$\mathbin{{\dashv}y_{n}}$$\Sigma

An equivalent disjunctive functional VA has at least $2^{n}$ accepting states since the states encode the variable configurations.

We record this in the following proposition.

Proposition 3.11.

For every natural number $n$ the following hold:

1. (1)

There exists a sequential regex formula $\gamma$ of that is the concatenation of $n$ regex formulas of constant length such that each of its equivalent disjunctive functional regex formulas includes at least $2^{n}$ disjuncts. 2. (2)

There exists a sequential VA $A$ with $3n+1$ states such that each of its equivalent disjunctive functional VA has at least $2^{n}$ states.

That is, the translation from sequential to disjunctive functional might necessitate an exponential blow-up. Although the translation cannot be done efficiently in the general case, the advantage of using disjunctive functional VAs lies in the fact that we can compile the join of two disjunctive functional VAs efficiently into a disjunctive functional VA.

Proposition 3.12.

Given two disjunctive functional VAs $A_{1}$ and $A_{2}$, one can construct in polynomial time a disjunctive functional VA $A$ that is equivalent to ${A_{1}\bowtie A_{2}}$.

To prove this we can perform a pairwise join between the set of functional components of $A_{1}$ and those of $A_{2}$ and obtain a set of functional VAs for the join (DBLP:conf/pods/FreydenbergerKP18, , Lemma 3.10).

Since disjunctive functional is a restricted type of sequential VA, we conclude the following.

Corollary 3.13.

Given two disjunctive functional VAs $A_{1}$ and $A_{2}$ and a input document $\mathbf{d}$, one can enumerate the mappings of $\mathord{\llceil{A_{1}\bowtie A_{2}}\rrceil}(\mathbf{d})$ in polynomial delay.

4. The Difference Operator

When we consider the class of functional VAs, we know that we can compile all of the positive operators efficiently (i.e., in polynomial time) into a functional VA (DBLP:conf/pods/FreydenbergerKP18, ). In the case of NFAs or regular expressions, compiling the complement into an NFA necessitates an exponential blowup in size (DBLP:journals/jalc/EllulKSW05, ; DBLP:journals/tcs/Jiraskova05, ). Since NFAs and regular expressions are the Boolean functional VA and Boolean regex formulas, respectively, we conclude that constructing a VA that is equivalent to the difference of two functional VAs, or two functional regex formulas, entails an exponential blowup. Therefore, the static compilation fails to yield tractability results for the difference.

In the case of NFAs and regular expressions, the membership of a string in the difference can be tested in polynomial time. In contrast, the following theorem states that, for functional regex formulas (and VAs), this is no longer true under the conventional complexity assumption $\mathrm{P}\neq\mathrm{NP}$.

Theorem 4.1.

The following problem is $\mathrm{NP}$-complete. Given two functional regex formulas $\gamma_{1}$ and $\gamma_{2}$ with $\mathsf{Vars}(\gamma_{1})=\mathsf{Vars}(\gamma_{2})$ and an input document $\mathbf{d}$, is $\mathord{\llceil{\gamma_{1}\setminus\gamma_{2}}\rrceil}(\mathbf{d})$ nonempty?

Proof.

Membership in $\mathrm{NP}$ is straightforward: for functional regex formulas, membership can be decided in polynomial time (DBLP:conf/icdt/Freydenberger17, ). Hence, we focus on $\mathrm{NP}$-hardness. We use a reduction from 3SAT as in the proof of Theorem 3.1. Here, however, we are restricted to functional regex formulas and therefore we cannot use the domains of the resulting mappings to encode the assignments. Recall that the input is a formula $\varphi$ with the free variables $x_{1},\ldots,x_{n}$ such that $\varphi$ has the form $C_{1}\wedge\cdots\wedge C_{m}$, where each $C_{i}$ is a clause. In turn, each clause is a disjunction of three literals, where a literal has the form $x_{i}$ or $\neg x_{i}$. Given a 3CNF formula, we construct two functional regex formulas $\gamma_{1}$ and $\gamma_{2}$, and an input document $\mathbf{d}$, such that there is a satisfying assignment for $\varphi$ if and only if $\mathord{\llceil{\gamma_{1}\bowtie\gamma_{2}}\rrceil}(\mathbf{d})\neq\emptyset$.

We begin with the document $\mathbf{d}$, which is defined by $\mathbf{d}:=\mathtt{a}^{n}$. The regex formulas $\gamma_{1}$ and $\gamma_{2}$ are constructed as follows. We associate every free variable $x_{i}$ with a capture variable $x_{i}$. We start by defining the auxiliary regex formulas

[TABLE]

for $1\leq i\leq n$ and then define

[TABLE]

Intuitively, $\gamma_{1}$ encodes all of the legal assignments for $\varphi$ in such a way that if $x_{i}$ captures the substring ‘$\mathtt{a}$’ then it corresponds with assigning $\mathsf{t}$ to the free variable $x_{i}$, and otherwise (in case it captures $\epsilon$), it corresponds with assigning to it $\mathsf{f}$. Before defining $\gamma_{2}$, for each $1\leq i\leq m$ we denote the indices of the literals that appear in $C_{i}$ by $i_{1}<i_{2}<i_{3}$ and define $\gamma_{2}^{i}$ as follows:

[TABLE]

where $\delta_{\ell}$ is defined as $(\mathop{x_{\ell}\{\epsilon\}}\cdot\mathtt{a})$ if $x_{\ell}$ appears as a literal in $C_{i}$ or as $(\mathop{x_{\ell}\{\mathtt{a}\}})$ if $\neg x_{\ell}$ appears as a literal in $C_{i}$ Intuitively, $\gamma_{2}^{i}$ encodes the assignments for which clause $C_{i}$ is not satisfied. We then set

[TABLE]

To emphasize the differences between this reduction and that in the proof of Theorem 4.1, we consider the same formula:

[TABLE]

We have $\mathbf{d}:=\mathtt{a}^{3}$ since we have three variables $\{x,y,z\}$ and

[TABLE]

For the first clause we have

[TABLE]

and for the second

[TABLE]

It is left to show that $\mathord{\llceil{\gamma_{1}\setminus\gamma_{2}}\rrceil}(\mathbf{d})\neq\emptyset$ if and only if $\varphi$ has a satisfying assignment. Note that for every assignment $\mu\in\mathord{\llceil{\gamma_{1}}\rrceil}(\mathbf{d})$ and for every $1\leq j\leq n$, it holds that $\mu(x_{j})$ is either $[j,j\rangle$ or $[j,j+1\rangle$. Note also that the same is true also for $\mu\in\mathord{\llceil{\gamma_{2}}\rrceil}(\mathbf{d})$. Let us assume that there exists a satisfying assignment $\tau$ for $\varphi$. We define $\mu$ to be the mapping that is defined as follows: $\mu(x_{i}):=[i,i\rangle$ if $\tau(x_{i})=\mathsf{f}$ and $\mu(x_{i}):=[i,i+1\rangle$, otherwise (if $\tau(x_{i})=\mathsf{t}$). It then follows immediately from the definition of $\gamma_{2}$ that $\mu\in\mathord{\llceil{\gamma_{1}\setminus\gamma_{2}}\rrceil}(\mathbf{d})$. On the other hand, assume that $\mu\in\mathord{\llceil{\gamma_{1}\setminus\gamma_{2}}\rrceil}(\mathbf{d})$. We can define an assignment $\tau$ is such a way that $\tau(x_{i})=\mathsf{t}$ if $\mu(x_{i})=[i,i+1\rangle$ and $\tau(x_{i})=\mathsf{f}$ otherwise (if $\mu(x_{i})=[i,i\rangle$). It follows directly from the way we defined $\gamma_{1}$ and $\gamma_{2}$ that $\tau$ is a satisfying assignment for $\varphi$.

In our example, the assignment $\tau$ defined by $\tau(x)=\tau(y)=\mathsf{t}$ and $\tau(z)=\mathsf{f}$ is a satisfying assignment. Indeed, the mapping $\mu$ corresponds to this assignment that is defined by $\mu(x)=[1,2\rangle$, $\mu(y)=[2,3\rangle$ and $\mu(z)=[3,3\rangle$ is in $\mathord{\llceil{\gamma_{1}}\rrceil}(\mathtt{a}^{n})$ but is not in $\mathord{\llceil{\gamma_{2}}\rrceil}(\mathtt{a}^{n})$ since either (a) $\mu(x)=[1,1\rangle$ and $\mu(y)=[2,2\rangle$) or (b) $\mu(x)=[1,2\rangle$ and $\mu(y)=[2,2\rangle$. Note also that the assignment $\mu$ defined by $\mu(x)=[1,2\rangle$, $\mu(y)=[2,3\rangle$ and $\mu(z)=[3,4\rangle$ is in $\mathord{\llceil{\gamma_{1}\setminus\gamma_{2}}\rrceil}(\mathtt{a}^{n})$ since it is in $\mathord{\llceil{\gamma_{1}}\rrceil}(\mathtt{a}^{n})$ and not in $\mathord{\llceil{\gamma_{2}}\rrceil}(\mathtt{a}^{n})$. Indeed, the assignment $\tau$ for which $\tau(x)=\tau(y)=\tau(z)$ is a satisfying assignment for $\varphi$.

We conclude the $\mathrm{NP}$-hardness of determining the nonemptiness of $\mathord{\llceil{\gamma_{1}\setminus\gamma_{2}}\rrceil}(\mathbf{d})$. ∎

We can conclude from Theorem 4.1 that, in contrast to the tractability of the natural join of disjunctive functional VAs (Corollary 3.13), here we are facing $\mathrm{NP}$-hardness already for functional VAs. In the remainder of this section, we discuss syntactic conditions that allow to avoid this hardness.

4.1. Bounded Number of Common Variables

Theorem 4.1 implies that no matter what approach we choose to tackle the evaluation of the difference, without imposing any restrictions we hit $\mathrm{NP}$-hardness. In this section, we investigate the restriction of an upper bound on the number of common variables shared between the operands. Recall that this restriction leads to an FPT static compilation for the natural join (Lemma 3.2). Yet, we observed at the beginning of Section 4, in the case of difference, such static compilation necessitates an exponential blow-up, even if there are no variables at all,

Therefore, instead of static compilation that is independent of the document, we apply an ad-hoc compilation that depends on the specific document at hand. In this case, we refer to the resulting automaton as an ad-hoc VA since it is valid only for that specific document. Ad-hoc VAs were introduced (without a name) by Freydenberger et al. (DBLP:conf/pods/FreydenbergerKP18, ) as a tool for evaluating functional VAs with polynomial delay. The next lemma is based on this idea.

Lemma 4.2.

Let $k$ be a fixed natural number. Given two sequential VAs $A_{1}$ and $A_{2}$ where $|\mathsf{Vars}(A_{1})\cap\mathsf{Vars}(A_{2})|\leq k$ and a document $\mathbf{d}$, one can construct in polynomial time a sequential VA $A_{\mathbf{d}}$ with $\mathord{\llceil{A_{\mathbf{d}}}\rrceil}(\mathbf{d})=\mathord{\llceil{A_{1}\setminus A_{2}}\rrceil}(\mathbf{d})$.

Since we can enumerate the result of sequential VA with polynomial delay (Theorem 2.5), we can conclude the following.

Theorem 4.3.

Let $k$ be a fixed natural number. Given two sequential VAs $A_{1}$ and $A_{2}$ where $|\mathsf{Vars}(A_{1})\cap\mathsf{Vars}(A_{2})|\leq k$ and a document $\mathbf{d}$, one can enumerate $\mathord{\llceil{A_{1}\setminus A_{2}}\rrceil}(\mathbf{d})$ with polynomial delay.

We now present the proof sketch of Lemma 4.2.

Proof Sketch.

We construct two sequential VAs $A$ and $B$ (that share a bounded number of variables) such that evaluating the difference of $A_{1}$ and $A_{2}$ on $\mathbf{d}$ is the same as evaluating the natural join of $A$ and $B$ on $\mathbf{d}$. This natural join can be compiled into a sequential VA in polynomial time when the number of common variables is bounded by a constant (Theorem 3.3), and therefore, we establish the desired result.

Yet, unlike the schema-based model, difference in the schemaless case cannot be translated straightforwardly into a natural join (e.g., via complementation). For illustration, let us consider the case where there are $\mu_{1}\in\mathord{\llceil{A_{1}}\rrceil}(\mathbf{d})$ and $\mu_{2}\in\mathord{\llceil{A_{2}}\rrceil}(\mathbf{d})$ such that $\mathsf{dom}(\mu_{1})\cap\mathsf{dom}(\mu_{2})=\emptyset$. In this case, the assignment $\mu_{1}$ is not in $\mathord{\llceil{A_{1}\setminus A_{2}}\rrceil}(\mathbf{d})$ since it is compatible with $\mu_{2}$. Nevertheless, $\mu_{1}$ will occur in the natural join of $A_{1}$ with every VA $A_{2}^{\prime}$, unless $A_{1}$ and $A_{2}^{\prime}$ share one or more common variables.

As a solution, we construct a VA that encodes information about the domains of the mappings $\mu$, within the variables shared by $A_{1}$ and $A_{2}$, using new shared dummy variables. Specifically, we have a dummy variable $\hat{x}$ for every shared variable $x$. If $x\in\mathsf{dom}(\mu)$, then $\hat{x}$ is assigned the first empty span $[1,1\rangle$, and if $x\notin\mathsf{dom}(\mu)$, then $\hat{x}$ is assigned the last empty span $[|\mathbf{d}|+1,|\mathbf{d}|+1\rangle$. (Here, we assume that $\mathbf{d}$ is nonempty; we deal separately with the case $\mathbf{d}=\epsilon$.)

We construct a VA $A$ for the above extended mappings of $A_{1}$. In addition, we construct a VA $B$ by iterating through all possible extended mappings over the shared variables, and for each such a mapping, if it is incompatible with all of the extended mappings of $\mathord{\llceil{A_{2}}\rrceil}(\mathbf{d})$, then we include it in $B$. This construction can be done in polynomial time, since we assume that the number of common variables is bounded by a constant.

We conclude by showing that the extended mappings of $\mathord{\llceil{A}\rrceil}(\mathbf{d})$ that have compatible mappings in $\mathord{\llceil{B}\rrceil}(\mathbf{d})$ correspond to the mappings of $\mathord{\llceil{A_{1}}\rrceil}(\mathbf{d})$ that have no compatible mappings in $\mathord{\llceil{A_{2}}\rrceil}(\mathbf{d})$, and also that the extended mappings of $\mathord{\llceil{A}\rrceil}(\mathbf{d})$ that have compatible mappings in $\mathord{\llceil{B}\rrceil}(\mathbf{d})$ correspond to the mappings of $\mathord{\llceil{A_{1}}\rrceil}(\mathbf{d})$ that do not have compatible mappings in $\mathord{\llceil{A_{2}}\rrceil}(\mathbf{d})$. ∎

Theorem 4.2 shows that we can enumerate the difference with polynomial delay when we restrict the number of common variables. A natural question is whether the degree of this polynomial depends on this number; the next theorem answers this question negatively, under the conventional assumptions of parameterized complexity.

Theorem 4.4.

The following problem is $\mathrm{W[1]}$-hard parametrized by $|\mathsf{Vars}(\gamma_{1})\cap\mathsf{Vars}(\gamma_{2})|$. Given two functional regex formulas $\gamma_{1}$ and $\gamma_{2}$ and an input document $\mathbf{d}$, is $\mathord{\llceil{\gamma_{1}\setminus\gamma_{2}}\rrceil}(\mathbf{d})$ nonempty?

This result contrasts our $\mathrm{FPT}$ result for the natural join (Theorem 3.3). The proof uses a reduction from the problem determining whether a 3-SAT formula has a satisfying assignment with at most $p$ ones, where $p$ is the parameter (downey2012parameterized, ).

4.2. Restricting the Disjunctions

We now propose another restriction that guarantees a tractable evaluation, this time allowing the number of common variables to be unbounded. We begin with some definitions.

Let $\gamma$ be a sequential regex formula and let $x\in\mathsf{Vars}$ be a variable. Then $\gamma$ is synchronized for $x$ if, for every subexpression of $\gamma$ of the form $\gamma_{1}\vee\gamma_{2}$, we have that $x$ appears neither in $\gamma_{1}$ nor in $\gamma_{2}$. A regex formula $\gamma$ is called synchronized for $X\subseteq\mathsf{Vars}$ if it is synchronized for every $x\in X$.

This notion generalizes to sequential VAs: A state $q$ of a sequential VA $A$ is called a unique target state for the variable operation $\omega\in\Gamma_{\mathsf{Vars}(A)}$, if for every state $p$ of $A$ we have that $(p,\omega,q)\in\delta$ implies $q=q_{\omega}$ where $\delta$ is the transition relation of $A$. In other words, $q_{\omega}$ is the only state that can be reached by processing $\omega$. We say that $A$ is synchronized for a variable $x\in\mathsf{Vars}$ if each of $\mathop{x{\vdash}}$ and $\mathbin{{\dashv}x}$ has a unique target state and either all accepting runs of $A$ open and close $x$, or no accepting run of $A$ operates on $x$. Finally, $A$ is synchronized for $X\subseteq\mathsf{Vars}$ if it is synchronized for every $x\in X$.

Example 4.5.

Consider the regex formula $(\mathop{x\{\Sigma^{*}\}}\vee\epsilon)\cdot\mathop{y\{\Sigma^{*}\}}$ and this equivalent VA:

\mathop{x{\vdash}}$$\Sigma$$\mathbin{{\dashv}x}$$\epsilon$$\mathop{y{\vdash}}$$\Sigma$$\mathbin{{\dashv}y}

Both are synchronized for $y$ and not for $x$: The regex formula has a subexpression of the form $(x\{\Sigma^{*}\}\vee\epsilon)$, whereas the variable $y$ does not appear under any disjunction. In the VA, although each variable operation has a unique target state, not all of the accepting runs include the variable operations $\mathop{x{\vdash}}$ and $\mathbin{{\dashv}x}$ (as opposed to $\mathop{y{\vdash}}$ and $\mathbin{{\dashv}y}$, which are included in every accepting run).

The following result states that conversions from regex formulas to VAs can preserve the property of being synchronized for $X$.

Lemma 4.6.

Let $\gamma$ be a sequential regex formula that is synchronized for $X\subseteq\mathsf{Vars}$. One can convert $\gamma$ in linear time into an equivalent sequential VA $A$ that is synchronized for $X$.

As one might expect, VAs that are synchronized (for some nonempty set $X$ of variables) are less expressive than sequential or semi-functional VAs (that are defined in Section 3.1). In fact, even functional regex formulas can express spanners that are not expressible with VAs that are synchronized for all their variables:

Proposition 4.7.

Let $\gamma:=(\mathsf{a}\cdot x\{\epsilon\}\cdot\mathsf{a})\vee(\mathsf{b}\cdot x\{\epsilon\}\cdot\mathsf{b})$. There is no sequential VA $A$ that is synchronized for $x$ and equivalent to $\gamma$.

Hence, by using synchronized VAs, we sacrifice expressive power. But this restriction also allows us to state the following positive result on the difference of VAs:

Theorem 4.8.

Given an input document $\mathbf{d}$ and two sequential VAs $A_{1}$ and $A_{2}$ such that, for $X:=\mathsf{Vars}(A_{1})\cap\mathsf{Vars}(A_{2})$, $A_{1}$ is semi-functional for $X$ and $A_{2}$ is synchronized for $X$, one can construct a sequential VA $A_{\mathbf{d}}$ with $\mathord{\llceil{A_{\mathbf{d}}}\rrceil}(\mathbf{d})=\mathord{\llceil{A_{1}\setminus A_{2}}\rrceil}(\mathbf{d})$ in polynomial time.

The full proof can be found in the Appendix; we discuss some of its key ideas. The first key observation is that $A_{2}$ can be treated as a functional VA that uses only the common variables (similarly to the proof of Lemma 3.8). This allows us to work with the variable configurations of $A_{2}$, and construct the match structure $M(A_{2},\mathbf{d})$ of $A_{2}$ on $\mathbf{d}$. This model was introduced (without a name) by Freydenberger et al. (DBLP:conf/pods/FreydenbergerKP18, ) to evaluate functional VAs with polynomial delay. As explained there, every element of $\mathord{\llceil{A_{2}}\rrceil}(\mathbf{d})$ can be uniquely expressed as a sequence of $|\mathbf{d}|+1$ variable configurations of $A_{2}$.

Every accepting run of $A_{2}$ on $\mathbf{d}$ can be mapped into such a sequence by taking the variable configurations of the states just before a symbol of $\mathbf{d}$ is read (and the configuration of the final state). The match structure $M(A_{2},\mathbf{d})$ is an NFA that has the set of variable configurations of $A_{2}$ as its alphabet; and its language is exactly the set of sequences of variables configurations that correspond to elements of $\mathord{\llceil{A_{2}}\rrceil}(\mathbf{d})$.

While determinizing match structures is still hard, the fact that $A_{2}$ is synchronizing on the common variables allows us to construct a deterministic match structure $D_{2}$ from $M(A,\mathbf{d})$. Using a variant of the proof of Lemma 3.8, we can then combine $A_{1}$ and $A_{2}$ into an ad-hoc VA $A_{\mathbf{d}}$ with $\mathord{\llceil{A_{\mathbf{d}}}\rrceil}(\mathbf{d})=\mathord{\llceil{A_{1}\setminus A_{2}}\rrceil}(\mathbf{d})$.

After creating $A_{\mathbf{d}}$ according to Theorem 4.8, we can use Theorem 2.5 to obtain the following tractability result:

Corollary 4.9.

Given an input document $\mathbf{d}$ and two sequential VAs $A_{1}$ and $A_{2}$ such that, for $X:=\mathsf{Vars}(A_{1})\cap\mathsf{Vars}(A_{2})$, $A_{1}$ is semi-functional for $X$ and $A_{2}$ is synchronized for $X$, one can enumerate the mappings in $\mathord{\llceil{A_{1}\setminus A_{2}}\rrceil}(\mathbf{d})$ in polynomial delay.

We saw that disallowing disjunctions over the variables leads to tractability. Can we relax this restriction by allowing a fixed number of such disjunctions? Our next result is a step towards answering this question. A disjunction-free regex formula is a regex formula that does not contain any subexpression of the form $\gamma_{1}\vee\gamma_{2}$.

Proposition 4.10.

The following decision problem is $\mathrm{NP}$-complete. Given two sequential regex formulas $\gamma_{1}$ and $\gamma_{2}$ with $\mathsf{Vars}(\gamma_{1})=\mathsf{Vars}(\gamma_{2})$ and an input document $\mathbf{d}$ such that

• •

$\gamma_{1}$* is functional,*

• •

$\gamma_{2}$* is a disjunction of regex formulas $\gamma_{2}^{i}$ such that each is disjunction-free,*

• •

for every variable $x\in\mathsf{Vars}(\gamma_{2})$, it holds that $x$ appears in at most 3 disjuncts $\gamma_{2}^{i}$ of $\gamma_{2}$,

is $\mathord{\llceil{\gamma_{1}\setminus\gamma_{2}}\rrceil}(\mathbf{d})$ nonempty?

Proof.

This proof is an adaption of the proof of Theorem 4.1, using mostly the same notation. Instead of a general 3CNF formula, let $\varphi=C_{1}\wedge\ldots\wedge C_{m}$ be a CNF formula, such that every clause $C_{i}$ contains either 2 or 3 literals, and each of the variables appears in at most $3$ clauses. Deciding satisfiability for such a formula is still NP-complete (DBLP:journals/dam/Tovey84, ).

For $\gamma_{1}$ to not have any disjunctions, we first set set $\mathbf{d}=(\mathtt{bab})^{n}$ for some $\mathtt{a},\mathtt{b}\in\Sigma$. We then define

[TABLE]

Intuitively $\gamma_{1}$ encodes all of the possible assignments. The regex formula $\gamma_{2}$ is defined analogously to $\gamma_{2}$ in the proof of Theorem 4.1 with an adaptation to the new input document and a slight simplification of the $\gamma_{2}^{i}$s (since we do not need $\gamma_{2}$ to be functional any more). Formally, we set

[TABLE]

if only variables $x_{i_{1}},x_{i_{2}}$ with $i_{1}<i_{2}$ appear in clause $C_{i}$, and

[TABLE]

if variables $x_{i_{1}},x_{i_{2}},x_{i_{3}}$ with $i_{1}<i_{2}<i_{3}$ appear in clause $C_{i}$.

By the choice of the 3CNF formula $\varphi$, every variable $x_{j}$ appears in at most three regex formulas of the form $\gamma_{2}^{i}$. Correctness of this reduction can be shown analogously to that of Theorem 4.1. ∎

We conclude that evaluating $\gamma_{1}\setminus\gamma_{2}$ remains hard even if $\gamma_{1}$ is functional (and hence also semi-functional for the common variables) and $\gamma_{2}$ is a disjunction of disjunction-free regex formulas, and each of $\gamma_{2}$’s variables appears in at most three such disjuncts.

It is open whether the problem becomes tractable if the variables are limited to at most one or two disjuncts.

5. Extraction Complexity

In this section, we discuss queries that are defined as RA expressions over schemaless spanners given in a representation language $\mathcal{L}$ (e.g., regex formulas), which we refer to as the language of the atomic spanners. Formally, an RA tree is a directed and ordered tree whose inner nodes are labeled with RA operators, the out-degree of every inner node is the arity its RA operator, and each of the leaves is a placeholder for a schemaless spanner. For illustration, Figure 2 shows an RA tree $\tau$, where the placeholders are the rectangular boxes with the question marks; the dashed arrows should be ignored for now. The RA tree corresponds to the relational concept of a query tree or a logical query plan (DBLP:books/ph/Garcia-MolinaUW99, ; DBLP:journals/cacm/SmithC75, ). As in the rest of the paper, we restrict the discussion to the RA operators projection, union, natural join, and difference.

Let $\mathcal{L}$ be a representation language for atomic spanners, and let $\tau$ be an RA tree. An instantiation of $\tau$ assigns a schemaless spanner representation from $\mathcal{L}$ to every placeholder, and a set of variables to every projection. For example, Figure 2 shows an instantiation $I$ for $\tau$ via the dashed arrows; here, we can think of $\mathcal{L}$ as the class of sequential regex formulas, and so, each $\alpha$ expression is a sequential regex formula.

An instantiation $I$ of $\tau$ transforms $\tau$ into an actual schemaless spanner representation, where $\tau$ is the parse tree of its algebraic expression. We denote this representation by $I[\tau]$. As usual, by $\mathord{\llceil{I[\tau]}\rrceil}$ we denote the actual schemaless spanner that $I[\tau]$ represents.

Example 5.1.

Assume that the input document $\mathbf{d}_{\mathsf{Students}}$ from the earlier examples is now extended and contains additional information about the students, including recommendations they got from their professors and previous hires. Let us assume that every line begins with a student’s name and contains information about that student. Let us also assume that we have the following functional regex formulas:

• •

regex formula $\alpha_{\mathsf{sm}}$ with capture variables $x_{\textit{stdnt}},x_{\textit{ml}}$ that extracts names with their corresponding email addresses;

• •

regex formula $\alpha_{\mathsf{sp}}$ with variables $x_{\textit{stdnt}},x_{\textit{phn}}$ that extracts names with their corresponding phone numbers;

• •

regex formula $\alpha_{\mathsf{nr}}$ with variables $x_{\textit{stdnt}},x_{\textit{rcmnd}}$ that extracts names with their corresponding recommendations.

Note that all of the regex formulas are functional, that is, they do not output partial mappings. The following query extracts the students that not have recommendations.

[TABLE]

This query is $I[\tau]$ for the RA tree $\tau$ and the instantiation $I$ of Figure 2. This query defines the spanner $\mathord{\llceil{I[\tau]}\rrceil}$, and the set of extracted spans is $\mathord{\llceil{I[\tau]}\rrceil}(\mathbf{d}_{\mathsf{Students}})$.

We present a complexity measure that is unique to spanners, namely the extraction complexity, where the RA tree $\tau$ is regarded fixed and the input consists of both the instantiation $I$ and the input document $\mathbf{d}$. Specifically, the evaluation problem for an RA tree $\tau$ is that of evaluating $\mathord{\llceil{I[\tau]}\rrceil}(\mathbf{d})$, given $I$ and $\mathbf{d}$. Similarly, the nonemptiness problem for an RA tree $\tau$ is that of deciding whether $\mathord{\llceil{I[\tau]}\rrceil}(\mathbf{d})$ is nonempty, given $I$ and $\mathbf{d}$.

Clearly, some RA trees have an intractable nonemptiness and, consequently, an intractable evaluation. For example, if $\mathcal{L}$ is the class of sequential regex formulas and $\tau$ is the RA tree that consists of a single natural-join node, then the nonemptiness problem for $\tau$ is NP-complete (Theorem 3.1). Also, if $\mathcal{L}$ is the class of functional regex formulas and $\tau$ is the RA tree that consists of a single difference node, then the nonemptiness problem for $\tau$ is NP-complete (Theorem 4.1). In contrast, by composing the positive results established in Sections 3 and 4, we obtain the following theorem, which is a consequence of Lemma 3.2 and Lemma 4.2.

Theorem 5.2.

Let $\mathcal{L}$ be the class of sequential VAs. Let $k$ be a fixed natural number and $\tau$ an RA tree. The evaluation problem for $\tau$ is solvable with polynomial delay, assuming that for all join and difference nodes $v$ of $I[\tau]$, the left and right subtrees under $v$ share at most $k$ variables.

We restate that, while static compilation suffices for the positive operators, we need ad-hoc compilation to support the difference. Interestingly, the ad-hoc approach allows us to incorporate into the RA tree other representations of schemaless spanners, which can be treated as black-box schemaless spanners, as long as these spanners can be evaluated in polynomial time and are of a bounded degree. In turn, the degree of a schemaless spanner $S$ is the maximal cardinality of a mapping produced over all possible documents, that is, $\max\{|\mathsf{dom}(\mu)|\mid\mathbf{d}\in\Sigma^{*},\mu\in S(\mathbf{d})\}$.

Formalizing the above, we can conclude from Theorem 5.2 a generalization that allows for black-box schemaless spanners. To this end, we call a representation language $\mathcal{L}^{\prime}$ for schemaless spanners tractable if $\mathord{\llceil{\beta}\rrceil}(\mathbf{d})$ can be evaluated in polynomial time (for some fixed polynomial), given $\beta\in\mathcal{L}^{\prime}$ and $\mathbf{d}\in\Sigma^{*}$, and we call $\mathcal{L}^{\prime}$ degree bounded if there is a fixed natural number that bounds the degree of all the schemaless spanners represented by expressions in $\mathcal{L}^{\prime}$.

Corollary 5.3.

Let $\mathcal{L}^{\prime}$ be a tractable and degree-bounded representation system for schemaless spanners, and let $\mathcal{L}$ be the union of $\mathcal{L}^{\prime}$ and the class of all sequential VAs. Let $k$ be a fixed natural number and let $\tau$ be an RA tree. The evaluation problem of $\tau$ is solvable with polynomial delay, assuming that for all join and difference nodes $v$ of $I[\tau]$, the left and right subtrees under $v$ share at most $k$ variables.

Combining such black-box schemaless spanners in the instantiated RA tree increases the expressiveness, as it allows us to incorporate spanners that are not (and possibly cannot be) described as RA expressions over VAs, such as string equalities (DBLP:journals/jacm/FaginKRV15, ). Other examples of such spanners are part of speech (POS) taggers, dependency parsers, sentiment analysis modules, and so on.

Example 5.4.

Following Example 5.1, suppose that we now wish to extract the students that do not have any positive recommendations. Assume we have a black-box spanner for sentiment analysis, namely PosRec, with the variables $x_{\textit{stdnt}}$ and $x_{\textit{posrec}}$, that extract names and their corresponding positive recommendation. Note that this spanner has the degree $2$. We can replace $\alpha_{\mathsf{nr}}$ in the instantiation $I$ of Figure 2 with PosRec, and thereby obtain the desired result. If PosRec can be computed in polynomial time, then the resulting query can be evaluated in polynomial delay.

6. Conclusions

We have studied the complexity of evaluating algebraic expressions over schemaless spanners that are represented as sequential regex formulas and sequential VAs. We have shown that we hit computational hardness already in the evaluation of the natural join and difference of two such spanners. In contrast, we have shown that we can compile the natural join of two sequential VAs (and regex formulas) into a single sequential VA, in polynomial time, if we assume a constant bound on the number of common variables of the joined spanners; hence, under this assumption, we can evaluate the natural join with polynomial delay. As an alternative to this assumption, we have proposed and investigated a new normal form for sequential spanners, namely disjunctive functional, that allows for such efficient compilation and evaluation.

Bounding the number of common variables between the involved spanners also allows to evaluate the difference with polynomial delay, even though this cannot be obtained by compiling into a VA—an exponential blowup in the number of states is necessary already for Boolean spanners. Evaluation with polynomial delay is then obtained via an ad-hoc compilation of both the spanners and the document into a VA. We have shown how the ad-hoc approach can be used for establishing upper bounds on general RA trees over regex formulas, VAs, and even black-box spanners of a bounded dimension. This has been done within the concept of extraction complexity that we have proposed as new lens to analyzing the complexity of spanners.

We believe that our analysis has merely touched the tip of the iceberg on the algorithms that can be devised under the guarantee of tractable extraction complexity. In particular, we have proposed sufficient conditions to avoid the inherent hardness of the natural join and difference, but it is quite conceivable that less restrictive conditions already suffice. Alternatively, are there conditions of extractors (possibly incomparable to ours) that are both common in practice and useful to bound the extraction complexity?

Appendix A Proofs for Section 3

A.1. Proof of Lemma 3.6

Lemma 3.6.* Given a sequential VA $A$ and $X\subseteq\mathsf{Vars}(A)$, one can construct in $O(2^{|X|}(n+m))$ time a sequential VA $A^{\prime}$ that is equivalent to $A$ and semi-functional for $X$ where $n$ is the number of states of $A$ and $m$ is the number of its transitions.*

To prove this lemma, we use the following auxiliary lemma.

Lemma A.1.

Let $A$ be a sequential VA that is semi-funcitonal for $Y\subseteq\mathsf{Vars}(A)$ and let $x\in\mathsf{Vars}(A)\setminus Y$. There is an algorithm that outputs a sequential VA $A^{\prime}$ that is equivalent to $A$ and semi-functional for $Y\cup\{x\}$ with $O(2|Q|)$ states and $O(2|\delta|)$ transitions in $O(2(|Q|+|\delta|))$ steps where $Q$ is the set of states of $A$ and $\delta$ is $A$’s transition function.

Lemma 3.6 follows directly from the previous lemma as we can invoke the algorithm iteratively for all of the variables of $X$. It is therefore left to present the proof of this lemma. The intuition is simple, we create a new VA that is equivalent to $A$ by replacing each state $q$ for which $\tilde{c}_{q}(x)=\mathsf{d}$ with two states $q^{\mathsf{u}}$ and $q^{\mathsf{c}}$ and re-wiring the transition in a way that in all of the paths from $q_{0}$ to $q^{\mathsf{u}}$ variable $x$ does not appear and in all of the paths from $q_{0}$ to $q^{\mathsf{c}}$ variable $x$ was closed. We then prove that the resulting automaton remains semi-functional for $Y$ and is also semi-functional for $\{x\}$ (in contrast to $A$).

Since we consider two VAs $A$ and $A^{\prime}$, to avoid ambiguity we add to the variable configuration function $\tilde{c}_{q}$ an upper script that indicates to which automaton we refer to. Formally, let us denote $A:=(Q,q_{0},\delta,F)$. We denote by $\tilde{Q}$ the subset of $Q$ that consists of those states $\tilde{q}$ for which $\tilde{c}^{A}_{\tilde{q}}(x)=\mathsf{d}$, these are the states that we wish to replace. We then define that the VA $A^{\prime}:=(Q^{\prime},q^{\prime}_{0},\delta^{\prime},F^{\prime})$ such that

1. (1)

$Q^{\prime}=(Q\setminus\tilde{Q})\cup\{\tilde{q}^{\mathsf{c}}|\tilde{q}\in\tilde{Q}\}\cup\{\tilde{q}^{\mathsf{u}}|\tilde{q}\in\tilde{Q}\}$ and 2. (2)

if $q_{0}\in\tilde{Q}$ then $q^{\prime}_{0}=q^{\mathsf{u}}_{0}$, otherwise $q^{\prime}_{0}=q_{0}$, and 3. (3)

$F^{\prime}=\{q|q\in F\setminus\tilde{Q}\}\cup\{q^{\mathsf{c}},q^{\mathsf{u}}|q\in F\cap\tilde{Q}\}$,

and the transition function $\delta^{\prime}$ is as described now. For every $(p,o,q)\in\delta$ with $\tilde{c}^{A}_{q}(x)\neq\mathsf{d}$, we set $(p,o,q)\in\delta^{\prime}$. In addition, for every $(p,o,q)\in\delta$ with $\tilde{c}^{A}_{q}(x)=\mathsf{d}$,

• •

if $\tilde{c}^{A}_{p}(x)=\mathsf{d}$ then $(p^{\mathsf{u}},o,q^{\mathsf{u}})\in\delta^{\prime}$ and $(p^{\mathsf{c}},o,q^{\mathsf{c}})\in\delta^{\prime}$;

• •

if $\tilde{c}^{A}_{p}(x)=\mathsf{o}$ then $(p^{\mathsf{o}},o,q^{\mathsf{c}})\in\delta^{\prime}$ ( and in this case $o=\mathbin{{\dashv}x}$);

• •

if $\tilde{c}^{A}_{p}(x)=\mathsf{c}$ then $(p^{\mathsf{c}},o,q^{\mathsf{c}})\in\delta^{\prime}$.

It is left to show that

1. (1)

$A^{\prime}$ is equivalent to $A$ and 2. (2)

$A^{\prime}$ is semi-functional for $Y\cup\{x\}$.

To prove (1) we show that every mapping in $\mathord{\llceil{A^{\prime}}\rrceil}(\mathbf{d})$ is also a mapping in $\mathord{\llceil{A}\rrceil}(\mathbf{d})$ and vice versa. Indeed, for every accepting run of $A^{\prime}$ there might be two case $(a)$ it includes only states from $Q$ or $(b)$ it includes also states in $Q^{\prime}\setminus Q$. If $(a)$ then the claim is straightforward as \delta^{\prime}\cap\Big{(}Q\times(\Sigma\cup\{\epsilon\}\cup\Gamma_{\mathsf{Vars}(A)})\times Q\Big{)} equals $\delta$. If $(b)$ then we can divide the corresponding run according to the variable configuration of $x$ and obtain that the claim is a direct consequence of the definition of $\delta^{\prime}$. To show that every mapping in $\mathord{\llceil{A}\rrceil}(\mathbf{d})$ is also a mapping in $\mathord{\llceil{A^{\prime}}\rrceil}(\mathbf{d})$ we take the corresponding map in $A$ and divide into segments according to the variable configuration of $A$, we can then use $\delta^{\prime}$ definition to construct an accepting run in $A^{\prime}$ for the same mapping.

To prove (2), we first observe that $A^{\prime}$ is semi-functional for $Y$ (since otherwise it would imply that $A$ is not semi-functional for $Y$). It is therefore left to show that it is semi-functional for $\{x\}$. Note that from the way we defined $\delta^{\prime}$ we can conclude that for every state $p\in\{\tilde{q}^{\mathsf{u}}|\tilde{q}\in\tilde{Q}\}$ it holds that $\tilde{c}^{A^{\prime}}_{p}(x)=\mathsf{w}$ and for every state $p\in\{\tilde{q}^{\mathsf{c}}|\tilde{q}\in\tilde{Q}\}$ it holds that $\tilde{c}^{A^{\prime}}_{p}(x)=\mathsf{c}$. For all other states $r$ the $\tilde{c}^{A^{\prime}}_{r}$ is identical to $\tilde{c}^{A}_{r}$.

A.2. Proof of Proposition 3.9

Proposition 3.9.* The following hold:*

1. (1)

For every sequential regex formula there exists an equivalent disjunctive functional regex formula. 2. (2)

For every sequential VA there exists an equivalent disjunctive functional VA.

Sequential regex formulas to disjunctive functional regex formulas

Let $\alpha$ be a sequential regex formula. We translate it into an equivalent disjunctive functional by defining the set $A(\alpha)$ of its disjuncts recursively as follows:

• •

if $\alpha=\emptyset$ then $A(\alpha)=\emptyset$;

• •

if $\alpha=\sigma$ then $A(\alpha)=\{\sigma\}$;

• •

if $\alpha=\epsilon$ then $A(\alpha)=\{\epsilon\}$;

• •

if $\alpha=\alpha_{1}\vee\alpha_{2}$ then

• –

if $\mathsf{Vars}(\alpha_{1})=\mathsf{Vars}(\alpha_{2})=\emptyset$ then $A(\alpha)=\{\alpha_{1}\vee\alpha_{2}\}$

• –

otherwise $A(\alpha)=A(\alpha_{1})\cup A(\alpha_{2})$;

• •

if $\alpha=\alpha_{1}\cdot\alpha_{2}$ then $A(\alpha)=\{\beta_{1}\cdot\beta_{2}|\beta_{1}\in A\{\alpha_{1}\},\beta_{2}\in A\{\alpha_{2}\}\}$;

• •

if $\alpha=(\alpha_{1})^{*}$ then $A(\alpha)=\{\beta_{0}\cdots\beta_{n}|n\in\mathbb{N},\beta_{i}\in A(\alpha_{1})\}$;

• •

if $\alpha=x\{\alpha_{1}\}$ then $A(\alpha)=\{x\{\beta\}|\beta\in A(\alpha_{1})\}$

We can then conclude the desired by proving the following lemma using a simple induction on $\alpha$’s structure.

Lemma A.2.

If $\alpha$ is a sequential regex formula then $\mathord{\llceil{\alpha}\rrceil}=\bigvee_{\gamma\in{A(\alpha)}}\mathord{\llceil{\gamma}\rrceil}$ where each $\gamma$ is functional. In addition, $A(\alpha)$ is finite if $\alpha$ is finite.

Sequential VA to disjunctive functional VA

Let $A$ be a sequential automaton with set of variables $\mathsf{Vars}(A)$. We iterate through all of the possible subsets $V$ of $\mathsf{Vars}(A)$ and for each such subset we create a new VA $A_{V}$ that consists of all of the accepting runs of $A$ that include exactly the variables of $V$ and only them (this can be done, for instance, by a BFS on $A$ starting from its initial state $q_{0}$). We then construct a new automaton $A^{\prime}$ that is the disjoint union of all of those $A_{V}$’s by adding a new initial state with transitions to all of the initial states of the $A_{V}$’s.

Appendix B Proofs for Section 4

B.1. Proof of Lemma 4.2

Lemma 4.2.* Let $k$ be a fixed natural number. Given two sequential VAs $A_{1}$ and $A_{2}$ where $|\mathsf{Vars}(A_{1})\cap\mathsf{Vars}(A_{2})|\leq k$ and a document $\mathbf{d}$, one can construct in polynomial time a sequential VA $A_{\mathbf{d}}$ with $\mathord{\llceil{A_{\mathbf{d}}}\rrceil}(\mathbf{d})=\mathord{\llceil{A_{1}\setminus A_{2}}\rrceil}(\mathbf{d})$.*

First, if $\mathbf{d}=\epsilon$ then there two possible cases:

• •

$\mathord{\llceil{A_{2}}\rrceil}(\mathbf{d})=\emptyset$ and then we set $A_{\mathbf{d}}$ to be $A_{1}$.

• •

$\mathord{\llceil{A_{2}}\rrceil}(\mathbf{d})\neq\emptyset$ and then $\mathord{\llceil{A_{1}\setminus A_{2}}\rrceil}(\mathbf{d})=\emptyset$ since any two mappings are compatible. In this case we set $A_{\mathbf{d}}$ to automaton for for $\emptyset$.

Note that determining whether $\mathord{\llceil{A_{2}}\rrceil}(\mathbf{d})=\emptyset$ can be done in polynomial time (see Theorem 2.5).

Second, we can assume that $A_{2}$ has no other variables except those that are in $A_{1}$ due to the following: $\mathord{\llceil{A_{1}\setminus A_{2}}\rrceil}(\mathbf{d})=\mathord{\llceil{A_{1}\setminus\pi_{\mathsf{Vars}(A_{1})}A_{2}}\rrceil}(\mathbf{d})$ since if a mapping $\mu_{1}\in\mathord{\llceil{A_{1}}\rrceil}(\mathbf{d})$ has a compatible mapping $\mu_{2}\in\mathord{\llceil{A_{2}}\rrceil}(\mathbf{d})$ then \mu_{2}\scalebox{1.0}[0.85]{\raisebox{8.99994pt}{\mbox{\rotatebox{270.0}{\leftharpoonup}}} }\mathsf{Vars}(A_{1}) is also compatible for $\mu_{1}$ and is in $\mathord{\llceil{\pi_{\mathsf{Vars}(A_{1})}{A_{2}}}\rrceil}(\mathbf{d})$. On the other hand, if a mapping $\mu_{1}\in\mathord{\llceil{A_{1}}\rrceil}(\mathbf{d})$ has a compatible mapping $\mu_{2}\in\mathord{\llceil{\pi_{\mathsf{Vars}(A_{1})}A_{2}}\rrceil}(\mathbf{d})$, then no matter how we extend $\mu_{2}$, as long as we extend it with variables that are not in $\mathsf{Vars}(A_{1})$, it remains compatible to that $\mu_{1}$. Thus, if a mapping $\mu_{1}\in\mathord{\llceil{A_{1}}\rrceil}(\mathbf{d})$ does not have a compatible mapping $\mu_{2}\in\mathord{\llceil{A_{2}}\rrceil}(\mathbf{d})$ then it also does not have such in $\pi_{\mathsf{Vars}(A_{1})}\mathord{\llceil{A_{2}}\rrceil}(\mathbf{d})$. Therefore, we may assume that $\mathsf{Vars}(A_{1})\supseteq\mathsf{Vars}(A_{2})$ and we denote $\mathsf{Vars}(A_{2})$ by $V$.

Third, we can also assume that $A_{1}$ is semi-functional for $V:=\mathsf{Vars}(A_{2})$ since if it is not we can translate it into such in polynomial time (see Lemma 3.6).

We now move to the construction of sequential VAs $A$ and $B$ with fixed $|\mathsf{Vars}(A)\cap\mathsf{Vars}(B)|$ for which we will prove that $\mathord{\llceil{A\bowtie B}\rrceil}(\mathbf{d})=\mathord{\llceil{A_{1}\setminus A_{2}}\rrceil}(\mathbf{d})$.

Constructing $A$:

For every $X\subseteq V$ we define $X^{\prime}=\{x^{\prime}|x\in X\}$. We construct a VA $A$ that extends the $A_{1}$ such that for every accepting run $\rho$ of $A_{1}$ that closes exactly the set $X$ of variables, there is a corresponding accepting run in $A$ that assigns the same values as $\rho$ to $X$, and in addition, assigns to each of the variables of $X^{\prime}$ the span $[1,1\rangle$ (indicating that the variables of $X$ were closed throughout $\rho$) and to each of the variables in $V^{\prime}\setminus X^{\prime}$ the span $[|\mathbf{d}|+1,|\mathbf{d}|+1\rangle$ (indicating that the variables of $V\setminus X$ were unseen throughout $\rho$).

More formally, the VA $A$ consists of the disjoint copies $A_{1}^{X}$ of $A_{1}$ for $X\subseteq V$, along with their transitions, of an initial state $q_{0}$, and of an accepting state $q_{f}$. We then extend $A$ as follows:

• •

for every $X$, we add a path from $q_{0}$ to the initial state of $A_{1}^{X}$ that opens and then immediately closes all of the variables in $X^{\prime}$;

• •

for every $X$ and every accepting state $q$ of $A_{1}^{X}$ with $c_{q}(x)=\mathsf{c}$ for every $x\in X$, we add a path from $q$ to the final state $q_{f}$ that opens and then immediately closes all of the variables in $V^{\prime}\setminus X^{\prime}$.

We then remove from $A$ all of the states that are not reachable throughout any accepting run and obtain a sequential VA.

Let $\mu$ be a mapping with $\mathsf{dom}(\mu)\subseteq V$. A mapping $\mu^{\prime}$ is called the marked extension of $\mu$ if the following hold:

• •

$\mathsf{dom}(\mu^{\prime})=\mathsf{dom}(\mu)\cup V^{\prime}$

• •

for every $x\in\mathsf{dom}(\mu)$ it holds that $\mu^{\prime}(x)=\mu(x)$;

• •

for every $x\in\mathsf{dom}{(\mu)}$ it holds that $\mu^{\prime}(x^{\prime})=[1,1\rangle$;

• •

for every $x\not\in\mathsf{dom}{(\mu)}$ it holds that $\mu^{\prime}(x^{\prime})=[|\mathbf{d}|+1,|\mathbf{d}|+1\rangle$.

We observe that the following holds:

Lemma B.1.

Let $\mu_{1},\mu_{2}$ be two mappings with $\mathsf{dom}(\mu_{1})=\mathsf{dom}(\mu_{2})\subseteq V$ and let $\mu^{\prime}_{1}$ be the marked extension of $\mu_{1}$ and $\mu^{\prime}_{2}$ the marked extension of $\mu_{2}$. The following holds: $\mu_{1}$ is compatible with $\mu_{2}$ if and only if $\mu^{\prime}_{1}$ is compatible with $\mu^{\prime}_{2}$

Proof.

Assume that $\mu_{1}$ is compatible with $\mu_{2}$. It is enough to show that \mu^{\prime}_{1}\scalebox{1.0}[0.85]{\raisebox{8.99994pt}{\mbox{\rotatebox{270.0}{\leftharpoonup}}} }{V^{\prime}} for and that \mu^{\prime}_{2}\scalebox{1.0}[0.85]{\raisebox{8.99994pt}{\mbox{\rotatebox{270.0}{\leftharpoonup}}} }{V^{\prime}} are compatible. This follows directly from the fact that $\mathsf{dom}(\mu_{1})=\mathsf{dom}(\mu_{2})$. On the other hand, assume that $\mu^{\prime}_{1}$ is compatible with $\mu^{\prime}_{2}$. Since \mu^{\prime}_{1}\scalebox{1.0}[0.85]{\raisebox{8.99994pt}{\mbox{\rotatebox{270.0}{\leftharpoonup}}} }V is identical to $\mu_{1}$ and \mu^{\prime}_{2}\scalebox{1.0}[0.85]{\raisebox{8.99994pt}{\mbox{\rotatebox{270.0}{\leftharpoonup}}} }V is identical to $\mu_{2}$ we conclude that $\mu_{1}$ is compatible with $\mu_{2}$. ∎

The following lemmas describe the connection between $A_{1}$ and $A$:

Lemma B.2.

$\mu_{1}\in\mathord{\llceil{A_{1}}\rrceil}(\mathbf{d})$* if and only if $\mu\in\mathord{\llceil{A}\rrceil}(\mathbf{d})$ where $\mu$ is the marked extension of $\mu_{1}$.*

Proof.

Assume that $\mu_{1}\in\mathord{\llceil{A_{1}}\rrceil}(\mathbf{d})$. Then there is an accepting run on $A_{1}$ on $\mathbf{d}$ that corresponds with $\mu_{1}$. We build an accepting run of $A$ that corresponds with the marked extension $\mu$ of $\mu_{1}$ in the following way. We denote $X=\mathsf{dom}(\mu_{1})$. The run starts with a path from $q_{0}$ to the initial state of the copy $A_{1}^{X}$ that opens and then closes all of the variables in $X^{\prime}$. We then continue with the accepting run of $A_{1}^{X}$ on $\mu_{1}$ (there is such since $A_{1}^{X}$ is a copy of $A_{1}$). Then from the accepting state $q$ we reached we continue with a path that opens and then closes all of the variables in $V^{\prime}\setminus X^{\prime}$. We can do that from the way we constructed $A$. Hence, we conclude that $\mu\in\mathord{\llceil{A}\rrceil}(\mathbf{d})$.

Assume that $\mu\in\mathord{\llceil{A}\rrceil}(\mathbf{d})$ is the marked extension of $\mu_{1}$. There is a run of $A$ on $\mathbf{d}$ that corresponds with $\mu$. We take the sub-run on the copy of $A_{1}$ in $A$ and obtain an accepting run of $A$ on $\mathbf{d}$ that corresponds with $\mu_{1}$.Thus, we can conclude that $\mu_{1}\in\mathord{\llceil{A_{1}}\rrceil}(\mathbf{d})$. ∎

Constructing $B$:

We construct $B$ as follows: We compute all of the assignments in $\mathord{\llceil{A_{2}}\rrceil}(\mathbf{d})$ and replace each of those with their marked extension to obtain the set $M$ of extended mappings. We then compute all of the possible extended mappings whose domain is contained in $V\cup V^{\prime}$. For every such mapping if it is not in $M$ then we insert it to the set $\bar{M}$.

We now define the VA $B$ as the disjoint union of all of the paths that correspond with the assignments in $\bar{M}$ along with an initial state $q_{0}$ and an accepting state $q$. For every path $P$ that corresponds with the assignment $\mu$ with $X:=\mathsf{dom}(\mu)$ we extend $B$ as follows:

• •

we add a path that connects $q_{0}$ with the first state of $P$ that opens and then immediately closes all of the variables in $X^{\prime}$;

• •

we add a path that connects the last state of $P$ to the accepting state $q$, that opens and then immediately closes all of the variables in $Y^{\prime}$ where $Y=V\setminus X$.

Note that $B$ is a sequential VA.

Correctness:

The following lemma that describes the connection between $A_{1},A_{2}$, $B$ and $\mathbf{d}$.

Lemma B.3.

A mapping $\mu_{1}\in\mathord{\llceil{A_{1}}\rrceil}(\mathbf{d})$ has a compatible mapping $\mathord{\llceil{A_{2}}\rrceil}(\mathbf{d})$ if and only if the marked extension $\mu$ of $\mu_{1}$ does not have a compatible mapping in $\mathord{\llceil{B}\rrceil}(\mathbf{d})$.

Proof.

Assume $\mu_{1}\in\mathord{\llceil{A_{1}}\rrceil}(\mathbf{d})$ has a compatible mapping $\mu_{2}\in\mathord{\llceil{A_{2}}\rrceil}(\mathbf{d})$ and assume by contradiction that $\mu$ has a compatible mapping $\mu^{\prime}_{2}\in\mathord{\llceil{B}\rrceil}(\mathbf{d})$. Then by Lemma B.1 the restriction \mu^{\prime}_{2}\scalebox{1.0}[0.85]{\raisebox{8.99994pt}{\mbox{\rotatebox{270.0}{\leftharpoonup}}} }V is compatible with $\mu_{2}$ which is impossible due to $B$’s definition.

Assume that $\mu_{1}\in\mathord{\llceil{A_{1}}\rrceil}(\mathbf{d})$ does not have a compatible mapping $\mu_{2}\in\mathord{\llceil{A_{2}}\rrceil}(\mathbf{d})$. By Lemma B.2, the extension $\mu$ of $\mu_{1}$ is in $\mathord{\llceil{A}\rrceil}(\mathbf{d})$. Lemma B.1 and $B$’s definition implies that there is a mapping in $\mathord{\llceil{B}\rrceil}(\mathbf{d})$ that is compatible to $\mu$. ∎

We can conclude that $\mathord{\llceil{\pi_{V}(A\bowtie B)}\rrceil}(\mathbf{d})=\mathord{\llceil{A_{1}\setminus A_{2}}\rrceil}(\mathbf{d})$

Note that $\mathsf{Vars}(A)=\mathsf{Vars}(B)=V\cup V^{\prime}$ and since $|V|$ is fixed we this is also the case for $\mathsf{Vars}(A)\cap\mathsf{Vars}(B)=\mathsf{Vars}(A)$. Therefore, we can use Lemma 3.2 to conclude the desired.

Complexity:

Note that throughout our construction we performed only polynomial time steps: In case the document is empty, checking the emptiness of $\mathord{\llceil{A_{2}}\rrceil}(\mathbf{d})$ can be done in polynomial time. Transforming $A_{1}$ into a semi-functional VA for $V$ requires polynomial time assuming $|V|$ is fixed. Constructing the extension $A$ of $A_{1}$ requires polynomial time since we fix $|V|$. Computing the set of extended mappings that are not compatible with any of the extended mappings that correspond with $\mathord{\llceil{A_{2}}\rrceil}(\mathbf{d})$ also requires polynomial time since $|V|$ is fixed.

B.2. Proof of Theorem 4.4

Theorem 4.4.* The following problem is $\mathrm{W[1]}$-hard parametrized by $|\mathsf{Vars}(\gamma_{1})\cap\mathsf{Vars}(\gamma_{2})|$. Given two functional regex formulas $\gamma_{1}$ and $\gamma_{2}$ and an input document $\mathbf{d}$, is $\mathord{\llceil{\gamma_{1}\setminus\gamma_{2}}\rrceil}(\mathbf{d})$ nonempty?*

Let $\varphi=C_{0}\wedge\cdots\wedge C_{m}$ be a 3CNF fomula with variables $x_{1},\ldots,x_{n}$ and denote by $l_{i,1},l_{i,2},l_{i,3}$ the literals of the $i$-th clause for $0\leq i\leq m$. To show the reduction, we use the same idea of the reduction in the proof of Theorem 3.1.

Set $\mathbf{d}=s_{1}\ldots s_{n}$, where every $s_{i}$ is unique (this can be achieved by two distinct elements in $\Sigma$ such that each $|s_{i}|\in O(\log(n))$) and let $S=\{s_{1},\ldots,s_{n}\}$. Further set $\alpha_{S}=\bigvee_{\sigma\in S}\sigma$. We define the regex formula $\alpha_{1}$, using only the $k$ variables $V=\{y_{1},\ldots,y_{k}\}$ as

[TABLE]

We next define for every $0\leq i\leq m$ the regex $\alpha_{C_{i}}$, such that $\mathord{\llceil{\alpha_{C_{i}}}\rrceil}(\mathbf{d})$ corresponds to all possible assignments of $\varphi$ of weight $k$, such that $C_{i}$ is not satisfied. Let $S_{i}\subseteq S$ such that such that $s_{j}\in S_{i}$ iff $x_{j}$ is contained in clause $C_{i}$. Further let $S_{i}^{-}$ denote the variables in $S_{i}$ which appear negated in $C_{i}$, i.e. $s_{j}\in S_{i}^{-}$ iff $x_{j}$ is a negated variable in $C_{i}$, and define $S_{i}^{+}=S_{i}\setminus S_{i}^{-}$. Let $\mathit{ind}(S_{i}^{-}):=\{1\leq j\leq n\mid s_{i}\in S_{i}^{-}\}$ and similarly $\mathit{ind}(S_{i}^{+}):=\{1\leq j\leq n\mid s_{i}\in S_{i}^{+}\}$. Further define the regular expression $\alpha_{S\setminus S_{i}^{+}}:=\bigvee_{\sigma\in S\setminus S_{i}^{+}}\sigma$. In order to define $\alpha_{C_{i}}$, we need to consider different cases on the number of positive and negative literals in $C_{i}$.

• •

First, assume that $|S_{i}^{+}|=3$, i.e. $C_{i}$ is a clause containing three positive variables. Then we set

[TABLE]

• •

Next assume that $|S_{i}^{+}|=2$ and thus there is some $1\leq j\leq n$ such that $\{j\}=\mathsf{ind}(S_{i}^{-})$. For every $1\leq u\leq k$, we define

[TABLE]

Note that the regex $\alpha_{C_{i}}^{u}$ is built similar to $\alpha_{C_{i}}$ in the first case, except for the part $\mathop{y_{u}\{s_{j}\}}$ instead of $\mathop{y_{u}\{S\setminus S_{i}^{+}\}}$. We set $\alpha_{C_{i}}=\bigvee_{1\leq u\leq k}\alpha_{C_{i}}^{u}$.

• •

The case where $|S_{i}^{+}|=1$ is defined similar to the case where $|S_{i}^{+}|=2$. Assume that there are $1\leq j_{1}<j_{2}\leq n$ such that $\{j_{1},j_{2}\}=\mathsf{ind}(S_{i}^{-})$. Then for every $1\leq u_{1}<u_{2}\leq k$ we define

[TABLE]

and set $\alpha_{C_{i}}=\bigvee_{1\leq u_{1}<u_{2}\leq k}\alpha_{C_{i}}^{u}$.

• •

If $|S_{i}^{+}|=0$, then $\alpha_{C_{i}}$ is defined analogously to the case $|S_{i}^{+}|=1$ but with three indices $u_{1},u_{2},u_{3}$ instead of $u_{1},u_{2}$.

To define $\alpha_{2}$, we set $\alpha_{2}=\bigvee_{1\leq i\leq m}\alpha_{C_{i}}$. Note that $|\alpha_{2}|\leq(m+1)\cdot n^{3}\cdot(m+1)^{3}$. It remains to show that $\varphi$ is satisfyable (with weight $k$) iff $\mathord{\llceil{\alpha_{1}-\alpha_{2}}\rrceil}(\mathbf{d})\neq\emptyset$.

So let $\tau$ be a satisfying truth assignment of weight $k$. We claim that there exists some $\mu\in\mathord{\llceil{\alpha_{1}-\alpha_{2}}\rrceil}(\mathbf{d})$, with $\mu(y_{i})=[j,j+1\rangle$ for every $1\leq i\leq k$ and $1\leq j\leq n$ such that $\tau(x_{j})=\mathsf{t}$, and $x_{j}$ is the $i$-th such variable (i.e. among the variables $x_{1},\ldots,x_{j-1}$, $i-1$ are set to true via $\tau$). First consider the set $\mathord{\llceil{\alpha_{1}}\rrceil}(\mathbf{d})$. It is easy to see that $\mu^{\prime}\in\mathord{\llceil{\alpha_{1}}\rrceil}(\mathbf{d})$ iff there are $1\leq j_{1}<\ldots<j_{k}\leq n$ with $\mu^{\prime}=\{y_{1}\mapsto[j_{1},j_{1}+1\rangle,\ldots,y_{k}\mapsto[j_{k},j_{k}+1\rangle\}$, hence $\mu\in\mathord{\llceil{\alpha_{1}}\rrceil}(\mathbf{d})$. To see that $\mu\not\in\mathord{\llceil{\alpha_{2}}\rrceil}(\mathbf{d})$, we need to consider the set $\mathord{\llceil{\alpha_{2}}\rrceil}(\mathbf{d})$. By definition of $\alpha_{2}$, we have that $\mu^{\prime}\in\mathord{\llceil{\alpha_{2}}\rrceil}(\mathbf{d})$ iff there is some $1\leq i\leq m$ such that $\mu^{\prime}\in\mathord{\llceil{\alpha_{C_{i}}}\rrceil}(\mathbf{d})$. The containment $\mu^{\prime}\in\mathord{\llceil{\alpha_{C_{i}}}\rrceil}(\mathbf{d})$ can be characterized as follows: There exist $j_{1},\ldots,j_{k}\in\{1,\ldots,n\}\setminus\mathsf{ind}(S_{i}^{+})$ such that $1\leq j_{1}<\ldots<j_{k}\leq n$, $\mathsf{ind}(S_{i}^{-})\subset\{j_{1},\ldots,j_{k}\}$ and $\mu^{\prime}=\{y_{1}\mapsto[j_{1},j_{1}+1\rangle,\ldots,y_{k}\mapsto[j_{k},j_{k}+1\rangle\}$.

For some $1\leq i\leq n$, first assume that $C_{i}$ is a clause only containing positive variables. Since $\tau$ is a satisfying assignment, there is some $i^{\prime}$ such that and $\tau(x_{i^{\prime}})=1$. Hence $\mu(y_{i^{\prime}})=[j,j+1\rangle$ for some $1\leq j\leq n$, and since $i^{\prime}\in\mathsf{S_{i}^{+}}$ we have that $\mu\not\in\mathord{\llceil{\alpha_{2}}\rrceil}(\mathbf{d})$.

Fix some $1\leq i\leq n$. Since $\tau$ is a satisfying assignment, $\tau(x_{i^{\prime}})=1$ for some $i^{\prime}\in\mathsf{ind}(S_{i}^{+})$ or $\tau(x_{i^{\prime}})=0$ for some $i^{\prime}\in\mathsf{ind}(S_{i}^{-})$. In the first case, $j_{1},\ldots,j_{k}\in\{1,\ldots,n\}\setminus\mathsf{ind}(S_{i}^{+})$ does not hold , and in the second case $\mathsf{ind}(S_{i}^{-})\subset\{j_{1},\ldots,j_{k}\}$ is violated. Thus $\mu\not\in\mathord{\llceil{\alpha_{C_{i}}}\rrceil}(\mathbf{d})$ for every $1\leq i\leq n$, and hence $\mu\not\in\mathord{\llceil{\alpha_{2}}\rrceil}(\mathbf{d})$.

For the other direction, let $\mu\in\mathord{\llceil{\alpha_{1}-\alpha_{2}}\rrceil}(\mathbf{d})$. Using the same arguments as above, we can show that the truth assignment $\tau$ is a satisfying truth assignment of $\varphi$ with weight $k$, where $\tau(x_{i})=1$ iff $y_{i}=[j,j+1\rangle$ for some $1\leq j\leq n$.

B.3. Proof of Lemma 4.6

Lemma 4.6.* Let $\gamma$ be a sequential regex formula that is synchronized for $X\subseteq\mathsf{Vars}$. One can convert $\gamma$ in linear time into an equivalent sequential VA $A$ that is synchronized for $X$.*

Proof.

Like in Lemma 3.4 in (DBLP:conf/pods/FreydenbergerKP18, ), we obtain $A$ using the Thompson construction (cf. e.g., (hop:int, )) for converting a regular expression into an $\epsilon$-NFA, where we treat variable operations like symbols. More specifically, each occurrence of a variable binding $\mathop{x\{\gamma\}}$ is interpreted like $\mathop{x{\vdash}}\cdot\gamma\cdot\mathbin{{\dashv}x}$.

As $\gamma$ is sequential, $A$ is also sequential. A feature of the Thompson construction is that for every occurrence of a symbol the regular expression is converted into an initial state and a final state, where the former has a transition to the latter that is labeled with that symbol. All new transitions that enter to this sub-automaton pass through the initial state; the final state can only be reached through the original transition. This creates a new target state for every variable operation.

The other condition for VAs that are synchronizing for $X$ follows directly from the fact that $\gamma$ has no disjunctions over $x$. Either $\gamma$ never uses $x$ (e. g., if $\gamma=\mathop{x\{\emptyset\}}$, if $\gamma=\mathop{x\{a\}}\cdot\emptyset$, or due to other uses of $\emptyset$); then the accepting runs of $A$ also never operate on $x$. Or $\gamma$ always uses $x$; then the same also holds for all accepting runs of $A$. ∎

B.4. Proof of Proposition 4.7

Proposition 4.7.* Let $\gamma:=(\mathsf{a}\cdot x\{\epsilon\}\cdot\mathsf{a})\vee(\mathsf{b}\cdot x\{\epsilon\}\cdot\mathsf{b})$. There is no sequential VA $A$ that is synchronized for $x$ and equivalent to $\gamma$.*

Proof.

Note that $\mathord{\llceil{\gamma}\rrceil}(\mathsf{a}\mathsf{a})\neq\emptyset$ and that also $\mathord{\llceil{\gamma}\rrceil}(\mathsf{b}\mathsf{b})\neq\emptyset$; however, $\mathord{\llceil{\gamma}\rrceil}(\mathsf{a}\mathsf{b})=\emptyset$.

Note also that $\mathord{\llceil{\gamma}\rrceil}(\mathsf{a}\mathsf{a})=\{\mu_{1}\}$ where $\mu_{1}$ is the mapping that maps $x$ to $[2,3\rangle$ and that $\mathord{\llceil{\gamma}\rrceil}(\mathsf{b}\mathsf{b})=\{\mu_{2}\}$ where $\mu_{2}$ is the mapping that maps $x$ to $[2,3\rangle$.

We can conclude that there exists states $q_{1},q_{2},q_{3}\in Q$ and $q_{4}\in F$ such that that the following is a valid an accepting run of $A$ (on $\mathsf{aa}$):

[TABLE]

Similarly, there exists states $p_{1},p_{2},p_{3}\in Q$ and $p_{4}\in F$ such that that the following is a valid an accepting run of $A$ (on $\mathsf{bb}$):

[TABLE]

Let us assume by contradiction that $A$ is synchronized for the variable $x$. Thus, we obtain that $p_{2}=q_{2}$. So we can take the first run of $\rho_{1}$ and the second of $\rho_{2}$ and glue them together to obtain a new run:

[TABLE]

This run is valid and accepting and therefore we can conclude that $\mathord{\llceil{A}\rrceil}(\mathsf{ab})$ is not empty (as it contains a mapping that maps $x$ to the span $[2,3\rangle$). However, we already noted that $\mathord{\llceil{\gamma}\rrceil}(\mathsf{a}\mathsf{b})=\emptyset$ and we assumed that $A$ and $\gamma$ are equivalent, which lead us to the desired contradiction. ∎

B.5. Proof of Theorem 4.8

Theorem 4.8.* Given an input document $\mathbf{d}$ and two sequential VAs $A_{1}$ and $A_{2}$ such that, for $X:=\mathsf{Vars}(A_{1})\cap\mathsf{Vars}(A_{2})$, $A_{1}$ is semi-functional for $X$ and $A_{2}$ is synchronized for $X$, one can construct a sequential VA $A_{\mathbf{d}}$ with $\mathord{\llceil{A_{\mathbf{d}}}\rrceil}(\mathbf{d})=\mathord{\llceil{A_{1}\setminus A_{2}}\rrceil}(\mathbf{d})$ in polynomial time. *

For two sequential VAs $A_{1}$ and $A_{2}$, let $X:=\mathsf{Vars}(A_{1})\cap\mathsf{Vars}(A_{2})$, and assume that $A_{1}$ is semi-functional for $X$ and that $A_{2}$ is synchronized for $X$. Before we consider the document, we fix some assumptions on $A_{2}$ that are possible without loss of generality.

Assumptions on $A_{2}$:

First of all, we assume that $\mathsf{Vars}(A_{2})=X$. Variables in $\mathsf{Vars}(A_{2})\setminus\mathsf{Vars}(A_{1})$ play no rule for the difference $A_{1}\setminus A_{2}$, so this assumption does not change the correctness of the construction. We can guarantee this by replacing all transitions with operations for variables from $\mathsf{Vars}(A_{2})\setminus\mathsf{Vars}(A_{1})$ with $\epsilon$-transitions.

As pointed out in Lemma 3.8 of (DBLP:conf/pods/FreydenbergerKP18, ), this change turns $A_{2}$ into an automaton for $\pi_{X}(A_{2})$. Although (DBLP:conf/pods/FreydenbergerKP18, ) only remarks this for functional automata, the same argument translates to sequential automata; and the translation can be performed in time that is linear in the size of the transition relation of $A_{2}$.

As $A_{2}$ is synchronized for $X$, for every $x\in X$, either all accepting runs of $A_{2}$ operate on $x$, or no accepting run of $A$ operates on $x$. We assume without loss of generality that the former is the case (this can be achieved by trimming $A_{2}$ by removing all states that are not reachable from the initial state, and from which no final state can be reached). As every variable operation from $\boldsymbol{\Gamma}_{X}$ has a unique target state, we know that $A_{2}$ is semi-functional for $X$. But as we assume that $\mathsf{Vars}(A_{2})=X$, and we have established that every accepting run of $A_{2}$ acts on all variables of $X$, this implies that $A_{2}$ is in fact functional.

Match structures in general:

We are now ready for the main part of the proof, which uses constructions from the proof of Theorem 3.3 in (DBLP:conf/pods/FreydenbergerKP18, ), which describes a polynomial delay algorithm for $\mathord{\llceil{A}\rrceil}(\mathbf{d})$ for functional VAs $A$ and documents $\mathbf{d}$. We shall sketch all details that are relevant to the present proof; more information can be found in Section 4 of that paper, and also its preprint at http://arxiv.org/abs/1703.10350.

Given a document $\mathbf{d}=\sigma_{1}\cdots\sigma_{\ell}$ with $\ell\geq 1$, the key idea of the construction is to represent each element of $\mathord{\llceil{A}\rrceil}(\mathbf{d})$ as a sequence $c_{0},\dots,c_{\ell+1}$ of variable configurations of $A$. Each $c_{i}$ is the last variable configuration before $\sigma_{i}$ is processed; and $c_{\ell+1}$ is the configuration where all variables have been closed.

To obtain these configurations, we construct the match graph of $A$ on $\mathbf{d}$, a directed acyclic graph $G(A,\mathbf{d})$ that has one designated source node, and nodes of the form $(i,q)$, where $0\leq i\leq\ell$ and $q$ is a state of $A$. Intuitively, the node $(i,q)$ represents that after processing $\sigma_{1}\dots\sigma_{i}$ (the first $i$ letters of $\mathbf{d}$), $A$ can be in state $q$.

Consequently, an edge from $(i,p)$ to $(i+1,q)$ represents that if $A$ is in state $p$ and reads the symbol $\sigma_{i+1}$, it can enter state $q$ (not necessarily directly; it may process arbitrarily many variable operations or $\epsilon$-transitions after reading $\sigma_{i+1})$. The match graph $G(A,\mathbf{d})$ can be constructed from $A$ and $\mathbf{d}$ directly from the transition relation of $A$ and by using standard reachability algorithms. These reachability algorithms can also be used to trim the match graph: We remove all nodes that cannot be reached from the source, and all nodes that cannot reach a node $(\ell+1,q)$ where $q$ is a final state of $A$. As $A$ is functional, each state $q$ has a well-defined variable configuration ${c}_{q}$, which means that we can also associate each node $(i,q)$ with that variable configuration.

We can now interpret the match graph $G$ as an NFA $M(A,\mathbf{d})$ over the alphabet of variable configurations in the following way: All nodes of $G(A,\mathbf{d})$ become states of $M(A,\mathbf{d})$. The source of $G(A,\mathbf{d})$ becomes the initial state, and the final state of $M(A,\mathbf{d})$ are those nodes $(\ell+1,q)$ where $q$ is a final state of $A$. Finally, each edge from a node $v$ to a node $(i,q)$ in $G(A,\mathbf{d})$ becomes a transition with the letter ${c}_{q}$. We call the automaton $M(A,\mathbf{d})$ the match structure of $A$ on $\mathbf{d}$.

As shown in (DBLP:conf/pods/FreydenbergerKP18, ), there is a one-to-one correspondence between the elements of $\mathord{\llceil{A}\rrceil}(\mathbf{d})$ and the words in the language of $A_{G}$.

Preliminaries to determinizing $M(A_{2},\mathbf{d})$:

In the present proof, we start with the same construction, and first construct the match structure $M(A_{2},\mathbf{d})$ of $A_{2}$ on $\mathbf{d}$, where $\mathbf{d}=\sigma_{1}\cdots\sigma_{\ell}$ with $\ell\geq 1$ is our specific input document.

We have already established that we can assume that $A_{2}$ is functional; hence, we can directly use the construction that we discussed previously. Our next goal is to determinize $M(A_{2},\mathbf{d})$, and use this to implicitly construct a VA for the complement of $\mathord{\llceil{A_{2}}\rrceil}(\mathbf{d})$. This can then be combined with $A_{1}$ using a minor variation of the join-construction from Lemma 3.8.

In general, determinizing a match structure is not a viable approach (this was already observed in (DBLP:conf/pods/FreydenbergerKP18, )). But in our specific case, we can use that $A_{2}$ is synchronizing for $X$ (which we assume to be identical with $\mathsf{Vars}(A_{2})$. As every variable operation from $\boldsymbol{\Gamma}_{X}$ has a unique target state (and as $A_{2}$ is functional), we know that every accepting run of $A_{2}$ executes the operations in the same order. Hence, we can define $\omega_{1},\ldots,\omega_{2k}\in\boldsymbol{\Gamma}_{X}$ with $k:=|X|$ according to this order (i. e., the $i$-th operation in every accepting run is $\omega_{i}$). Accordingly, we define a sequence $c_{0},c_{1}\ldots,c_{2k}$ of variable configurations, where $c_{0}$ is the configuration that assigns $\mathsf{w}$ to all variables of $x$, and $c_{i+1}$ is the configuration that is obtained from applying operation $\omega_{i}$ to configuration $c_{i}$. Note that, as $A_{2}$ is functional, we use variable configurations (that use $\mathsf{w}$) instead of extended variable configurations (that use $\mathsf{u}$). Later on, this will help us distinguish between the configurations of the match structure of $A_{2}$ and the extended configurations of $A_{1}$.

As the order of the possible variable operations of $A_{2}$ is fixed, the language of the match structure of $A_{2}$ on any document (not just our specific document $\mathbf{d}$) is a subset of $c_{0}^{*}\cdot c_{1}^{*}\cdots c_{2k}^{*}$. Note that this does not mean that every configuration would appear; if two variable operations are always performed without consuming a symbol between them, the intermediate variable configuration would never appear. Consider the following functional VA that is synchronized for all it variables:

q_{0}$$q_{1}$$q_{2}$$q_{3}$$q_{4}$$\mathop{x{\vdash}}$$\Sigma$$\mathbin{{\dashv}x}$$\mathop{y{\vdash}}$$\mathbin{{\dashv}y}

Then on every document $\mathbf{d}$, its match structure accepts only a single sequence of variable configurations, namely ${c}_{q_{0}}\cdot{c}_{q_{2}}^{|\mathbf{d}|}\cdot{c}_{q_{3}}$. If we replace the state $q_{2}$ with an arbitrarily complicated NFA, the same observation would hold for all documents that belong to the language of the NFA (all others would be rejected).

The fact that $M(A_{2},\mathbf{d})$ expresses a concatenation of unary languages is not enough to allow for determinization; for this, we need to use the unique target states. Assume that $M(A_{2},\mathbf{d})$ reads a new variable configuration. More formally and more specifically, assume that $G(A_{2},\mathbf{d})$ contains edges

• •

from some $(i,p)$ to some $(i+1,q)$ with ${c}_{p}\neq{c}_{q}$, and

• •

from some $(i,p^{\prime})$ to some $(i^{\prime}+1,q^{\prime})$ with ${c}_{p^{\prime}}\neq{c}_{q^{\prime}}$,

such that ${c}_{q}={c}_{q^{\prime}}$. Choose $j$ such that $c_{j}={c}_{q}$. Then the transition from $p$ to $q$ and the transition from $p^{\prime}$ to $q^{\prime}$ both execute $\omega_{j}$, which means that they pass through its unique target state. Both can then execute different sequences $\epsilon$-transitions, which means that $q\neq q^{\prime}$ may hold. But we can switch these sequences of transitions, and observe that $G(A_{2},\mathbf{d})$ must also contain an edge from $(i,p)$ to $(i+1,q^{\prime})$, and an edge from $(i,p^{\prime})$ to $(i^{\prime}+1,q)$.

Thus, for every variable configuration $j$, we can define a set $I_{j}$ that contains exactly those $q$ such that $M(A_{2},\mathbf{d})$ contains an edge from some node $(i,p)$ to some node $(i+1,q)$ with ${c}_{p}\neq{c}_{q}$. In other words, $I_{j}$ contains those states that of $A_{2}$ that are encoded in states that $M(A_{2},\mathbf{d})$ can enter when first reading variable configuration $c_{j}$. And as we established in the previous paragraph, $A_{2}$ being synchronized for all its variables ensures that each $I_{j}$ is well-defined. We also use $Q_{q}$ to refer to that $I_{j}$ for which $c_{j}={c}_{q}$ holds.

Determinizing $M(A_{2},\mathbf{d})$:

We use this to these insights to turn the NFA $M(A_{2},\mathbf{d})$ into a DFA $D_{2}$ over the alphabet $c_{0},\dots,c_{2k}$. Apart from the special initial state, each state of $D_{2}$ is a triple $(i,s,Q)$, where

• •

$i$ has the same role as in $G(A_{2},\mathbf{d})$, meaning that it encodes how many positions of $\mathbf{d}$ have been consumed,

• •

$s\leq i$ denotes when the current variable configuration was consumed the first time,

• •

$Q$ is a set of states of $A_{2}$, and all its elements have the same variable configuration.

We construct $D_{2}$ by performing a variant of the power set construction that, in addition to taking the special structure of $M(A_{2},\mathbf{d})$ into account, also includes a reachability analysis.

The first set of states and transitions is computed as follows: For every variable configuration $c_{j}$ with $0\leq j\leq 2k$, we compute a set $Q_{j}$ that contains exactly those states $q$ such that ${c}_{q}=c_{j}$ and $M(A_{2},\mathbf{d})$ contains a state $(0,q)$. If $Q_{j}\neq\emptyset$, we extend $D_{2}$ with the state $(0,0,Q_{j})$ and a transition with label $c_{j}$ from the initial state to $(0,0,Q_{j})$. In this case, $Q_{j}=I_{j}$ holds.

To compute the successor states of some state $(i,s,P)$, we proceed as follows: First, we consider successors with the same variable configuration. Let $c_{j}$ be the variable configuration of all states in $P$. We compute the set $Q_{j}$ of all states $q$ such that $M(A_{2},\mathbf{d})$ contains a transition with label $c_{j}$ from some state $(i,p)$ to $(i+1,q)$. If $Q_{j}\neq\emptyset$, we extend $D_{2}$ with the state $(i+1,s,Q_{j})$ and a transition with label $c_{j}$ from the $(i,s,P)$ to $(i+1,s,Q_{j})$.

To determine successors with different configurations, we consider all $j^{\prime}$ with $j<j^{\prime}\leq 2k$. For every such $c_{j}$, we check if $M(A_{2},\mathbf{d})$ contains a transition from some $(i,p)$ with $p\in P$ to some $(i+1,q)$ with ${c}_{q}=c_{j^{\prime}}$. If this is the case, we extend $D_{2}$ with an edge from the state $(i,s,P)$ to the state $(i+1,i+1,I_{j^{\prime}})$ that is labeled with $c_{j^{\prime}}$, and add the state $(i+1,i+1,I_{j^{\prime}})$ to $D_{2}$ if it does not exist yet.

Intuitively, $D_{2}$ simulates $M(A_{2},\mathbf{d})$ by disassembling it into sub-automata for the unary languages of each $c_{j}$. It needs to keep track of where in $\mathbf{d}$ the current part started (using the middle component of each state triple) to avoid accepting words of the wrong length. Note that for each state $(i,s,Q)$ with $s=0$, we have $Q=I_{j}$ for some variable configuration $c_{j}$ with $0\leq j\leq 2k$. Hence, and by the definition of $D_{2}$, we observe that each component $Q$ is determined by the combination of $i$, $s$, and $c_{j}$. Therefore, we can bound the number of states in $D_{2}$ by $O(\ell^{2}k)$, as $1\leq s\leq i\leq\ell+1$ and $0\leq j\leq 2k$ hold. Likewise, the number of transitions is bounded by $O(\ell^{2}k^{2})$.

Combining $A_{1}$ and $D_{2}$:

The actual construction of $A_{\mathbf{d}}$ is a variant of the proof of Lemma 3.8: To fix identifiers, we declare that $A_{1}=(V_{1},Q_{1},q_{0,1},F_{1},\delta_{1})$ and $D_{2}=(Q_{2},q_{0,2},F_{2},\delta_{2})$. As $A_{2}$ is sequential, every state $q_{i}\in Q_{1}$ has an extended variable configuration $\tilde{c}_{q}$; and as $A_{1}$ is semi-functional for $X$, we know that $\tilde{c}_{q}(x)\in\{\mathsf{u},\mathsf{o},\mathsf{c}\}$ for all $x\in X$. Although $D_{2}$ is technically a DFA over the alphabet of variable configurations $c_{0},\ldots,c_{2k}$, its definition allows us to associate each state in $Q_{2}$ with a variable configuration ${c}_{q}$ by considering the incoming transitions of $q$ (which is equivalent to considering the variable configuration of the states that are encoded in $q$), and by setting ${c}_{q_{0,2}}(x)=\mathsf{w}$ for all $x\in X$.

Recall that our goal is to construct a sequential VA $A$ with $\mathord{\llceil{A_{\mathbf{d}}}\rrceil}(\mathbf{d})=\mathord{\llceil{A_{1}\setminus A_{2}}\rrceil}(\mathbf{d})$. In principle, $A_{\mathbf{d}}$ simulates $A_{1}$ and $A_{2}$ in parallel on the input document $\mathbf{d}$. But instead of using $A_{2}$, we use its deterministic representation $D_{2}$. If $A_{1}$ picks a next state $q_{1}$, its opponent $D_{2}$ tries to counter that by picking an a state that has consistent behavior on the variables of $X$ (as $D_{2}$ has variable configurations as input, it hides the states of $A_{2}$ behind a layer of abstraction). If $D_{2}$ can pick such a state of its states that is consistent, it can follow $A_{1}$ for the current input. If all available states are inconsistent, it cannot follow $A_{2}$ and has to enter the trap state. This means that the finishing the current run of $A_{1}$ will lead to an element of $\mathord{\llceil{A_{1}\setminus A_{2}}\rrceil}(\mathbf{d})$.

One technical problem is that $A_{1}$ may choose to leave variables from $X$ undefined; and by definition, $\mathord{\llceil{A_{1}\setminus A_{2}}\rrceil}(\mathbf{d})$ only takes the common variables into account. Thus, if $D_{2}$ is forced at some point to open a variable that was not yet processed, it cannot decide whether $A_{1}$ will open that variable later in the run; and using the finite control to keep track of the arising combinations would lead to an exponential blowup.

But we can work around this problem: As $A_{1}$ is semi-functional for $X$, each final state contains information which variables were skipped in every run that ends in that state. For each $q\in F_{1}$ and each variable $x\in X$, we say that $q$ skips $x$ if $\tilde{c}_{q}(x)=\mathsf{u}$. Let $S(q)$ denote the set of skipped variables in $q$. The idea is to decompose $A_{1}$ into sub-automata that all skip the same variables. There are at most $|F_{1}|$ different sets $S(q)$ with $q\in F_{1}$. We call these $S_{1},\ldots,S_{f}$ with $f\leq|F_{1}|$.

For each $S_{j}$, we create a sub-automaton $A_{i,j}=(V_{1},Q_{1,j},q_{0,1},F_{1,j},\delta_{i,j})$ that accepts exactly those runs of $A_{1}$ that end in a state $q$ with $S(q)=S_{j}$. We first mark $q_{0,1}$ and all $q\in F_{1}$ with $S(q)=S_{f}$. Then we use a standard reachability algorithm to mark all states and transitions that lead from $q_{0,1}$ to some marked finite state. Finally, we obtain $A_{i,j}$ by removing all unmarked states and transitions. The resulting automaton is sequential and semi-functional for $X$, and all accepting runs skip the same variables. Furthermore, we observe that $\mathord{\llceil{A_{1}}\rrceil}=\bigcup_{j=1}^{f}\mathord{\llceil{A_{i,j}}\rrceil}$.

The last major step is creating automata $A_{\mathbf{d},j}$ such that $\mathord{\llceil{A_{\mathbf{d},j}}\rrceil}(\mathbf{d})=\mathord{\llceil{A_{1,j}\setminus A_{2}}\rrceil}(\mathbf{d})$. Recall that each $A_{1,j}$ skips exactly the variables from $S_{j}$.

For this, we define the notions of consistent and inconsistent state pairs. For each $q_{1}\in Q_{i,j}$, each $q_{2}\in Q_{2}$, and each $x\in X$, we say $q_{1}$ and $q_{2}$ are consistent for $x$ if one of the following conditions holds:

• •

$x\in S_{j}$,

• •

$\tilde{c}_{q_{1}}(x)=\mathsf{u}$ and ${c}_{q_{2}}(x)=\mathsf{w}$,

• •

$\tilde{c}_{q_{1}}(x)={c}_{q_{2}}(x)\in\{\mathsf{o},\mathsf{c}\}$.

Otherwise, $q_{1}$ and $q_{2}$ are inconsistent. Building on these definitions, we say that

• •

$q_{1}$ and $q_{2}$ are consistent if they are consistent for all $x\in X$,

• •

$q_{1}$ and $q_{2}$ are inconsistent if they are inconsistent for at least one $x\in X$.

The set of states of $A_{\mathbf{d},j}$ shall consist of states of the following type:

• •

consistent pairs $(q_{1},q_{2})$, with $q_{1}\in Q_{1,j}$ and $q_{2}\in Q_{2}$,

• •

pairs from $Q_{1,j}\times\{\mathsf{trap}\}$,

• •

a number of unnamed helper states.

Here $\mathsf{trap}$ is a special trap state that we shall use to denote that something happened that made the parallel simulations of $A_{1,j}$ and $D_{2}$ inconsistent.

We define the initial state of $A_{\mathbf{d},j}$ as $(q_{0,1},q_{0,2})$ and its set of final states as $F_{1}\times\{\mathsf{trap}\}$. This matches the intuition that $\mathsf{trap}$ denotes that the two automata have inconsistent behavior on the variable operations. Following the same intuition, we define that for every transition from some state $p$ to some state $q$ in $A_{1,j}$, we have a transition with the same label from $(p,\mathsf{trap})$ to $(q,\mathsf{trap})$ in $A_{\mathbf{d},j}$.

To define the “main behavior” of $A_{\mathbf{d},j}$, we use the notion of a the variable-$\epsilon$-closure as in Lemma 3.8: For every $p\in Q_{1,j}$, we define $\mathcal{VE}(p)$ as the that of all $q\in Q_{1,j}$ that can be reached from $p$ by using only transitions from $\{\epsilon\}\cup\boldsymbol{\Gamma}_{V_{i}}$.

Let $\hat{Q}_{2}\subset Q_{2}$ be the set of all states to which $q_{0,2}$ has a transition. For each $q_{1}\in\mathcal{VE}(q_{0,1})$, we distinguish the following cases:

• •

If there is some $q_{2}\in\hat{Q}$ such that $q_{1}$ and $q_{2}$ are consistent, we add a state $(q_{1},q_{2})$ to $A_{\mathbf{d}}$.

• •

If all $q_{2}\in\hat{Q}$ are inconsistent with $q_{1}$, we add a state $(q_{1},\mathsf{trap})$ to $A_{\mathbf{d}}$.

In both cases, we connect $q_{0,1}$ with the new state using a sequence of helper states that has exactly the same variable operations and $\epsilon$-transitions as one that takes $Q_{1}$ from $q_{0,1}$ to $q_{1}$. We consider all states $(q_{1},q_{2})$ and $(q_{1},P)$ that were introduced in this step states on level 0. Intuitively, state of the form $(q_{1},q_{2})$ describe cases where $D_{2}$ can follow the behavior of $A_{1,j}$, states of the form $(q_{1},\mathsf{trap})$ describe cases where it cannot follow the behavior and has to give up.

Now, we successively process the symbols $\sigma_{i}$ of $\mathbf{d}=\sigma_{1}\cdots\sigma_{\ell}$. For each $i$ with $1\leq i\leq\ell$, we process the states on level $i-1$ and compute their successors as follows:

For each state $(p_{1},p_{2})$ on level $i-1$, we define $\hat{Q}\subset Q_{2}$ as the set of all states to which $p_{2}$ has a transition. We then consider each $q_{1}\in\bigcup_{(p_{1},\sigma_{i},q^{\prime})\in\delta_{1,j}}\mathcal{VE}(q^{\prime})$ and distinguish the same cases as for level [math]. Now, the new non-trap states are on level $i$; and in each case, $(p_{1},p_{2})$ is connected with the new state using a sequence of helper states that processes $\sigma_{i}$ and then exactly the same variable operations and $\epsilon$-transitions that take $A_{1}$ from $q^{\prime}$ to $q_{1}$.

Now observe that every run of $A_{1,j}$ maps into a run of $A_{\mathbf{d},j}$ (and, likewise, every run of $A_{1}$ that skips exactly the variables of $S_{j}$ maps into a run of $A_{i,j}$). If this run can be matched by any run of $A_{2}$ (as realized by $D_{2}$), it ends in a state $(q_{1},q_{2})$ and is not an accepting run of $A_{\mathbf{d},j}$. But if it is not matched, then $D_{2}$ will be forced to admit the inconsistency, and redirect the simulation into the copy of $A_{1,j}$ that assigns the $\mathsf{trap}$ state to $D_{2}$. In other words, $\mathord{\llceil{A_{\mathbf{d},j}}\rrceil}(\mathbf{d})=\mathord{\llceil{A_{1,j}\setminus A_{2}}\rrceil}(\mathbf{d})$.

Finally, we obtain $A_{\mathbf{d}}$ by taking all $A_{\mathbf{d},j}$ and adding a new initial state that has an $\epsilon$-transition to each initial state of some $A_{\mathbf{d},j}$. Then

[TABLE]

As $A_{1}$ is sequential, every $A_{1,j}$ is sequential. Therefore, every $A_{\mathbf{d},j}$ is sequential, and so is $A_{\mathbf{d}}$.

Complexity:

Let $m_{i}$ and $n_{i}$ denote the number of transitions and states of $A_{i}$, let $\ell=|\mathbf{d}|$, and $k=|\mathsf{Vars}(A_{1})\cap\mathsf{Vars}(A_{2})$. Let $v:=|\mathsf{Vars}(A_{1})$. The match structure $M(A_{2},\mathbf{d})$ can be constructed in $O(\ell n_{2}^{2})$, see (DBLP:conf/pods/FreydenbergerKP18, ). Recall that $D_{2}$ has $O(\ell^{2}k)$ states and $O(\ell^{2}k^{2})$ transitions. Each transition can be computed in $O(n_{2})$, which means that we can obtain $D_{2}$ from $M(A_{2},\mathbf{d})$ in $O(\ell^{2}k^{2}n_{2})$. Hence, the total time of computing $D_{2}$ from $A_{2}$ and $\mathbf{d}$ is $O(\ell^{2}k^{2}n_{2}+\ell n_{2}^{2})$.

To construct an automata $A_{\mathbf{d},j}$, we first pre-compute the comparisons of variable configurations in time $O(n_{1}k^{2})$ (as $D_{2}$ has $O(k)$ different variable configurations, and variable configurations can be compared in $O(k)$). We also pre-compute the variable-$\epsilon$-closures, which takes $O(m_{1}n_{1})$.

As $A_{1}$ has at most $|F_{1}|$ different skip sets $S_{j}$, we need to compute $O(n_{1})$ sub-automata $A_{j}$. Each can be constructed with a standard reachability analysis in time $O(m_{1}+n_{1})$. Hence, computing all automata $A_{1,j}$ takes time $O(m_{1}n_{1}+n_{1}^{2})$. This will be subsumed by the complexity of the next steps.

For each $A_{\mathbf{d},j}$, we have to compute $\ell+1$ levels; in each level, we combine $O(n_{1}k)$ state pairs $(p_{1},p_{2})$ with $O(n_{1}k)$ state pairs $(q_{1},q_{2})$ (as $D_{2}$ is deterministic and over an alphabet of size $2k$, each state has $O(k)$ outgoing transitions), and we need to include $O(v)$ helper states. Hence, each $A_{\mathbf{d},j}$ can be constructed in time $O(\ell k^{2}n_{1}^{2}v)$ without pre-computations, and $O(\ell k^{2}n_{1}^{2}v+m_{1}n_{1})$ including pre-computations.

Hence, computing $A_{\mathbf{d}}$ by constructing $O(n_{1})$ many $A_{\mathbf{d}_{j}}$, each in time $O(\ell k^{2}n_{1}^{2}v+m_{1}n_{1})$, takes a total time of $O(\ell k^{2}n_{1}^{3}v+m_{1}n_{1}^{2})$.

This combines to a total time of $O(\ell^{2}k^{2}n_{2}+\ell n_{2}^{2}+\ell k^{2}n_{1}^{3}v+m_{1}n_{1}^{2})$. For a less precise estimation, let $n\in O(n_{1}+n_{2})$ and observe that $k\leq v$ and $m_{1}\in O(n^{2})$. Then the complexity becomes $O(\ell^{2}v^{2}n+\ell n^{2}+\ell v^{3}n^{3}+n^{4})=O(\ell^{2}v^{2}n+\ell v^{3}n^{3}+n^{4})$.