Note on the Infiniteness and Equivalence Problems for Word-MIX Languages

TL;DR

This paper presents a decidable, graph-based method to determine the infiniteness and equivalence of certain word languages defined by equal subword counts, providing a self-contained proof without relying on constrained automata theory.

Contribution

It introduces a new, self-contained graph-structural characterization for deciding infiniteness and equivalence of specific word languages, simplifying previous complex automata-based approaches.

Findings

Decidable graph-structural characterization of language infiniteness

Decidable equivalence criteria for the languages

Self-contained proof avoiding automata theory

Abstract

In this note we provide a (decidable) graph-structural characterisation of the infiniteness of $L(w_1, ..., w_k)$, where $L(w_1, ..., w_k) = \{w \in A^* | |w|_{w_1} = \cdots = |w|_{w_k}\}$ is the set of all words that contain the same number of subword occurrences of parameter words $w_1, ..., w_k$. We also provide the decidable characterisation of the equivalence for those languages. Although those two decidability results are also obtained from more general known decidability results on unambiguous constrained automata, this note tries to give a self-contained (without the knowledge about constrained automata) proof of the decidability.

Figures

Equations

$$|w{|}_v\stackrel{\mathtt{def}}{=}\#\left(\{(w_1,w_2)\in A^{\ast }\times A^{\ast }\mid w_{1}v{w}_2=w\}\right).$$

$$L(w_1,\dots ,w_k)\stackrel{\mathtt{def}}{=}\{w\in A^{\ast }\mid |w{|}_{w_1}=\cdots =|w{|}_{w_k}\}$$

$$\mathrm{pref}(w)$$

$$\mathrm{suff}(w)$$

$$(x_1,y_1){\le }_{1,2}(x_2,y_2)\stackrel{\mathtt{def}}{⟺}{x}_1{\le }_1{x}_2\text{ and }{y}_1{\le }_2{y}_2.$$

$$E\stackrel{\mathtt{def}}{=}\{(av,vb)\mid a,b\in A,v\in A^{N-1}\}.$$

$$|w{|}_{(w_1,\dots ,w_k)}^{\mathtt{suff}}\stackrel{\mathtt{def}}{=}$$

$$|w{|}_{(w_1,\dots ,w_k)}\stackrel{\mathtt{def}}{=}$$

$$|vw{|}_{(w_1,\dots ,w_k)}=|v{|}_{(w_1,\dots ,w_k)}+|\omega {|}_{(w_1,\dots ,w_k)}$$

$${\Phi }_{\mathcal{G}}((v))$$

$${\Phi }_{\mathcal{G}}(\omega \odot (v,v^{\prime }))$$

$$\text{ where }(\pi ,\Gamma )={\Phi }_{\mathcal{G}}(\omega ).$$

$${\Psi }_{\mathcal{G}}(\omega ,\varnothing )$$

$${\Psi }_{\mathcal{G}}(\omega ,\gamma .\Gamma )$$

$$|\omega {|}_{(w_1,\dots ,w_k)}=|\pi {|}_{(w_1,\dots ,w_k)}+\sum _{i=1}^{|\Gamma |}|\Gamma (i){|}_{(w_1,\dots ,w_k)}$$

$$(\mathrm{NTr}(\omega ))(\pi )\stackrel{\mathtt{def}}{=}\{\begin{array}{ll}1& \text{ if }\pi ={\pi }_{\omega }\\ 0& \text{ otherwise}\end{array}$$

$$({\pi }_{\omega },\Gamma )={\Phi }_{\mathcal{G}}(\omega ).$$

$$\mathrm{Tr}(\omega )\stackrel{\mathtt{def}}{=}\{\pi \in \mathcal{P}(\mathcal{G})\mid (\mathrm{NTr}(\omega ))(\pi )\ne 0\}\cup \{\gamma \in \mathcal{C}(\mathcal{G})\mid (\mathrm{NTr}(\omega ))(\gamma )\ne 0\}.$$

$$|\omega {|}_{(w_1,\dots ,w_k)}=\sum _{\pi \in \mathcal{P}({\mathcal{G}}_A^N)}(\mathrm{NTr}(\omega ))(\pi )\cdot |\pi {|}_{(w_1,\dots ,w_k)}+\sum _{\gamma \in \mathcal{C}({\mathcal{G}}_A^N)}(\mathrm{NTr}(\omega ))(\gamma )\cdot |\gamma {|}_{(w_1,\dots ,w_k)}$$

$$|\omega |=\sum _{\pi \in \mathcal{P}({\mathcal{G}}_A^N)}(\mathrm{NTr}(\omega ))(\pi )\cdot |\pi |+\sum _{\gamma \in \mathcal{C}({\mathcal{G}}_A^N)}(\mathrm{NTr}(\omega ))(\gamma )\cdot |\gamma |.$$

$$\mathtt{diff}(\mathbf{v})\stackrel{\mathtt{def}}{=}\sum _{i=1}^k(\max (c_1,\dots ,c_k)-c_i).$$

$$\mathtt{diff}\left(|\mathtt{from}(\pi ){|}_{(w_1,\dots ,w_k)}+|\pi {|}_{(w_1,\dots ,w_k)}+\sum _{i=1}^m{x}_i\cdot |{\gamma }_i{|}_{(w_1,\dots ,w_k)}\right)=0.$$

$$\mathtt{diff}\left(\sum _{i=1}^m{y}_i\cdot |{\gamma }_i{|}_{(w_1,\dots ,w_k)}\right)=0.$$

$$\mathtt{diff}(|v{u}_0{|}_{(w_1,\dots ,w_k)})=\mathtt{diff}(|v{|}_{(w_1,\dots ,w_k)}+|{\omega }_0{|}_{(w_1,\dots ,w_k)})$$

$$=$$

$$\mathtt{diff}(|v{u}_n{|}_{(w_1,\dots ,w_k)})=\mathtt{diff}(|v{|}_{(w_1,\dots ,w_k)}+|{\omega }_n{|}_{(w_1,\dots ,w_k)})$$

$$=$$

$$(M_i{)}_{i\in \mathbb{N}}\stackrel{\mathtt{def}}{=}(\mathrm{NTr}({\mathtt{walk}}_{{\mathcal{G}}_A^N}(v_i,u_i)){)}_{i\in \mathbb{N}}.$$

$$M_{i_1}(\gamma )<M_{i_2}(\gamma )\text{ for some cycle }\gamma \in T_{i_2}.$$

$$\mathtt{diff}\left(|\mathtt{from}(\pi ){|}_{(w_1,\dots ,w_k)}+|\pi {|}_{(w_1,\dots ,w_k)}+\sum _{i=1}^m{x}_i\cdot |{\gamma }_i{|}_{(w_1,\dots ,w_k)}\right)$$

$$=$$

$$\mathtt{diff}\left(\sum _{i=1}^m(x_i-x_i^{\prime })\cdot |{\gamma }_i{|}_{(w_1,\dots ,w_k)}\right)=0$$

$$|ba{|}_{(ab,ba,a)}=(0,1,1)|{\pi }_1{|}_{(ab,ba,a)}=(1,0,0)|{\gamma }_1{|}_{(ab,ba,a)}=(1,1,1)$$

$$|ba{|}_{(ab,ba,a,b)}=$$

Full text

1 Introduction

Counting occurrences of letters in words is a major topic in formal language theory and much ink has been spent on this topic. Measuring the counting ability of a language class is in this topic. For example, Joshi et al. [1] suggested that the language MIX $\{w\in\{a,b,c\}^{*}\mid|w|_{a}=|w|_{b}=|w|_{c}\}$ should not be in the class of so-called mildly context-sensitive languages since it allows too much freedom in word order, so that relations between MIX and several language classes have been investigated (e.g., indexed languages [2], range concatenation languages [3], tree-adjoining languages [4], multiple context-free languages [5][6], etc.). The Parikh map is another rich example on this topic [7].

In the recent work [8] by Colbourn et. al., the counting feature of MIX is generalised from the counting letter occurrences to the counting of word occurrences. They considered several problems for languages of the form $L(w_{1},\ldots,w_{k})=\{w\in A^{*}\mid|w|_{w_{1}}=\cdots=|w|_{w_{k}}\}$ which we call Word-MIX languages (WMIX for short) in this note. It is interesting that the situation is drastically changed by this generalisation. The decidability of the infiniteness/equivalence turn to be non-trivial: $L(0,1,00,11)$ and $L(0,1,01,10)$ are finite but $L(00,11,000,111)$ is infinite over $A=\{0,1\}$ (example from [8]), and $L(ab,ba,a)$ is infinite but $L(ab,ba,a,b)$ is finite over $A=\{a,b\}$ (these two examples appear again in Section 4.1), for example. In addition, while $L(w_{1},w_{2})$ is always deterministic context-free (DCFL), it can also be regular ($L(ab,ba)\subseteq\{a,b\}^{*}$ is regular, for example) [8]. This kind of generalisation (from letter occurrences to word occurrences) is also considered in the context of the Parikh map [9]. Colbourn et. al. [8] provided a necessary and sufficient condition for $w_{1}$ and $w_{2}$ for these languages to be regular, and gave a polynomial time algorithm for testing that condition. The finiteness of $L(w_{1},w_{2})$ is also considered in [8] and they proved that, for any non-empty words $w_{1},w_{2}\in A^{*}$, $L(w_{1},w_{2})$ is finite if and only if the alphabet $A$ consists of a single letter (${\it i.e.},A=\{a\}$) and $w_{1}\neq w_{2}$. For more general case, allowing more than two parameter words $L(w_{1},\ldots,w_{k})\,(k\geq 2)$, they give a sufficient condition for the infiniteness of $L(w_{1},\ldots,w_{k})$ (Theorem 8 in [8]): if all of $w_{1},\ldots,w_{k}$ have the same length, then $L(w_{1},\ldots,w_{k})$ is infinite.

For the fully general case, the decidability of both regularity and infiniteness for WMIX languages can be derived from some known results on constrained automata (CA for short), since $L(w_{1},\ldots,w_{k})$ is always recognised by a deterministic CA, and its regularity and Parikh image are effectively computable [10]. In this note, we provide a self-contained (without the knowledge about constrained automata) description of a decidable, necessary and sufficient condition for the infiniteness of $L(5w_{1},\ldots,w_{k})$, and give some open problems about the infiniteness. Our proof is based on a combinatorics on walks in the de Bruijn graph. The ($n$-dimensional) de Bruijn graph [11] can track all information of subword occurrences (of length at most $n$), hence it is a very useful tool for counting subword occurrences and related problems. The de Bruijn graph also played a key role in the proof of Theorem 8 in [8].

The rest of this note consists as follows. In Section 2, we give some preliminary definitions and propositions about words, orders, graphs and walks. Section 3 investigates a simple decomposition method which decomposes a walk into a path and a sequence of cycles. This decomposition is useful for the proof of our main theorem. The main result of this note (Theorem 4.1), which states a decidable characterisation of the infiniteness of a WMIX language, is stated and proved in Section 4, the decidability is explained with two examples in Section 4.1. The decidability of the equivalence for two WMIX languages is also explained in Section 5. We end this note with list of open problems in Section 6.

2 Preliminaries

For a set $X$, we denote by $\#\!\left(X\right)$ the cardinality of $X$. We write $\#\!\left(X\right)=\infty$ if $X$ is an infinite set, and write $\#\!\left(X\right)<\infty$ otherwise. We denote by $\mathbb{N}$ the set of natural numbers including [math]. We call a mapping $M:X\rightarrow\mathbb{N}$ multiset over $X$.

2.1 Words and Orders

For an alphabet $A$, we denote the set of all (resp. non-empty) words over $A$ by $A^{*}$ (resp. $A^{+}$). We write $A^{n}$ (resp. $A^{<n}$) the set of all words of length $n$ (resp. less than $n$). For a pair of words $v,w\in A^{*}$, $|w|_{v}$ denotes the number of subword occurrences of $v$ in $w$

[TABLE]

For words $w_{1},\ldots,w_{k}\in A^{*}$, we define

[TABLE]

and call it the Word-MIX language of $k$ parameter words $w_{1},\ldots,w_{k}$ ((k-)WMIX for short). For a word $w\in A^{*}$, we denote the set of prefixes and suffixes of $w$ by

[TABLE]

and denote the length-$n$ ($n\leq|w|$) prefix and suffix of $w$ by $\mathrm{pref}_{n}(w)$ and $\mathrm{suff}_{n}(w)$, respectively.

A quasi order $\leq$ on a set $X$ is called well-quasi-order (wqo for short) if any infinite sequence $(x_{i})_{i\in\mathbb{N}}\,(x_{i}\in X)$ contains an increasing pair $x_{i}\leq x_{j}$ with $i<j$. Let $\leq_{1}$ be a quasi order on a set $X_{1}$ and $\leq_{2}$ be a quasi order on a set $X_{2}$. The product order $\leq_{1,2}$ is a quasi order on $X_{1}\times X_{2}$ defined by

[TABLE]

Lemma 1 (cf. Proposition 6.1.1 in [12])

Let $\leq_{1}$ be a wqo on a set $X_{1}$ and $\leq_{2}$ be a wqo on a set $X_{2}$. The product order $\leq_{1,2}$ is again a wqo on $X_{1}\times X_{2}$.

We list some examples of wqos below:

1. (1)

The identity relation $=$ on any finite set $X$ is a wqo (the pigeonhole principle). 2. (2)

The usual order $\leq$ on $\mathbb{N}$ is a wqo. 3. (3)

The product order $\leq_{m}$ on $\mathbb{N}^{m}$ is a wqo for any $m\geq 1$ (Dickson’s lemma), which is a direct corollary of Lemma 1. 4. (4)

The point-wise order $\leq_{\mathtt{pt}}$ on the multisets $\mathbb{N}^{X}$ ($M\leq_{\mathtt{pt}}M^{\prime}\stackrel{{\scriptstyle\mathtt{def}}}{{\Longleftrightarrow}}M(x)\leq M^{\prime}(x)$ for all $x\in X$) over a finite set $X$ is a wqo (just a paraphrase of Dickson’s lemma).

2.2 Graphs and Walks

Let ${\mathcal{G}}=(V,E)$ be a (directed) graph. We call a sequence of vertices $\omega=(v_{1},\ldots,v_{n})\in V^{n}\,(n\geq 1)$ walk (from $v_{1}$ into $v_{n}$ in ${\mathcal{G}}$) if $(v_{i},v_{i+1})\in E$ for each $i\in\{1,\ldots,n-1\}$, and define the length of $\omega$ as $n-1$ and denote it by $|\omega|$. We denote by $\mathtt{from}(\omega)$ and $\mathtt{into}(\omega)$ the source $\mathtt{from}(\omega)\xlongequal{\!\!\!\mathtt{def}\!\!\!}v_{1}$ and the target $\mathtt{into}(\omega)\xlongequal{\!\!\!\mathtt{def}\!\!\!}v_{n}$ of $\omega$. $\omega$ is called an empty walk if $|\omega|=0$. If two walks $\omega_{1}=(v_{1},\ldots,v_{m}),\omega_{2}=(v^{\prime}_{1},\ldots,v^{\prime}_{n})$ is connectable (i.e., $\mathtt{into}(\omega_{1})=\mathtt{from}(\omega_{2})$), we write $\omega_{1}\odot\omega_{2}$ for the connecting walk $\omega_{1}\odot\omega_{2}\xlongequal{\!\!\!\mathtt{def}\!\!\!}(v_{1},\ldots,v_{m},v^{\prime}_{2},\ldots,v^{\prime}_{n})$. A non-empty walk $\omega$ is called loop (on $\mathtt{from}(\omega)$) if $\mathtt{from}(\omega)=\mathtt{into}(\omega)$. A walk $(v_{1},\ldots,v_{n})$ is called path if $v_{i}\neq v_{j}$ for every $i,j\in\{1,\ldots,n\}$ with $i\neq j$. A loop $(v,v_{1},\ldots,v_{n},v)$ is called cycle if $(v,v_{1},\ldots,v_{n})$ is a path. We use the metavariable $\pi$ for a path, and the metavariable $\gamma$ for a cycle. For a cycle $\gamma$ and $n\geq 1$, we write $\gamma^{n}$ for the loop which is an $n$-times repetition of $\gamma$. We denote by ${\mathcal{W}}({\mathcal{G}}),{\mathcal{P}}({\mathcal{G}}),$ and by ${\mathcal{C}}({\mathcal{G}})$ the set of all walks, paths and cycles in ${\mathcal{G}}$. Note that ${\mathcal{W}}({\mathcal{G}})$ is infinite in general, but ${\mathcal{P}}({\mathcal{G}})$ and ${\mathcal{C}}({\mathcal{G}})$ are both finite if ${\mathcal{G}}$ is finite (i.e., $\#\!\left(V\right)<\infty$).

The $N$-dimensional de Bruijn graph ${\mathcal{G}}^{N}_{A}=(A^{N},E)$ over $A$ is a graph whose vertex set $A^{N}$ is the set of words of length $N$ and the edge set $E$ is defined by

[TABLE]

The case $N=2$ is depicted in Fig. 1.

Let $v$ be a vertex of ${\mathcal{G}}^{N}_{A}$. A word $w=a_{1}\cdots a_{m}\in A^{+}$ induces the walk $(v,v_{1},\ldots,v_{m})$ (where $v_{i}=\mathrm{suff}_{n}(v\,\mathrm{pref}_{i}(w))$) in ${\mathcal{G}}^{N}_{A}$, and we denote it by $\mathtt{walk}_{{\mathcal{G}}^{N}_{A}}(v,w)$. Conversely, a walk $\omega=(v_{1},\ldots,v_{n})$ in ${\mathcal{G}}^{N}_{A}$ induces the word $v_{1}\mathrm{suff}_{1}(v_{2})\cdots\mathrm{suff}_{1}(v_{n})\in A^{*}$, and we denote it by $\mathtt{word}_{{\mathcal{G}}^{N}_{A}}(\omega)$ (see Fig. 1). For words $w,w_{1},\ldots,w_{k}\in A^{*}$ and a walk $\omega=(v_{0},v_{1},\ldots,v_{n})\in{\mathcal{W}}({\mathcal{G}}^{N}_{A})$, we define the following vectors in $\mathbb{N}^{k}$:

[TABLE]

We notice that the range of the summation in the above definition of $|\omega|_{(w_{1},\ldots,w_{k})}$ does not contain [math], hence $|\omega|_{(w_{1},\ldots,w_{k})}=(0,\ldots,0)$ if $\omega$ is an empty walk $\omega=(v_{0})$. The next proposition states a basic property of ${\mathcal{G}}^{N}_{A}$.

Proposition 1

Let $w_{1},\ldots,w_{k}\in A^{*}$ and $N=\max(|w_{1}|,\ldots,|w_{k}|)$. For any pair of words $v,w\in A^{*}$ such that $|v|=N$, we have

[TABLE]

where $\omega=\mathtt{walk}_{{\mathcal{G}}^{N}_{A}}(v,w)$.

Proof

Straightforward induction on the length of $w$.

3 Path-Cycle Decomposition of Walks

In this section, we provide a simple method which decomposes, in left-to-right manner, a walk $\omega$ into a (possibly empty) path $\pi$ and a sequence of cycles $\Gamma$ (Fig. 2). This decomposition is probably folklore but useful for our main proof in the next section. We also introduce in this section the notion of multi-traces and traces of walks, which play crucial role in the characterisation of the infiniteness and equivalence for WMIX languages.

Let ${\mathcal{G}}=(V,E)$ be a graph. For a pair of sequences of cycles $\Gamma_{1}=(\gamma_{1},\ldots,\gamma_{n}),\Gamma_{2}=(\gamma^{\prime}_{1},\ldots,\gamma^{\prime}_{m})$, we write $\Gamma_{1}.\Gamma_{2}$ for the concatenation $(\gamma_{1},\ldots,\gamma_{n},\gamma^{\prime}_{1},\ldots,\gamma^{\prime}_{m})$. When $\Gamma_{1}=(\gamma)$ we simply write $\gamma.\Gamma_{2}$ for $\Gamma_{1}.\Gamma_{2}$ We write $\emptyset$ for the empty sequence of cycles. For $\Gamma=(\gamma_{1},\ldots,\gamma_{n})$, we denote by $\Gamma(i)$ for the $i$-th component $\gamma_{i}$ of $\Gamma$, and denote by $|\Gamma|_{\gamma}$ the number $\#\!\left(\{i\mid\Gamma(i)=\gamma\}\right)$ of occurrences of $\gamma$ in $\Gamma$. For a walk $\omega=(v_{1},\ldots,v_{n})$, we denote by $V(\omega)$ the set of all vertices appeared in $\omega$: $V(\omega)\xlongequal{\!\!\!\mathtt{def}\!\!\!}\{v_{1},\ldots,v_{n}\}$.

We then define a decomposition function $\Phi_{{\mathcal{G}}}:{\mathcal{W}}({\mathcal{G}})\rightarrow{\mathcal{P}}({\mathcal{G}})\times{\mathcal{C}}({\mathcal{G}})^{*}$ inductively as follows:

[TABLE]

Conversely, we define a composition (partial) function $\Psi_{{\mathcal{G}}}:{{\mathcal{W}}({\mathcal{G}})\times{\mathcal{C}}({\mathcal{G}})^{*}\rightharpoonup{\mathcal{W}}({\mathcal{G}})}$ inductively as follows:

[TABLE]

We list some important properties of $\Phi_{{\mathcal{G}}}$ and $\Psi_{{\mathcal{G}}}$.

Proposition 2

Let ${\mathcal{G}}=(V,E)$ be a graph and $\omega\in{\mathcal{W}}({\mathcal{G}})$. Then the followings hold for $(\pi,\Gamma)=\Phi_{{\mathcal{G}}}(\omega)$:

1. (1)

$\pi$* is a path in ${\mathcal{G}}$.* 2. (2)

$\Gamma$* is a sequence of cycles in ${\mathcal{G}}$.* 3. (3)

$|\omega|=|\pi|+\sum_{i=1}^{|\Gamma|}|\Gamma(i)|$** 4. (4)

$\omega=\Psi_{{\mathcal{G}}}(\pi,\Gamma)$, i.e., $\Psi_{{\mathcal{G}}}\circ\Phi_{{\mathcal{G}}}$ is the identity function on ${\mathcal{W}}({\mathcal{G}})$.

Proof

(1)–(2) are obvious by the definition. (3)–(4) can be shown by an easy induction on the length of $\omega$.

Proposition 3

Let $w_{1},\ldots,w_{k}\in A^{*}$ and $N=\max(|w_{1}|,\ldots,|w_{k}|)$. For any $\omega$ in ${\mathcal{W}}({\mathcal{G}}^{N}_{A})$,

[TABLE]

holds where $(\pi,\Gamma)=\Psi_{{\mathcal{G}}^{N}_{A}}(\pi,\Gamma)$.

Proof

Straightforward induction on the length of $\omega$.

Example 1

Consider the complete graph ${\mathcal{K}}_{4}=(V_{4}=\{1,2,3,4\},E_{4}=V_{4}\times V_{4})$ of order $4$ and a walk $\omega=(1,2,3,2,3,4,3,4,2,4)$. The result of decomposition is $\Phi_{{\mathcal{K}}_{4}}(\omega)=(\omega=(1,2,4),\Gamma=((2,3,2),(3,4,3),(2,3,4,2)))$. All intermediate computation step of $\Phi_{{\cal K}_{4}}(\omega)$ and $\Psi_{{\mathcal{K}}_{4}}(\Phi_{{\mathcal{K}}_{4}}(\omega))$ are drawn in Fig. 2 (in the figure we denote by $\pi\&\Gamma$ a pair $(\pi,\Gamma)$ for visibility). It is clear that the all conditions in Proposition 2 are satisfied ($|\omega|=9=2+2+2+3=|\omega|+\sum_{i=1}^{3}|\Gamma(i)|$, in particular).

3.1 Multi-Traces and Traces

For a walk $\omega$ in a graph ${\mathcal{G}}$, we define the multi-trace $\operatorname{\mathbb{N}Tr}(\omega):{\mathcal{P}}({\mathcal{G}})\cup{\mathcal{C}}({\mathcal{G}})\rightarrow\mathbb{N}$ of a walk $\omega$ as the following multiset over paths and cycles:

[TABLE]

We define the trace $\operatorname{Tr}(\omega)$ of a walk $\omega$ in ${\mathcal{G}}$ as the following set of paths and cycles:

[TABLE]

Intuitively, the multi-trace of $\omega$ in ${\mathcal{G}}$ is obtained by forgetting the ordering of the decomposition result $(\omega,\Gamma)=\Phi_{{\mathcal{G}}}(\omega)$ of $\omega$, and the trace of $\omega$ is obtained by forgetting the multiplicity from the original multi-trace (see Fig. 3 for the relation).

Since Item 3 in Proposition 2 and Proposition 3 do not depend on the order of a sequence $\Gamma$, one can easily observe that the following proposition holds by the definition of $\operatorname{\mathbb{N}Tr}(\omega)$.

Proposition 4

Let $w_{1},\ldots,w_{k}\in A^{*}$ and $N=\max(|w_{1}|,\ldots,|w_{k}|)$. For any $\omega$ in ${\mathcal{W}}({\mathcal{G}}^{N}_{A})$, we have

[TABLE]

and

[TABLE]

For a set $T\subseteq{\mathcal{P}}({\mathcal{G}})\cup{\mathcal{C}}({\mathcal{G}})$, the following lemma states that we can effectively test whether $T$ is a trace or not (see Fig. 4 for the intuition).

Lemma 2

Let $T\subseteq{\mathcal{P}}({\mathcal{G}})\cup{\mathcal{C}}({\mathcal{G}})$ be a set of paths and cycles in a graph ${\mathcal{G}}=(V,E)$. The followings are equivalent:

1. (1)

$T$* is a trace of some walk in ${\mathcal{G}}$.* 2. (2)

$T$* can be written as $T=\{\pi\}\cup\{\gamma_{1},\ldots,\gamma_{m}\}$ such that (i) $V(\gamma_{m})\cap V(\pi)=\{\mathtt{from}(\gamma_{m})\}$ and (ii) for every $i\in\{1,\ldots,m-1\}$, $V(\pi^{\prime})\cap V(\gamma_{i})=\{\mathtt{from}(\gamma_{i})\}$ where $\pi^{\prime}\odot(\mathtt{from}(\gamma_{i}))\odot\omega=\Psi_{{\mathcal{G}}}(\pi,(\gamma_{i+1},\ldots,\gamma_{m})).$*

4 Characterisation of the Finiteness

For a vector $\bm{v}=(c_{1},\ldots,c_{k})\in\mathbb{N}^{k}$, we define

[TABLE]

Observe that $w\in L(w_{1},\ldots,w_{k})$ if and only if $\mathtt{diff}(|w|_{(w_{1},\ldots,w_{k})})=0$.

We now ready to state and prove the main result.

Theorem 4.1

Let $w_{1},\ldots,w_{k}\in A^{*}$ and $N=\max(|w_{1}|,\ldots,|w_{k}|)$. Then the followings are equivalent:

1. (1)

$L(w_{1},\ldots,w_{k})$* is infinite.* 2. (2)

*There exists a trace $T\subseteq{\mathcal{P}}({\mathcal{G}}^{N}_{A})\cup{\mathcal{C}}({\mathcal{G}}^{N}_{A})$ that satisfies the following two conditions. By Lemma 2, we can assume without loss of generality that $T$ is of the form $T=\{\pi\}\cup\{\gamma_{1},\ldots,\gamma_{m}\}$ that satisfies Condition (2) in Lemma 2. *

(balance condition)* there exist positive coefficients $x_{1},\ldots,x_{m}\in\mathbb{N}$, $x_{i}\geq 1$ for each $i\in\{1,\ldots,m\}$, such that*

[TABLE]

(pumping condition)* there exist coefficients $y_{1},\ldots,y_{m}\in\mathbb{N}$, not all zero, such that*

[TABLE]

Proof

The direction $(2)\Rightarrow(1)$ is relatively easy. Intuitively, the balance condition ensures the existence of a word $vu_{0}\in A^{*}$ such that $|v|=N$, $vu_{0}\in L(w_{1},\ldots,w_{k})$, and the pumping condition further ensures $vu_{0}$ is “pumpable” in some sense, which implies the infiniteness of $L(w_{1},\ldots,w_{k})$. We prove this intuition. Assume a trace $T=\{\pi\}\cup\{\gamma_{1},\ldots,\gamma_{m}\}$ satisfies the balance and pumping conditions and let $v=\mathtt{from}(\pi)$. Since $T$ is a trace, $\omega=\Psi_{{\mathcal{G}}^{N}_{A}}(\pi,(\gamma_{1},\ldots,\gamma_{m}))$ is defined by Lemma 2. Let $x_{1},\ldots,x_{m}$ be positive coefficients that satisfy the balance condition, and $y_{1},\ldots,y_{m}$ be coefficients, not all zero, that satisfy the pumping condition. For each $n\geq 0$, we define $\omega_{n}\xlongequal{\!\!\!\mathtt{def}\!\!\!}\Psi_{{\mathcal{G}}^{N}_{A}}(\pi,(\gamma_{1}^{x_{1}+n\cdot y_{1}},\ldots,\gamma_{m}^{x_{m}+n\cdot y_{m}}))$ and $vu_{n}\xlongequal{\!\!\!\mathtt{def}\!\!\!}\mathtt{word}_{{\mathcal{G}}^{N}_{A}}(\omega_{n})$. Note that, since every cycle has at least length one and by Item 3 in Proposition 2, $|\omega_{n}|<|\omega_{n^{\prime}}|$ and hence $|vu_{n}|<|vu_{n^{\prime}}|$ holds for every $n<n^{\prime}$. Combining Proposition 1, Proposition 3 and the balance condition, we obtain

[TABLE]

Moreover, by Proposition 1, Proposition 3 and the pumping condition we have

[TABLE]

for any $n\geq 1$. This means that every distinct word $vu_{n}$ is in $L(w_{1},\ldots,w_{k})$, hence $L(w_{1},\ldots,w_{k})$ is infinite.

We then prove the opposite direction $(1)\Rightarrow(2)$. Assume $\#\!\left(L(w_{1},\ldots,w_{k})\right)=\infty$. Since $L(w_{1},\ldots,w_{k})$ is infinite and hence it contains an arbitrary long word, we can take an infinite sequence $(v_{i}u_{i})_{i\in\mathbb{N}}$ of words from $L(w_{1},\ldots,w_{k})$ that satisfies $|v_{i}|=N\,(i\in\mathbb{N})$ ,$v_{i}u_{i}\in L(w_{1},\ldots,w_{k})\,(i\in\mathbb{N})$ and $|v_{i}u_{i}|<|v_{j}u_{j}|\,(i<j)$. Now consider an infinite sequence of multi-traces

[TABLE]

Since the point-wise order on the multisets over any finite set is a wqo (thanks to Dickson’s lemma) and ${\mathcal{P}}({\mathcal{G}}^{N}_{A})\cup{\mathcal{C}}({\mathcal{G}}^{N}_{A})$ is finite, $(M_{i})_{i\in\mathbb{N}}$ contains an increasing pair $M_{i_{1}}\leq_{\mathtt{pt}}M_{i_{2}}$ with $i_{1}<i_{2}$. Define $T_{i}\xlongequal{\!\!\!\mathtt{def}\!\!\!}\operatorname{Tr}(\mathtt{walk}_{{\mathcal{G}}^{N}_{A}}(v_{i},u_{i})\,(i\in\{i_{1},i_{2}\})$. We notice that, since $M_{i_{1}}\leq_{\mathtt{pt}}M_{i_{2}}$ and every multi-trace contains exactly one path as its element, $T_{i_{1}}\subseteq T_{i_{2}}$ and $(T_{i_{1}}\cap T_{i_{2}})\cap{\mathcal{P}}({\mathcal{G}}^{N}_{A})=\{\pi\}$ for some path $\pi$. Since $|v_{i}u_{i}|<|v_{j}u_{j}|$ holds for every $i<j$ by definition, we can deduce $M_{i_{1}}\neq M_{i_{2}}$ by Proposition 4 and thus we have

[TABLE]

Let $m$ be the number of cycles in $T_{i_{2}}$ and write $T_{i_{2}}=\{\pi\}\cup\{\gamma_{1},\ldots,\gamma_{m}\}$. Define $x_{i}\xlongequal{\!\!\!\mathtt{def}\!\!\!}M_{i_{2}}(\gamma_{i})$ and $x^{\prime}_{i}\xlongequal{\!\!\!\mathtt{def}\!\!\!}M_{i_{1}}(\gamma_{i})$ for each $i\in\{1,\ldots,m\}$. Clearly $x_{i}\geq 1$ and $x_{i}\geq x^{\prime}_{i}$ hold for every $i\in\{1,\ldots,m\}$ by the definition. By Proposition 1 and Proposition 4, we have

[TABLE]

that is, the balance condition is satisfied. In addition, from the above equation we obtain

[TABLE]

because for any $\bm{v}$ such that $\mathtt{diff}(\bm{v})=0$, $\mathtt{diff}(\bm{v}+\bm{v^{\prime}})=0$ if and only if $\mathtt{diff}(\bm{v}^{\prime})=0$. By Condition ($\bigstar$ ‣ 4), coefficients $(x_{1}-x^{\prime}_{1})$, …, $(x_{m}-x^{\prime}_{m})$ are not all zero, the pumping condition is satisfied. This ends the proof. ∎

4.1 Decidability and Examples

The decision problem whether both balance and pumping condition are satisfied for a given trace $T\subseteq{\mathcal{P}}({\mathcal{G}}^{N}_{A})\cup{\mathcal{C}}({\mathcal{G}}^{N}_{A})$ in ${\mathcal{G}}^{N}_{A}$ can be reduced into $\mathrm{\Sigma}_{1}$-formula (existential formula) of Presburger arithmetic (see the examples in below). The set of traces in ${\mathcal{G}}^{N}_{A}$ is clearly finite and effectively enumerable (due to Lemma 2), in addition. Thus we obtain the following corollary.

Corollary 1

For all words $w_{1},\cdots,w_{k}\in A^{*}$, it is decidable whether $L(w_{1},\ldots,w_{k})$ is infinite or not.

Proof

Enumerate possible traces in ${\mathcal{G}}^{N}_{A}$ and check whether there is a trace that satisfies both balance and pumping condition.

Example 2

Consider the language $L(ab,ba,a)$ over $A=\{a,b\}$, $\max(|ab|,|ba|,|a|)=2$ and the 2-dimensional de Bruijn graph ${\mathcal{G}}^{2}_{A}$ shown in Fig. 1. We claim that a trace $T_{1}=\{\pi_{1}=(ba,ab)\}\cup\{\gamma_{1}=(ba,ab,ba)\}$ satisfies both balance and pumping condition. One can easily observe that

[TABLE]

and hence the coefficient $x_{1}=1$ simultaneously satisfies the two condition stated in (2) of Theorem 4.1. For each $n\geq 1$, by Proposition 1 the word $ba(ba)^{n}b=\mathtt{word}_{{\mathcal{G}}^{N}_{A}}(\gamma_{1}^{n}\odot\pi_{1})$ is in $\#\!\left(L(ab,ba,a)\right)=\infty$. Hence $ba(ba)^{+}b\subseteq L(ab,ba,a)$ and $\#\!\left(L(ab,ba,a)\right)=\infty$.

Example 3

Next consider another language $L(ab,ba,a,b)$ over $A=\{a,b\}$, $\max(|ab|,|ba|,|a|,|b|)=2$ and again the 2-dimensional de Bruijn graph ${\mathcal{G}}^{2}_{A}$ shown in Fig. 1. In contrast with Example 2, the trace $T_{1}=\{\pi_{1}=(ba,ab)\}\cup\{\gamma_{1}=(ba,ab,ba)\}$ does not satisfy the balance condition any more (even it still satisfies the pumping condition). We have

[TABLE]

We can formally prove that there is no positive coefficient $x_{1}\in\mathbb{N}\,(x_{1}>0)$ that satisfies the balance condition, since the existence of such coefficients can be expressed in the following $\mathrm{\Sigma}_{1}$-formula of Presburger arithmetic

[TABLE]

where $\theta_{T_{1}}^{w}$ is a subexpression defined by

[TABLE]

$\phi_{T_{1}}$ can be algorithmically verified to be not valid since the validity of a first-order formula of Presburger arithmetic is decidable (cf. Section 6.2 of [13]). We can algorithmically verify, by using the same reduction into $\mathrm{\Sigma}_{1}$-formulae of Presburger arithmetic, that no trace in ${\mathcal{G}}^{2}_{A}$ satisfies both balance and pumping condition. Thus $\#\!\left(L(ab,ba,a,b)\right)<\infty$ by Theorem 4.1.

5 Characterisation of the Equivalence

In the previous section, multi-traces and traces play crucial role for the characterisation of the finiteness. Multi-traces are also important for the characterisation of the equivalence of WMIX languages which is given here. Before stating the main statement, we lift the notion of traces of walks to one of languages. For a language $L\subseteq A^{*}$, we define $\operatorname{\mathbb{N}Tr}(L)$ the multi-trace of a language $L$ (of order $N$) as

[TABLE]

The following theorem states that any WMIX language is completely determined by its multi-trace (excluding shorter part $A^{<N}$).

Theorem 5.1

Let $w_{1},\ldots,w_{k},w^{\prime}_{1},\ldots,w^{\prime}_{k^{\prime}}\in A^{*}$ and $N=\max(|w_{1}|,\ldots,|w_{k}|,|w^{\prime}_{1}|,\ldots,|w^{\prime}_{k^{\prime}}|)$. Then $L(w_{1},\ldots,w_{k})=L(w^{\prime}_{1},\ldots,w^{\prime}_{k^{\prime}})$ if and only if

[TABLE]

and

[TABLE]

Proof

The “only-if”-part is trivial. We prove the “if”-part by contraposition. Assume $L(w_{1},\ldots,w_{k})\neq L(w^{\prime}_{1},\ldots,w^{\prime}_{k^{\prime}})$. Then we can assume that there is some word $w$ such that $w\in L(w_{1},\ldots,w_{k})$ but $w\notin L(w^{\prime}_{1},\ldots,w^{\prime}_{k^{\prime}})$ without loss of generality. If $|w|<N$ it is clear that

[TABLE]

and the “if”-part holds. Thus we consider the case $|w|\geq N$. Let $w=vu$ such that $|v|=N$ and $M=\operatorname{\mathbb{N}Tr}(\mathtt{walk}_{{\mathcal{G}}^{N}_{A}}(v,u))$. We now prove that $L(w^{\prime}_{1},\ldots,w^{\prime}_{k^{\prime}})$ does not contain any word $w^{\prime}=v^{\prime}u^{\prime}\,(|v^{\prime}|=N)$ that has the same multi-trace with $w$ (i.e., $\operatorname{\mathbb{N}Tr}(\mathtt{walk}_{{\mathcal{G}}^{N}_{A}}(v^{\prime},u^{\prime}))=M=\operatorname{\mathbb{N}Tr}(\mathtt{walk}_{{\mathcal{G}}^{N}_{A}}(v,u))$; $v^{\prime}=v$ holds in this case). By Proposition 1 and Proposition 4, any subword occurrences in a word is completely determined by its multi-trace. Thus if there is a word $w^{\prime}=vu^{\prime}$ in $L(w^{\prime}_{1},\ldots,w^{\prime}_{k^{\prime}})$ such that $\operatorname{\mathbb{N}Tr}(\mathtt{walk}_{{\mathcal{G}}^{N}_{A}}(v,u^{\prime}))=M$, then

[TABLE]

from which we obtain $w\in L(w^{\prime}_{1},\ldots,w^{\prime}_{k^{\prime}})$; this contradicts with the assumption. Therefore we can conclude that

[TABLE]

5.1 Decidability

By using Theorem 5.1, we can obtain an algorithm for deciding the equivalence of two WMIX languages. This algorithm also uses the decidability of Presburger arithmetic, as like the previous algorithm for the infiniteness, but in contrast to the case of inifiniteness, it is reduced into $\mathrm{\Pi}_{1}$-formula of Presburger arithmetic.

Corollary 2

For any word $w_{1},\ldots,w_{k},w^{\prime}_{1},\ldots,w^{\prime}_{k^{\prime}}\in A^{*}$, it is decidable whether $L(w_{1},\ldots,w_{k})=L(w^{\prime}_{1},\ldots,w^{\prime}_{k^{\prime}})$ or not.

Proof

Let $N=\max(|w_{1}|,\ldots,|w_{k}|,|w^{\prime}_{1}|,\ldots,|w^{\prime}_{k^{\prime}}|)$. We can effectively check $L(w_{1},\ldots,w_{k})\cap A^{<N}=L(w^{\prime}_{1},\ldots,w^{\prime}_{k^{\prime}})\cap A^{<N}$ holds or not, since $A^{<N}$ is finite. If $L(w_{1},\ldots,w_{k})\cap A^{<N}\neq L(w^{\prime}_{1},\ldots,w^{\prime}_{k^{\prime}})\cap A^{<N}$ then two languages are not equivalent. Otherwise, enumerate all possible traces in ${\mathcal{G}}^{N}_{A}$, then for each trace $T$, and check whether every multi-trace $M$ with $T=\{\omega\in{\mathcal{P}}({\mathcal{G}}^{N}_{A})\cup{\mathcal{C}}({\mathcal{G}}^{N}_{A})\mid M(\omega)\neq 0\}$ satisfies

[TABLE]

or not. If there is some multi-trace that does not satisfy Condition ($\blacklozenge$ ‣ 5.1) then $L(w_{1},\ldots,w_{k})\neq L(w^{\prime}_{1},\ldots,w^{\prime}_{k^{\prime}})$, otherwise $L(w_{1},\ldots,w_{k})=L(w^{\prime}_{1},\ldots,w^{\prime}_{k^{\prime}})$ holds. Since every multi-trace can be represented by a corresponding trace and its multiplicity (positive coefficients), for a trace $T=\{\pi\}\cup\{\gamma_{1},\ldots,\gamma_{m}\}$, the statement “every multi-trace $M$ with $T=\{\omega\in{\mathcal{P}}({\mathcal{G}}^{N}_{A})\cup{\mathcal{C}}({\mathcal{G}}^{N}_{A})\mid M(\omega)\neq 0\}$ satisfies Condition ($\blacklozenge$ ‣ 5.1)” can be represented by the following $\mathrm{\Pi}_{1}$-formula of Presburger arithmetic $\psi_{T}$:

[TABLE]

where $\theta_{T}^{w}$ is a subexpression defined by

[TABLE]

6 Open Problem

We would like to introduce the following open problem which asks the existence of a non-trivial finite WMIX language.

Problem 1 ([14])

Are there $w_{1},\ldots w_{k}\in A^{*}$ such that $L(w_{1},\ldots,w_{k})$ is finite but $|w|_{w_{1}}=\cdots=|w|_{w_{k}}\geq 1$ for some $w\in L(w_{1},\ldots,w_{k})$?

Note that all examples of finite WMIX languages in this note are not of this type. The complexity issue is also interesting.

Problem 2

What is the complexity of the infiniteness problem (resp. the equivalence problem) for WMIX languages?

Acknowledgment. The author would like to thank Thomas Finn Lidbetter (University of Waterloo) for telling me this topic. The author also thank to an anonymous reviewer for letting me know some known results on unambiguous CA [10] and pointing out that the decidability results presented in this note are also from those results.