Models for the Displacement Calculus
Oriol Valent\'in
TL;DR
This paper introduces models for the displacement calculus, extending the Lambek calculus to include intercalation, and demonstrates that its proof theory and models are natural extensions of those for $ extbf{L1}."
Contribution
It develops the model theory for the displacement calculus and proves completeness results, extending the proof theory of the Lambek calculus.
Findings
Models for $ extbf{D}$ are natural extensions of those for $ extbf{L1}$.
Completeness results for $ extbf{D}$ are established.
Proofs and model classes for $ extbf{D}$ mirror those of $ extbf{L1}$.
Abstract
The displacement calculus is a conservative extension of the Lambek calculus (with empty antecedents allowed in sequents). can be said to be the logic of concatenation, while can be said to be the logic of concatenation and intercalation. In many senses, it can be claimed that mimics in that the proof theory, generative capacity and complexity of the former calculus are natural extensions of the latter calculus. In this paper, we strengthen this claim. We present the appropriate classes of models for and prove some completeness results; strikingly, we see that these results and proofs are natural extensions of the corresponding ones for .
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Taxonomy
TopicsLogic, programming, and type systems · Logic, Reasoning, and Knowledge · Advanced Algebra and Logic
11institutetext: Universitat Politècnica de Catalunya
Authors’ Instructions
Models for the Displacement Calculus
Oriol Valentín Research partially supported by SGR2014-890 (MACDA) of the Generalitat de Catalunya, and MINECO project APCOM (TIN2014-57226-P).
Abstract
The displacement calculus is a conservative extension of the Lambek calculus (with empty antecedents allowed in sequents). can be said to be the logic of concatenation, while can be said to be the logic of concatenation and intercalation. In many senses, it can be claimed that mimics in that the proof theory, generative capacity and complexity of the former calculus are natural extensions of the latter calculus. In this paper, we strengthen this claim. We present the appropriate classes of models for and prove some completeness results; strikingly, we see that these results and proofs are natural extensions of the corresponding ones for .
1 Introduction
The displacement calculus is a quite well-studied extension of the Lambek calculus (with empty antecedents allowed in sequents). In many papers (see [9], [12] and [11]), has proved to provide elegant accounts of a variety of linguistic phenomena of English, and of Dutch, namely a processing interpretation of the so-called Dutch cross-serial dependencies.
The hypersequent format 111Not to be confused with the hypersequents of Avron ([1]). of displacement calculus is a pure sequent calculus free of structural rules which subsumes the sequent calculus for . The Cut elimination algorithm for provided in [12] mimics the one of Lambek’s [5] syntactic calculus (with some minor differences concerning the possibility of empty antecedents). Like , enjoys some nice properties such as the subformula property, decidablity, the finite reading property and the focalisation property ([7]).
Like , is known to be NP-complete [6]. Concerning (weak) generative capacity, recognises the class of well-nested multiple context-free languages ([13]). In this respect, the result on generative capacity generalises the result that states that recognises the class of context-free languages. One point of divergence in terms of generative capacity is that recognises the class of the permutation closures of context-free languages ([10]). Finally, it is important to note that a Pentus-like upper bound theorem for is not known.
In this paper we present natural classes of models for . Several strong completeness results are proved, in particular strong completeness w.r.t. the class of residuated displacement algebras (a natural extension of residuated monoids). Powerset frames for are of interest from the linguistic point of view because of their relation to language models. Powerset residuated displacement algebras over displacement algebras are given, which generalise the powerset residuated monoids over monoids, as well as over free monoids. Strong completeness results for the so-called implicative fragment of , which is very relevant linguistically, is proved in the spirit of Buszkowski ([2]), but the construction is more subtle.
The structure of the paper is as follows. In Section 2 we present the basic proof-theoretic tools (useful for the construction of canonical models) which we shall employ for the study of from a semantic point of view. In Section 3 we provide the proof of two strong completeness of what we call the implicative fragment w.r.t. powerset DAs over standard DAs (with a countably infinite set of generators) and L-models respectively.
2 The Categorical calculus and the Hypersequent Calculus
is model-theoretically motivated, and the key to its conception is the use of many-sorted universal algebra ([3]), namely -sorted universal algebra. Here, we assume a version of many-sorted algebra such that the sort domains of an -sorted algebra are non-empty. With this condition we avoid some pathologies which arise in a naïve version of many-sorted universal algebra (cf. [3], and [4]). Some definitions are needed. Let \mbox{\mathcal{M}}=(\mbox{|\mbox{}|},+, 0,\mbox{1}) be a free monoid where is a distinguished element of the set of generators of . We call such an algebra a separated monoid. Given an element a\in\mbox{|\mbox{}|}, we can associate to it a number, called its sort as follows: \enumsentence \begin{array}[t]{lll}s(\mbox{1})&=&1\\ s(a)&=&0\mbox{ if a\in Xa\neq\mbox{}}\\ s(w_{1}+w_{2})&=&s(w_{1})+s(w_{2})\end{array}
This induction is well-defined since is free and is a (distinguished) generator; the sort function in a separated monoid simply counts the number of separators an element contains.
Definition 1
(Sort Domains)
Where \mbox{\mathcal{M}}=(\mbox{|\mbox{}|},+,0,\mbox{1}) is a separated monoid, the sort domains \mbox{|\mbox{}|}_{i} of sort are defined as follows:
[TABLE]
It is readily seen that for every , \mbox{|\mbox{}|}_{i}\cap\mbox{|\mbox{}|}_{j}=\emptyset\mbox{ iff }i\neq j.
Definition 2
(Standard Displacement Algebra)
The standard displacement algebra (or standard DA) defined by a separated monoid (\mbox{|\mbox{}|},+,0,\mbox{1}) is the -sorted algebra with the -sorted signature with sort functionality :
[TABLE]
where:
[TABLE]
The sorted types of , which we will interpret residuating w.r.t the sorted operations in Definition 2, are defined by mutual recursion in Figure 1. We let \mbox{\mathbf{Tp}}=\bigcup_{i\geq 0}\mbox{\mathbf{Tp}}_{i}. A subset of |\mbox{\mathcal{M}}| is called a same-sort subset iff there exists an i\in\mbox{\omega} such that for every , . types are to be interpreted as same-sort subsets of |\mbox{\mathcal{M}}|.
I.e. every inhabitant of \mbox{\llbracket}A\mbox{\rrbracket} has the same sort. The intuitive semantic interpretation of the connectives is shown in Figure 2; this interpretation is called the standard interpretation. Observe that for any type A\in\mbox{\mathbf{Tp}}, the interpretation of , i.e. \mbox{\llbracket}A\mbox{\rrbracket}, is contained in , where the sort map for the set , is such that \enumsentence \begin{array}[t]{lllllllll}s(p)&=&i&\mbox{ for p}\in\mbox{\mathbf{Pr}}_{i}&\\ s(I)&=&0\\ s(J)&=&1\\ s(A\mbox{\bullet}B)&=&s(A)+s(B)\\ s(A\mbox{\backslash}B)&=&s(B)-s(A)\\ s(B/A)&=&s(B)-s(A)\\ s(A\odot_{k}B)&=&s(A)+s(B)-1\\ s(A\mbox{\downarrow}_{k}B)&=&s(B)-s(A)+1\\ s(B\mbox{{\uparrow}}_{k}A)&=&s(B)-s(A)+1\end{array}
2.1 and its Categorical Presentation
In [14] is presented as a categorical calculus:
\enumsentence
\begin{array}[t]{lll}A\rightarrow A\mbox{ Axiom}\\ A\mbox{\bullet}B\rightarrow C\mbox{ iff }A\rightarrow C/B\mbox{ iff }B\rightarrow A\mbox{\backslash}C\quad\quad\quad\mbox{ Res_{cont}}\\ A\odot_{i}B\rightarrow C\mbox{ iff }A\rightarrow C\mbox{{\uparrow}}_{i}B\mbox{ iff }B\rightarrow A\mbox{\downarrow}_{i}C\quad\ \mbox{ Res_{disc}}\\ \\ A\bullet I\mbox{;\leftrightarrow;}A\mbox{;\leftrightarrow;}I\bullet A\quad A\odot_{i}J\;\mbox{;\leftrightarrow;}\;A\;\mbox{;\leftrightarrow;}\;J\odot_{1}A\\ (A\bullet B)\bullet C\;\mbox{;\leftrightarrow;}\;A\bullet(B\bullet C)\quad\quad\quad\quad\quad\mbox{ Continuous associativity}\\ A\odot_{i}(B\odot_{j}C)\mbox{;\leftrightarrow;}(A\odot_{i}B)\odot_{i+j-1}C\quad\mbox{ Discontinuous associativity}\\ (A\odot_{i}B)\odot_{j}C\mbox{;\leftrightarrow;}A\odot_{i}(B\odot_{j-i+1}C)\mbox{, if }i\leq j\leq 1+s(B)-1\\ \\ (A\odot_{i}B)\odot_{j}C\mbox{;\leftrightarrow;}(A\odot_{j-s(B)+1}C)\odot_{i}B,\mbox{ if }j>i+s(B)-1\mbox{ Mixed permutation}\\ (A\odot_{i}C)\odot_{j}B\mbox{;\leftrightarrow;}(A\odot_{j}B)\odot_{i+s(B)-1}C,\mbox{ if }j<i\\ \\ (A\bullet B)\odot_{i}C\mbox{;\leftrightarrow;}(A\odot_{i}C)\bullet B\mbox{, if }1\leq i\leq S(A)\mbox{ Mixed associativity}\\ (A\bullet B)\odot_{i}C\mbox{;\leftrightarrow;}A\bullet(B\odot_{i-s(C)}C)\mbox{, if }s(A)+1\leq i\leq s(A)+s(B)\\ \mbox{From }A\rightarrow B\mbox{ and }B\rightarrow C\mbox{ we have }A\rightarrow C\quad\mbox{ Transitivity}\end{array}
In Figure 3 we find the axiomatisation of the class of DAs . Just as in the case of , the natural class of algebras is the class of residuated monoids , in the case of , the natural class of algebras is the class of residuated displacement algebras (residuated DAs) .
One can restrict the definition of the sorted types. Let be a subset of the connectives considered in the definition of types in Figure 1. We define \mbox{\mathbf{Tp}}[\mathbf{C}] as the least set of sorted types generated by and the set of connectives . If the context is clear, we will write instead of \mbox{\mathbf{Tp}}[\mathbf{C}].
Let us define the formal definition of a model. A model \mathcal{M}=(\mbox{\mathcal{A}},v) comprises a (residuated) -algebra and a -sorted mapping v:\mbox{\mathbf{Pr}}\rightarrow\mbox{\mathbf{Tp}}[\mathbf{C}] called a valuation. The mapping is the unique function which extends and which is such that (if * is a binary connective of ) and (if * is a unary connective of ). Finally, a [math]-ary connective is mapped into the corresponding unit of |\mbox{\mathcal{A}}|. Needless to say, the mappings and preserve the sorting regime.
Let us see that (with all the connectives) is strongly complete w.r.t. . Soundness is trivial because we are considering the categorical calculus . For completeness, we can define the well-known Lindenbaum-Tarski construction to see that is strongly complete w.r.t. . The canonical model is (\mbox{\mathcal{L}},v) where is (\mbox{\mathbf{Tp}}/\theta,\mbox{\circ},(\mbox{\circ_{i}})_{i>0},\mbox{\mbox{}!\mbox{}},\mbox{/!/},(\mbox{\downdownarrows}_{i})_{i>0}(\mbox{\upuparrows}_{i})_{i>0},\mathbb{I},\mathbb{J};\leq) where the interpretation of the new symbols is as expected. Let be the equivalence relation on defined as follows: iff R\vdash_{\mbox{\mathbf{cD}}}A\rightarrow B and R\vdash_{\mbox{\mathbf{cD}}}B\rightarrow A, where is a set of non-logical axioms. Using the usual tonicity properties for the connectives of , one proves that is a congruence. Where is a type, is an element of \mbox{\mathbf{Tp}}/\theta_{R}, i.e. modulo . We define \mbox{\overline{A}}\leq\mbox{\overline{B}} iff R\vdash_{\mbox{\mathbf{cD}}}A\rightarrow B. We define the valuation as v(p)=\mbox{\overline{p}} ( is a primitive type). We have that for every type , \widehat{v}(A)=\mbox{\overline{A}}. Finally, one has that (\mbox{\mathcal{L}},v)\models A\rightarrow B iff R\vdash_{\mbox{\mathbf{cD}}}A\rightarrow B. From this, we infer the following theorem:
Theorem 2.1
The calculus is strongly complete w.r.t. .
Since is a variety222The term equational class is sometimes used in the literature. (see Figure 3), it is closed by subalgebras, direct products and homomorphic images, which give additional DAs.
We have other interesting examples of DAs, for instance the *powerset DA over *\mbox{\mathcal{A}}=(\mbox{|\mbox{}|},+,\{\times_{i}\}_{i>0},0,1), which we denote . We have: \enumsentence \mbox{\mathcal{P}(A)}=(\mbox{|\mbox{\mathcal{\mbox{\mathcal{P}(A)}}}|},\cdot,\{\mbox{\circ_{i}}\}_{i>0},\mathbb{I},\mathbb{J})
The notation of the carrier set of presupposes that its members are same-sort subsets; notice that vacuously satisfies the same-sort condition. Where , and denote same-sort subsets of |\mbox{\mathcal{A}}|, the operations , , and are defined as follows: \enumsentence \begin{array}[t]{lll}\mathbb{I}&=&\{0\}\\ \mathbb{J}&=&\{1\}\\ A\cdot B&=&\{a+b:a\in A\mbox{ and }b\in B\}\\ A\mbox{\circ_{i}}B&=&\{a\times_{i}b:a\in A\mbox{ and }b\in B\}\mbox{ }\end{array}
It is readily seen that for every , is in fact a DA. Notice that every sort domain \mbox{|\mbox{\mathcal{\mbox{\mathcal{P}(A)}}}|}_{i} is a collection of same-sort subsets, that the sort domains of are non-empty, but no longer satisfy that \mbox{|\mbox{\mathcal{\mbox{\mathcal{P}(A)}}}|}_{i}\cap\mbox{|\mbox{\mathcal{\mbox{\mathcal{P}(A)}}}|}_{j}=\emptyset iff , since the empty set \emptyset\in\mbox{|\mbox{\mathcal{\mbox{\mathcal{P}(A)}}}|}_{i} for every . A residuated powerset displacement algebra over a displacement algebra is the following: \enumsentence \mbox{\mathcal{P}(A)}=(\mbox{|\mbox{\mathcal{\mbox{\mathcal{P}(A)}}}|},\cdot,\mbox{\mbox{}!\mbox{}},\mbox{/!/},\{\mbox{\circ_{i}}\}_{i>0},,\{\mbox{\upuparrows}_{i}\}_{i>0},\{\mbox{\downdownarrows}_{i}\}_{i>0},\mathbb{I},\mathbb{J};\subseteq)
where \backslash$$\backslash, , \mbox{\upuparrows}_{i} and \mbox{\downdownarrows}_{i} are defined as follows: \enumsentence \begin{array}[t]{lll}A\mbox{\mbox{}!\mbox{}}B&=&\{d:\mbox{ for every }a\in A,\mbox{ }a+d\in B\}\\ B\mbox{/!/}A&=&\{d:\mbox{ for every }a\in A,\mbox{ }d+a\in B\}\\ B\mbox{\upuparrows}_{i}A&=&\{d:\mbox{ for every }a\in A,\mbox{ }d\times_{i}a\in B\}\\ A\mbox{\downdownarrows}_{i}B&=&\{d:\mbox{ for every }a\in A,\mbox{ }a\times_{i}d\in B\}\\ \end{array}
The class of powerset residuated DAs over a DA is denoted . The class of powerset residuated DAs over a standard DA is denoted . Finally, the subclass of which is formed by powerset residuated algebras over finitely-generated standard DA are known simply as L-models.
Every standard DA has two remarkable properties, namely the property that sort domains \mbox{|\mbox{}|}_{i} (for ) can be defined in terms of \mbox{|\mbox{}|}_{0}, and the property that every element of a sort domain \mbox{|\mbox{}|}_{i} is decomposed uniquely around the separator : \enumsentence \begin{array}[t]{ll}\mbox{(S1) }\mbox{For i>0, }\mbox{|\mbox{}|}_{i}=\underbrace{\mbox{|\mbox{}|}_{0}\mbox{\circ}\{1\}\cdots\{1\}\mbox{\circ}\mbox{|\mbox{}|}_{0}}_{\mbox{(i-1)1^{\prime}s}}\\ \mbox{(S2) }\mbox{For i>0a_{0}+\mbox{}+\cdots+\mbox{}+a_{i}=b_{0}+\mbox{}+\cdots+\mbox{}+b_{i} then }\\ a_{k}=b_{k}\mbox{ for }0\leq k\leq i\end{array}
Standard DAs, as their name suggests, are particular cases of (general) DAs:
Lemma 1
The class of standard DAs is a subclass of the class of DAs.333Later we see that the inclusion is proper.
Proof
We define a useful notation which will help us to prove the lemma. Where \mathcal{A}=(\mbox{|\mbox{}|},+,(\times_{i})_{i>0},0,1) is a standard DA, let be an arbitrary element of sort . We associate to every a\in\mbox{|\mbox{}|} a sequence of elements . We have the following vectorial notation: \enumsentence \overrightarrow{a}_{i}^{j}=\left\{\begin{array}[]{l}a_{i}\mbox{, if }i=j\\ \overrightarrow{a}_{i}^{j-1}+\mbox{1}+a_{j}\mbox{, if }j-i>0\end{array}\right. Since is a standard DA, the associated to a given are unique (by freeness of the underlying monoid). We have that , and we write in place of . Consider arbitrary elements , and of |\mbox{\mathcal{A}}|:
Continuous associativity is obvious.
Discontinuous associativity. Let be such that :
[TABLE]
On the other hand, we have that:
[TABLE]
It follows that:
[TABLE]
By comparing the right hand side of (*) and (**), we have therefore:
[TABLE]
Mixed Permutation. Consider (\overrightarrow{a}\mbox{!\times!}_{i}\overrightarrow{b})\mbox{!\times!}_{j}\overrightarrow{c} and suppose that :
[TABLE]
It follows that:
[TABLE]
Since , then . Then we have that:
[TABLE]
It follows that:
[TABLE]
By comparing the right hand side of (*) and (**), we have therefore:
[TABLE]
Mixed associativity. There are two cases: or . Considering the first one, this is true for:
[TABLE]
The other case corresponding to is completely similar.
The case corresponding to the units is completely trivial.
∎
2.2 The Hypersequent Calculus hD
We will now consider the string-based hypersequent syntax from [8]. The reason for using the prefix hyper in the term sequent is that the data-structure used in hypersequent antecedents is quite nonstandard. A fundamental tool to build the data-structure of a sequent calculus for is the notion of type-segment. For any type of sort [math] . If then seg(A)=\{\mbox{\sqrt[0]{A}},\cdots,\mbox{\sqrt[s(A)]{A}}\}. We call the set of type-segments of . If is a set of connectives, we can now define the set of type-segments corresponding to the set \mbox{\mathbf{Tp}}[\mathbf{C}] of types generated by the connectives as seg(A). Type segments of sort [math] are types. But, type segments of sort greater than [math] are no longer types. Strings of type segments can form meaningful logical material like the set of configurations, which we now define. Where is a set of connectives the configurations \mbox{\mathcal{O}}[\mathbf{C}] are defined in BNF unambiguously by mutual recursion as follows, where is the empty string and is the metalinguistic separator: \enumsentence\footnotesize{\begin{array}[t]{rcll}\mbox{\mathcal{O}}[\mathbf{C}]&::=&\Lambda\\ \mbox{\mathcal{O}}[\mathbf{C}]&::=&\mbox{1},\mbox{\mathcal{O}}[\mathbf{C}]\\ \mbox{\mathcal{O}}[\mathbf{C}]&::=&A,\mbox{\mathcal{O}}[\mathbf{C}]\quad\mbox{for\ }s(A)=0\\ \mbox{\mathcal{O}}[\mathbf{C}]&::=&\mbox{\sqrt[0]{A}},\mbox{\mathcal{O}}[\mathbf{C}],\mbox{\sqrt[1]{A}},\cdots,\mbox{\sqrt[s(A)-1]{A}},\mbox{\mathcal{O}}[\mathbf{C}],\mbox{\sqrt[s(A)]{A}}_{s(A)},\mbox{\mathcal{O}}[\mathbf{C}]\quad\mbox{for\ }s(A)>0\end{array}} The intuitive semantic interpretation of the last clause from (2.2) consists of elements where \alpha_{0}+\mbox{1}+\alpha_{1}+\cdots+\alpha_{n{-}1}+\mbox{1}+\alpha_{n}\!\in\mbox{\llbracket}A\mbox{\rrbracket} and are the interpretations of the intercalated configurations.
If the context is clear we will write for \mbox{\mathcal{O}}[\mathbf{C}], and likewise , and .
The syntax in which has been defined is called string-based hypersequent syntax. An equivalent syntax for is called tree-based hypersequent syntax, which was defined in [9], [12]. For proof-search and human readability, the tree-based notation is more convenient than the string-based notation, but for semantic purposes, the string-based notation turns out to be very useful since the canonical model construction considered in Section 3 relies on the set of type-segments.
In string-based notation the figure of a type is defined as follows: \enumsentence\overrightarrow{A}=\left\{\begin{array}[]{ll}A&\mbox{if\ }s(A)=0\\ \mbox{\sqrt[0]{A}},\mbox{1},\mbox{\sqrt[1]{A}},\cdots,\mbox{\sqrt[s(A)-1]{A}},\mbox{1},\mbox{\sqrt[s(A)]{A}}&\mbox{if\ }s(A)>0\end{array}\right.
The sort of a configuration is the number of metalinguistic separators it contains. We have \mbox{\mathcal{O}}=\bigcup_{i\geq 0}\mbox{\mathcal{O}}_{i}, where \mbox{\mathcal{O}}_{i} is the set of configurations of sort . We define a more general notion of configuration, namely preconfiguration. If denotes \mathbf{seg}[\mathbf{C}]\cup\{\mbox{1}\}, a preconfiguration is simply a word of . Obviously, we have that \mbox{\mathcal{O}}\subsetneq V^{*}. A preconfiguration is proper iff \Delta\not\in\mbox{\mathcal{O}}. As in the case of configurations, preconfigurations have a sort.
Where and are configurations and the sort of is at least , () signifies the configuration which is the result of replacing the -th separator in by . The notation , which we call a configuration with a distinguished configuration abbreviates the following configuration: , where \Delta_{i}\in\mbox{\mathcal{O}} but and are possibly proper preconfigurations. When a type-occurrence in a configuration is written without vectorial notation, that means that the sort of is [math]. However, when one writes the metanotation for configurations , this does not mean that the sort of is necessarily greater than [math].
A hypersequent \Gamma\mbox{\ \Rightarrow\ }A comprises an antecedent configuration in string-based notation of sort and a succedent type of sort . The hypersequent calculus for is as shown in Figure 4.
The following lemma is useful for the strong completeness results of section 3:
Lemma 2
Recall that is a subset of . We have that:
- i)
* is closed by concatenation and intercalation.*
- ii)
If , \Gamma\in\mbox{\mathcal{O}}{}, and \Delta,\Gamma\in\mbox{\mathcal{O}}, then \Delta\in\mbox{\mathcal{O}}. Similarly, if we have \Gamma,\Delta\in\mbox{\mathcal{O}} instead of \Delta,\Gamma\in\mbox{\mathcal{O}}. Finally, If , \Gamma\in\mbox{\mathcal{O}}{}, and \Delta|_{i}\Gamma\in\mbox{\mathcal{O}}, then \Delta\in\mbox{\mathcal{O}}.
Proof
Propositions (i) and ii) are both proved via the BNF derivations of (2.2). The details of the proof are rather tedious but not difficult.
What is the connection between the calculi and ? In [14] a (faithful) embedding translation is proved. Let denote a configuration. We define its type-equivalent , which is a type which has the same algebraic meaning as . Via the BNF formulation of in (2.2) one defines recursively as follows:
[TABLE]
The semantic interpretation of a configuration (for a given valuation ) is \widehat{v}(\Delta)\mbox{;=;}\widehat{v}(\Delta^{\bullet}). The embedding translation is as follows. For any \Delta\in\mathcal{O},\mbox{\mathbf{cD}}\vdash\Delta^{\bullet}\rightarrow A iff \mbox{\mathbf{hD}}\vdash\Delta\mbox{\ \Rightarrow\ }A.
2.3 Some special DAs
The standard DA , induced by the separated monoid with generator set V=\mbox{\mathbf{seg}}\cup\{1\}, plays an important role. The interpretation of the signature in |\mbox{\mathcal{S}}| is: \enumsentence \mbox{\mathcal{S}}=(V^{*},+,\{|_{i}\}_{i>0},\Lambda,\mbox{1})
Here, denotes concatenation, and i-th intercalation. We have seen in Section 2 that is closed by concatenation and intercalation , , i.e. \mathcal{C}=(\mbox{\mathcal{O}},+,(|_{i})_{i>0},\Lambda,1) is a -subalgebra of the standard DA . Since is a variety,444We recall that varieties are closed by subalgebras, homomorphic images, and direct products. is a (general) DA, concretely a nonstandard DA. To see that cannot be standard we notice that the sort domains of are not separated by . Recall that \mbox{|\mbox{}|}=\displaystyle\bigcup_{i\geq 0}\mbox{\mathcal{O}}_{i} (\mbox{|\mbox{}|}_{i}=\mbox{\mathcal{O}}_{i}, for every ). We have that: \enumsentence\mbox{For i>0,}\mbox{|\mbox{}|}_{i}\neq\underbrace{\mbox{\mathcal{O}}_{0}\mbox{\circ}\cdots\mbox{\circ}\mbox{\mathcal{O}}_{0}}_{\mbox{i times}}
Because, for example, let us take \overrightarrow{p\mbox{{\uparrow}}_{1}p}=\mbox{\sqrt[0]{p\mbox{}{1}p}},1,\mbox{\sqrt[1]{p\mbox{}{1}p}}, where p\in\mbox{\mathbf{Pr}}_{0}. The type p\mbox{{\uparrow}}_{1}p has sort , but clearly neither \sqrt[0]{p\mbox{{\uparrow}}_{1}p} nor \sqrt[1]{p\mbox{{\uparrow}}_{1}p} are members of \mbox{\mathcal{O}}_{0}. In fact, we have the proper inclusion: \enumsentence\mbox{For i>0,}\underbrace{\mbox{\mathcal{O}}_{0}\mbox{\circ}\cdots\mbox{\circ}\mbox{\mathcal{O}}_{0}}_{\mbox{i times}}\subsetneq\mbox{|\mbox{}|}_{i}
It follows that the class of standard DAs is a proper subclass of the class of general DAs.
2.4 Synthetic Connectives and the Implicative fragment
From a logical point of view, synthetic connectives abbreviate formulas in sequent systems. They form new connectives with left and right sequent rules. Using a linear logic slogan, synthetic connectives help to eliminate some bureaucracy in Cut-free proofs and in the (syntactic) Cut-elimination algorithms (see [14]). We consider here a set of synthetic connectives which are of linguistic interest:
- •
The binary non-deterministic implications , and .
- •
The unary connectives , and (\mbox{\v{}{}^{{}_{k}}})_{k>0}, which are called respectively left projection, right projection, and split.
Together with the binary deterministic implications , /, (\mbox{{\uparrow}}_{i})_{i>0}, (\mbox{\downarrow}_{i})_{i>0}, these constitute what we call implicative connectives. These connectives are incorporated in the recursive definitions of , , and . We denote this implicative fragment as \mbox{\mathbf{D}}[\rightarrow]. We write also \mbox{\mathbf{Tp}}[\rightarrow], , and , although, as usual, if the context is clear we will avoid writing . The intuitive semantic interpretation of the implicative connectives can be found in Figure 5. Figure 6 and Figure 7 correspond to their hypersequent rules.
Besides the usual continuous and discontinuous implications, the nondeterministic discontinous implications are used to account for particle shift nondeterminism where the object can be intercalated between the verb and the particle, or after the particle. For a particle verb like call+\mbox{1}+up we can give the lexical assigment \mbox{\triangleleft^{-1}}(\mbox{\v{}{}^{{}{1}}}(N\mbox{\backslash}S)\mbox{{\Uparrow}}N). Projections can be used to account for the cross-serial dependencies of Dutch. The split connective can be used for parentheticals like fortunately with the type assignment \mbox{\v{}{}^{{}{1}}}S\mbox{\downarrow}_{1}S.
3 Strong Completeness of the implicative fragment w.r.t. L-models
In this section we prove two strong completeness theorems in relation to the implicative fragment. In order to prove them, we demonstrate first strong completeness of \mbox{\mathbf{hD}}[\rightarrow] w.r.t. powerset residuated DAs over standard DAs with a countable set of generators.
Let V=\mbox{\mathbf{seg}}[\rightarrow]\cup\{\mbox{1}\}. Clearly, is countably infinite since \mbox{\mathbf{seg}}[\rightarrow] is the countable union \displaystyle\bigcup_{i}\mbox{\mathbf{seg}}[\rightarrow]_{i}, where each \mbox{\mathbf{seg}}[\rightarrow]_{i} is also countably infinite. Let us consider the standard DA (from (2.3)), induced by the (countably) infinite set of generators :
[TABLE]
We define some notation:
Definition 3
For any type C\in\mbox{\mathbf{Tp}}[\rightarrow] and set of non-logical axioms:
[TABLE]
In practice, when the set of hypersequents is clear from the context, we simply write instead of \mbox{[C]}_{R}.
Lemma 3
(Truth Lemma)
Let be the powerset residuated DA over the standard DA from (2.3). Let be the following valuation on the powerset :
[TABLE]
Let \mbox{\mathcal{M}}=(\mbox{\mathcal{P}(S)},v_{R}) be called as usual the canonical model. The following equality holds:
[TABLE]
Proof
We proceed by induction on the structure of type ; we will write instead of , and instead of \mbox{[\cdot]}_{R}; we will say that an element is correct555Recall that a priori \Delta\in\mbox{|\mbox{}|}, which is equal to (\mbox{\mathbf{Seg}}[\rightarrow]\cup\{\mbox{1}\})^{*}. iff \Delta\in\mbox{\mathcal{O}}[\mathbf{C}].
is primitive. True by definition.
C=B\mbox{{\uparrow}}_{i}A. Let us see:
[TABLE]
Let be such that R\vdash\Delta\mbox{\ \Rightarrow\ }B\mbox{{\uparrow}}_{i}A. Let . By induction hypothesis (i.h.), \widehat{v}(A)=\mbox{[A]}. Hence, R\vdash\Gamma_{A}\mbox{\ \Rightarrow\ }A We have:
[TABLE]
By i.h., \widehat{v}(B)=\mbox{[B]}. It follows that , hence \Delta\in\widehat{v}(B\mbox{{\uparrow}}_{i}A). Whence, \mbox{[B\mbox{}_{i}A]}\subseteq\widehat{v}(B\mbox{{\uparrow}}_{i}A).
Conversely, let us see:
[TABLE]
Let \Delta\in\widehat{v}(B\mbox{{\uparrow}}_{i}A). By i.h. \widehat{v}(A)=\mbox{[A]}. For any type , we have eta-expansion, i.e. \overrightarrow{A}\mbox{\ \Rightarrow\ }A666By simple induction on the structure of types.. Hence, . We have that . By i.h., \Delta|_{i}\overrightarrow{A}\mbox{\ \Rightarrow\ }B. Since is correct, and by i.h. is correct, by Lemma 2, is correct. By applying the \mbox{{\uparrow}}_{i} right rule to the provable hypersequent \Delta|_{i}\overrightarrow{A}\mbox{\ \Rightarrow\ }B we get:
[TABLE]
This ends the case of B\mbox{{\uparrow}}_{i}A.
C=A\mbox{\downarrow}_{i}B. Completely similar to case B\mbox{{\uparrow}}_{i}A.
or A\mbox{\backslash}B. Similar to the disconcontuous case.
Nondeterministic connectives. Consider the case .
[TABLE]
Let . By i.h, \Gamma_{A}\mbox{\ \Rightarrow\ }A. Let \Delta\mbox{\ \Rightarrow\ }B{\Uparrow}A. By applications of left rule, we have
[TABLE]
By Cut applications with \Delta\mbox{\ \Rightarrow\ }B{\Uparrow}A, we get:
[TABLE]
Hence, for , by i.h. . Hence, .
Conversely, let us see:
[TABLE]
By i.h, we see that . Let . This means that for every . By i.h., \Delta|_{i}\overrightarrow{A}\mbox{\ \Rightarrow\ }B. By a similar reasoning to the deterministic case C=B\mbox{{\uparrow}}_{i}A, we see that is correct. We have that:
[TABLE]
is completely similar to the previous one.
C=\mbox{\triangleleft^{-1}}A. Let us see:
[TABLE]
Let \Delta\in[\mbox{\triangleleft^{-1}}A]. Hence, \Delta\mbox{\ \Rightarrow\ }\mbox{\triangleleft^{-1}}A. We have that:
[TABLE]
By i.h., . Hence, \Delta\in\widehat{v}(\mbox{\triangleleft^{-1}}A).
Conversely, let us see:
[TABLE]
Let \Delta\in\widehat{v}(\mbox{\triangleleft^{-1}}A). By definition, . By i.h., \Delta,1\mbox{\ \Rightarrow\ }A, and by lemma 2, is correct. By application of right rule, we get:
[TABLE]
This proves the converse.
C=\mbox{\triangleright^{-1}}A is completely similar to the previous one.
C=\mbox{\v{}{}^{{}_{k}}}\!A. Let us see:
[TABLE]
Let \Delta\mbox{\ \Rightarrow\ }\mbox{\v{}{}^{{}_{k}}}\!A. We have that:
[TABLE]
By i.h., \Delta\in\widehat{v}(\mbox{\v{}{}^{{}_{k}}}A).
Conversely, let us see that:
[TABLE]
Let \Delta\in\widehat{v}(\mbox{\v{}{}^{{}_{k}}}\!A). By definition, . By i.h. and lemma 2, is correct and \Delta|_{k}\Lambda\mbox{\ \Rightarrow\ }A. By application of the ˇ right rule:
[TABLE]
Hence, \Delta\in\mbox{[\mbox{\v{}}A]}.∎
By induction on the structure of , see (2.2), one proves the following lemma:
Lemma 4
(Identity lemma)
For any \Delta\in\mbox{\mathcal{O}}, .
Let be the sequence of type-occurrences in a configuration . Let \Delta\mbox{\left(\begin{array}[]{c}\Gamma_{1}\cdots\Gamma_{n}\ A_{1}\cdots A_{n}\ \end{array}\right)} be the result of replacing every type-occurrence with . Recall that we have fixed a set of hypersequents . We have the lemma:
Lemma 5
\mbox{\mathcal{M}}=(\mbox{\mathcal{P}(S)},v)\models R**
Proof
Let (\Delta\mbox{\ \Rightarrow\ }A)\in R. For every type-occurrence in (we suppose that the sequence of type occurrences in is ), we have by the Truth Lemma that \widehat{v}(A_{i})=\mbox{[A_{i}]}_{R}. For any , we have by the Truth Lemma that R\vdash\Gamma_{i}\mbox{\ \Rightarrow\ }A_{i}. Since (\Delta\mbox{\ \Rightarrow\ }A)\in R, we have then that R\vdash\Delta\mbox{\ \Rightarrow\ }A. By applications of the Cut rule with the premises we get from R\vdash\Delta\mbox{\ \Rightarrow\ }A that R\vdash\Delta\mbox{\left(\begin{array}[]{c}\Gamma_{1}\cdots\Gamma_{n}\ A_{1}\cdots A_{n}\ \end{array}\right)}\mbox{\ \Rightarrow\ }A. We have that \widehat{v}(\Delta)=\{\Delta\mbox{\left(\begin{array}[]{c}\Gamma_{1}\cdots\Gamma_{n}\ A_{1}\cdots A_{n}\ \end{array}\right)}:\Gamma_{i}\in\widehat{v}(A_{i})\}. Since, we have R\vdash\Delta\mbox{\left(\begin{array}[]{c}\Gamma_{1}\cdots\Gamma_{n}\ A_{1}\cdots A_{n}\ \end{array}\right)}\mbox{\ \Rightarrow\ }A, again by the Truth Lemma, \Delta\mbox{\left(\begin{array}[]{c}\Gamma_{1}\cdots\Gamma_{n}\ A_{1}\cdots A_{n}\ \end{array}\right)}\in\widehat{v}(A). We have then that . We are done.∎
Theorem 3.1
* is strongly complete w.r.t. the class .*
Proof
Suppose \mbox{\mathcal{PRSD}}(R)\models\Delta\mbox{\ \Rightarrow\ }A. Hence, in particular this is true of the canonical model . Since , it follows that . By the Truth Lemma, \widehat{v}(A)=\mbox{[A]}_{R}. Hence, R\vdash\Delta\mbox{\ \Rightarrow\ }A. We are done.∎
We shall also prove strong completeness w.r.t. L-models over the set of connectives , where \mathbf{split}=\{\mbox{\v{}{}^{{}_{k}}}:k>0\}. Since the canonical model is countably infinite, |\mbox{\mathcal{S}}| is in bijection with a set via a mapping . Let be the standard DA associated to . extends to an isomorphism of standard DAs between and , and then induces an isomorphism of residuated powerset DAs over standard DAs. Let be a standard DA generated by the finite set of generators . We have that \mbox{|\mbox{}|}=V_{1}^{*}, and \mbox{|\mbox{}|}=V_{2}^{*}. Let be the following injective mapping from into : \enumsentence \begin{array}[t]{lllllll}\rho(1)&=&1&&\rho(a_{i})&=&a+b^{i}+a\end{array}
The mapping extends recursively to the morphism of standard DAs.Clearly is injective by freeness777Since the underlying structures are free monoids we can apply left/right cancellation. of the underlying free monoids |\mbox{\mathcal{A}}| and |\mbox{\mathcal{B}}|. The mapping is a monomorphism of DAs which induces a monomorphism of residuated powerset DAs over DAs. Let , and range over subsets of |\mbox{\mathcal{A}}| such that they are non-empty and different from . Since is injective, so is . The following equalities hold: \enumsentence \begin{array}[t]{lllllllll}\bar{\rho}(A\cdot B)&=&\bar{\rho}(A)\cdot\bar{\rho}(B)&\mbox{ }\bar{\rho}(A\mbox{\circ_{i}}B)&=&\bar{\rho}(A)\mbox{\circ_{i}}\bar{\rho}(B)&\bar{\rho}(A\mbox{/!/}\mbox{\mathbb{J}})&=&\bar{\rho}(A)\mbox{/!/}\bar{\rho}(\mbox{\mathbb{J}})\\ \bar{\rho}(A\mbox{\mbox{}!\mbox{}}B)&=&\bar{\rho}(A)\mbox{\mbox{}!\mbox{}}\bar{\rho}(B)&\mbox{ }\bar{\rho}(B\mbox{/!/}A)&=&\bar{\rho}(B)\mbox{/!/}\bar{\rho}(A)&\bar{\rho}(\mbox{\mathbb{J}}\mbox{\mbox{}!\mbox{}}A)&=&\mbox{\mathbb{J}}\mbox{\mbox{}!\mbox{}}\bar{\rho}(A)\\ \bar{\rho}(A\mbox{\downdownarrows}_{i}B)&=&\bar{\rho}(A)\mbox{\downdownarrows}_{i}\bar{\rho}(B)&\mbox{ }\bar{\rho}(B\mbox{\upuparrows}_{i}A)&=&\bar{\rho}(B)\mbox{\upuparrows}_{i}\bar{\rho}(A)\\ \end{array} The equalities (3) are due to the fact that 1) is a monomorphism of DAs, 2) we can apply cancellation, and 3) the subsets considered are non-empty and different from . Since is injective, arbitrary families of (same-sort) subsets satisfy . Moreover, using (3) one proves: \enumsentence \begin{array}[t]{lll}\bar{\rho}(\displaystyle\bigcap_{i=1}^{s(B)-s(A)+1}B\mbox{\upuparrows}_{i}A)&=&\displaystyle\bigcap_{i=1}^{s(B)-s(A)+1}\bar{\rho}(B)\mbox{\upuparrows}_{i}\bar{\rho}(A)\\ \bar{\rho}(\displaystyle\bigcap_{i=1}^{s(B)-s(A)+1}A\mbox{\downdownarrows}_{i}B)&=&\displaystyle\bigcap_{i=1}^{s(B)-s(A)+1}\bar{\rho}(A)\mbox{\downdownarrows}_{i}\bar{\rho}(A)\end{array}
Recall that is the valuation of the canonical model . Consider the following composition of mappings: \Pr\overset{v}{\longrightarrow}\mbox{\mathcal{P}(S)}\overset{\bar{\Phi}}{\longrightarrow}\mbox{\mathcal{P}(A)}\overset{\bar{\rho}}{\longrightarrow}\mbox{\mathcal{P}(B)}. Put . We have that . In order to prove the last equality we have to see that is a monorphism of DAs.888There is a unique morphism of DAs extending . For example, if and are types, one has:
[TABLE]
Similar computations give the desired equalities for the remaining considered implicative connectives.999 Including also projection connectives. Given a set of non-logical axioms , R\vdash_{\mathbf{hD}}\Delta\mbox{\ \Rightarrow\ }A iff (we write instead of ) iff iff . We have proved:
Theorem 3.2
\mbox{\mathbf{D}}[\Sigma_{\rightarrow}{-}\mathbf{split}]* is strongly complete w.r.t. L-models.*
Corollary 1
\mbox{\mathbf{D}}[\Sigma_{\rightarrow}{-}\mathbf{split}]* is strongly complete w.r.t. powerset residuated DAs overs standard DAs with generators.*
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] A. Avron. Hypersequents, Logical Consequence and Intermediate Logic form Concurrency. Annals of Mathematics and Artificial Intelligence , 4:225–248, 1991.
- 2[2] W. Buszkowski. Completeness results for Lambek syntactic calculus. Zeitschrift für mathematische Logik und Grundlagen der Mathematik , 32:13–28, 1986.
- 3[3] J.A. Goguen and J. Meseguer. Completeness of Many-Sorted Equational Logic. Houston Journal of Mathematics , 11(3):307–334, 1985.
- 4[4] R. Lalement. Logique, réduction, résolution . études et recherches en informatique. Masson, Paris, 1990.
- 5[5] Joachim Lambek. The mathematics of sentence structure. American Mathematical Monthly , 65:154–170, 1958. Reprinted in Buszkowski, Wojciech, Wojciech Marciszewski, and Johan van Benthem, editors, 1988, Categorial Grammar , Linguistic & Literary Studies in Eastern Europe volume 25, John Benjamins, Amsterdam, 153–172.
- 6[6] Richard Moot. Extended Lambek calculi and first-order linear logic. In Claudia Casadio, Bob Coecke, Michael Moortgat, and Philip Scott, editors, Categories and Types in Logic, Language, and Physics , volume 8222 of Lecture Notes in Computer Science , pages 297–330. Springer Berlin Heidelberg, 2014.
- 7[7] G. Morrill and O. Valentín. Spurious ambiguity and focalisation. Manuscript, Submitted.
- 8[8] Glyn Morrill, Mario Fadda, and Oriol Valentín. Nondeterministic Discontinuous Lambek Calculus. In Jeroen Geertzen, Elias Thijsse, Harry Bunt, and Amanda Schiffrin, editors, Proceedings of the Seventh International Workshop on Computational Semantics, IWCS-7 , pages 129–141. Tilburg University, 2007.