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Language-Theoretic and Finite Relation Models for the (Full) Lambek Calculus

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Abstract

We prove completeness for some language-theoretic models of the full Lambek calculus and its various fragments. First we consider syntactic concepts and syntactic concepts over regular languages, which provide a complete semantics for the full Lambek calculus \({\mathbf {FL}}_\perp \). We present a new semantics we call automata-theoretic, which combines languages and relations via closure operators which are based on automaton transitions. We establish the completeness of this semantics for the full Lambek calculus via an isomorphism theorem for the syntactic concepts lattice of a language and a construction for the universal automaton recognizing the same language. Finally, we use automata-theoretic semantics to prove completeness of relation models of binary relations and finite relation models for the Lambek calculus without and with empty antecedents (henceforth: \(\mathbf L \) and \(\mathbf L1 \)), thus solving a problem left open by Pentus (Ann Pure Appl Log 75:179–213, 1995).

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Notes

  1. Whereas \(\mathbf L \) and \(\mathbf L1 \) are equally powerful in the sense of languages which are recognizable – except for languages containing the empty word, which cannot be recognized by \(\mathbf L \) under standard assumptions—Kanazawa (1992) shows that \(\mathbf {FL}\) is considerably more powerful than \(\mathbf L \): whereas \(\mathbf L \) only recognizes context-free languages by the classical result of Pentus (1993), \(\mathbf {FL}\) can recognize any finite intersection of context-free languages.

  2. It is well-known that for any algebra \(\mathbf A \), the set of its subidentity elements forms a closed subalgebra.

  3. Actually, we also have \(\gamma (X)\subseteq B^{*}\); but still we use \(\gamma \) as an embedding into a larger co-domain.

  4. We assume that \(\top \in B\)!

  5. But in principle, nothing prevents us from having \((x,y)\in \phi (\epsilon )\) with \(x\ne y\)—we just have to make sure that for all \(a\in \varSigma \), we have \(\phi (\epsilon )\mathbf ; \phi (a)=\phi (a)=\phi (a)\mathbf ; \phi (\epsilon )\). One might also think this gives rise to another problem, namely when we have automata with \(\epsilon \) transitions. However, \(\epsilon \)-transitions in a (classical) automaton have to be distinguished from \(\phi (\epsilon )\), which is algebraic in nature: assume we have an \(\epsilon \)-transition \(R_\epsilon \), in addition to transitions \(R_a:a\in \varSigma \). In the algebraic setting, this is part of primitive transitions: we have to define \(\phi (a)\) as the smallest set such that 1. \(R_a\subseteq \phi (a)\), and 2. \(\phi (a)\mathbf ; R_\epsilon \subseteq \phi (a)\), \(R_\epsilon \mathbf ; \phi (a)\subseteq \phi (a)\).

  6. This class is strictly smaller than the class of automata recognizing regular languages, as obviously there are infinite automata recognizing regular languages.

  7. Recall that sets of the form \(M^\triangleright \) etc. are always closed.

  8. Keep in mind that here \(m='n\) is an abbreviation for \( m\le ' n\ \& \ n\le ' m\), same for \(m=n\).

  9. This is because \(\psi \) is a pointwise map, see (23).

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  1. Heinrich Heine Universität Düsseldorf, Universitätsstr. 1, 40225, Düsseldorf, Germany

    Christian Wurm

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Wurm, C. Language-Theoretic and Finite Relation Models for the (Full) Lambek Calculus. J of Log Lang and Inf 26, 179–214 (2017). https://doi.org/10.1007/s10849-017-9249-z

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