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Register a new userLanguage Models for Some Extensions of the Lambek Calculus
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Stepan Kuznetsov
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We investigate language interpretations of two extensions of the Lambek calculus: with additive conjunction and disjunction and with additive conjunction and the unit constant. For extensions with additive connectives, we show that conjunction and disjunction behave differently. Adding both of them leads to incompleteness due to the distributivity law. We show that with conjunction only no issues with distributivity arise. In contrast, there exists a corollary of the distributivity law in the language with disjunction only which is not derivable in the non-distributive system. Moreover, this difference keeps valid for systems with permutation and/or weakening structural rules, that is, intuitionistic linear and affine logics and affine multiplicative-additive Lambek calculus. For the extension of the Lambek with the unit constant, we present a calculus which reflects natural algebraic properties of the empty word. We do not claim completeness for this calculus, but we prove undecidability for the whole range of systems extending this minimal calculus and sound w.r.t. language models. As a corollary, we show that in the language with the unit there exissts a sequent that is true if all variables are interpreted by regular language, but not true in language models in general.
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We consider the Lambek calculus, or non-commutative multiplicative intuitionistic linear logic, extended with iteration, or Kleene star, axiomatised by means of an $omega$-rule, and prove that the derivability problem in this calculus is $Pi_1^0$-hard. This solves a problem left open by Buszkowski (2007), who obtained the same complexity bound for infinitary action logic, which additionally includes additive conjunction and disjunction. As a by-product, we prove that any context-free language without the empty word can be generated by a Lambek grammar with unique type assignment, without Lambeks non-emptiness restriction imposed (cf. Safiullin 2007).
Formulae of the Lambek calculus are constructed using three binary connectives, multiplication and two divisions. We extend it using a unary connective, positive Kleene iteration. For this new operation, following its natural interpretation, we present two lines of calculi. The first one is a fragment of infinitary action logic and includes an omega-rule for introducing iteration to the antecedent. We also consider a version with infinite (but finitely branching) derivations and prove equivalence of these t
This paper shows connections between command injection attacks, continuations, and the Lambek calculus: certain command injections, such as the tautology attack on SQL, are shown to be a form of control effect that can be typed using the Lambek calculus, generalizing the double-negation typing of continuations. Lambeks syntactic calculus is a logic with two implicational connectives taking their arguments from the left and right, respectively. These connectives describe how strings interact with their left and right contexts when building up syntactic structures. The calculus is a form of propositional logic without structural rules, and so a forerunner of substructural logics like Linear Logic and Separation Logic.
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Descriptive set theory was originally developed on Polish spaces. It was later extended to $omega$-continuous domains [Selivanov 2004] and recently to quasi-Polish spaces [de Brecht 2013]. All these spaces are countably-based. Extending descriptive set theory and its effective counterpart to general represented spaces, including non-countably-based spaces has been started in [Pauly, de Brecht 2015]. We study the spaces $mathcal{O}(mathbb{N}^mathbb{N})$, $mathcal{C}(mathbb{N}^mathbb{N},2)$ and the Kleene-Kreisel spaces $mathbb{N}langlealpharangle$. We show that there is a $Sigma^0_2$-subset of $mathcal{O}(mathbb{N}^mathbb{N})$ which is not Borel. We show that the open subsets of $mathbb{N}^{mathbb{N}^mathbb{N}}$ cannot be continuously indexed by elements of $mathbb{N}^mathbb{N}$ or even $mathbb{N}^{mathbb{N}^mathbb{N}}$, and more generally that the open subsets of $mathbb{N}langlealpharangle$ cannot be continuously indexed by elements of $mathbb{N}langlealpharangle$. We also derive effecti
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