scholarly journals Involutive Nonassociative Lambek Calculus: Sequent Systems and Complexity

2017 ◽  
Vol 46 (1/2) ◽  
Author(s):  
Wojciech Buszkowski
Keyword(s):  
Cut Elimination ◽  
Lambek Calculus ◽  
Double Negation ◽  
Sequent Systems ◽  
Sequent System

In [5] we study Nonassociative Lambek Calculus (NL) augmented with De Morgan negation, satisfying the double negation and contraposition laws. This logic, introduced by de Grooté and Lamarche [10], is called Classical Non-Associative Lambek Calculus (CNL). Here we study a weaker logic InNL, i.e. NL with two involutive negations. We present a one-sided sequent system for InNL, admitting cut elimination. We also prove that InNL is PTIME.

scholarly journals One-Sided Sequent Systems for Nonassociative Bilinear Logic: Cut Elimination and Complexity

2020 ◽  
Author(s):  
Paweł Płaczek
Keyword(s):  
Analogous Result ◽  
Cut Elimination ◽  
Sequent Systems ◽  
Natural Symmetry ◽  
Sequent System

Bilinear Logic of Lambek [10] amounts to Noncommutative MALL of Abrusci [1]. Lambek [10] proves the cut–elimination theorem for a onesided (in fact, left-sided) sequent system for this logic. Here we prove an analogous result for the nonassociative version of this logic. Like Lambek [10], we consider a left-sided system, but the result also holds for its right-sided version, by a natural symmetry. The treatment of nonassociative sequent systems involves some subtleties, not appearing in associative logics. We also prove the PTIME complexity of the multiplicative fragment of NBL.

A proof-theoretic treatment of λ-reduction with cut-elimination: λ-calculus as a logic programming language

2011 ◽  
Vol 76 (2) ◽  
pp. 673-699 ◽  
Author(s):  
Michael Gabbay
Keyword(s):  
Logic Programming ◽  
Programming Language ◽  
Functional Programming ◽  
Order Logic ◽  
First Order Logic ◽  
Cut Elimination ◽  
Logic Programming Language ◽  
First Order ◽  
Functional Programming Language ◽  
Sequent System

AbstractWe build on an existing a term-sequent logic for the λ-calculus. We formulate a general sequent system that fully integrates αβη-reductions between untyped λ-terms into first order logic.We prove a cut-elimination result and then offer an application of cut-elimination by giving a notion of uniform proof for λ-terms. We suggest how this allows us to view the calculus of untyped αβ-reductions as a logic programming language (as well as a functional programming language, as it is traditionally seen).

scholarly journals Bunched Hypersequent Calculi for Distributive Substructural Logics

10.29007/ngp3 ◽  
2018 ◽  
Author(s):  
Agata Ciabattoni ◽  
Revantha Ramanayake
Keyword(s):  
Large Class ◽  
General Rule ◽  
Expressive Power ◽  
Substructural Logics ◽  
Cut Elimination ◽  
Lambek Calculus ◽  
Hypersequent Calculi

We introduce a new proof-theoretic framework which enhances the expressive power of bunched sequents by extending them with a hypersequent structure. A general cut-elimination theorem that applies to bunched hypersequent calculi satisfying general rule conditions is then proved. We adapt the methods of transforming axioms into rules to provide cutfree bunched hypersequent calculi for a large class of logics extending the distributive commutative Full Lambek calculus DFLe and Bunched Implication logic BI. The methodology is then used to formulate new logics equipped with a cutfree calculus in the vicinity of Boolean BI.

scholarly journals Reconciling Lambek’s restriction, cut-elimination and substitution in the presence of exponential modalities

2020 ◽  
Vol 30 (1) ◽  
pp. 239-256 ◽  
Author(s):  
Max Kanovich ◽  
Stepan Kuznetsov ◽  
Andre Scedrov
Keyword(s):  
Linear Logic ◽  
Cut Elimination ◽  
Lambek Calculus ◽  
Full Extent ◽  
Intuitionistic Linear Logic

Abstract The Lambek calculus can be considered as a version of non-commutative intuitionistic linear logic. One of the interesting features of the Lambek calculus is the so-called ‘Lambek’s restriction’, i.e. the antecedent of any provable sequent should be non-empty. In this paper, we discuss ways of extending the Lambek calculus with the linear logic exponential modality while keeping Lambek’s restriction. Interestingly enough, we show that for any system equipped with a reasonable exponential modality the following holds: if the system enjoys cut elimination and substitution to the full extent, then the system necessarily violates Lambek’s restriction. Nevertheless, we show that two of the three conditions can be implemented. Namely, we design a system with Lambek’s restriction and cut elimination and another system with Lambek’s restriction and substitution. For both calculi, we prove that they are undecidable, even if we take only one of the two divisions provided by the Lambek calculus. The system with cut elimination and substitution and without Lambek’s restriction is folklore and known to be undecidable.

Multiple conclusion linear logic: cut elimination and more

2020 ◽  
Vol 30 (1) ◽  
pp. 157-174 ◽  
Author(s):  
Harley Eades III ◽  
Valeria de Paiva
Keyword(s):  
Sequent Calculus ◽  
Linear Logic ◽  
Direct Proof ◽  
Cut Elimination ◽  
Tensor Model ◽  
Double Negation ◽  
Proof Nets ◽  
Small Change ◽  
Definition Of ◽  
Intuitionistic Linear Logic

Abstract Full intuitionistic linear logic (FILL) was first introduced by Hyland and de Paiva, and went against current beliefs that it was not possible to incorporate all of the linear connectives, e.g. tensor, par and implication, into an intuitionistic linear logic. Bierman showed that their formalization of FILL did not enjoy cut elimination as such, but Bellin proposed a small change to the definition of FILL regaining cut elimination and using proof nets. In this note we adopt Bellin’s proposed change and give a direct proof of cut elimination for the sequent calculus. Then we show that a categorical model of FILL in the basic dialectica category is also a linear/non-linear model of Benton and a full tensor model of Melliès’ and Tabareau’s tensorial logic. We give a double-negation translation of linear logic into FILL that explicitly uses par in addition to tensor. Lastly, we introduce a new library to be used in the proof assistant Agda for proving properties of dialectica categories.

scholarly journals Sequent Systems without Improper Derivations

2021 ◽  
Author(s):  
Katsumi Sasaki
Keyword(s):  
Propositional Logic ◽  
Natural Deduction ◽  
Inference Rules ◽  
Classical Propositional Logic ◽  
Deduction System ◽  
Structural Rules ◽  
Sequent Systems ◽  
Natural Deduction System ◽  
Sequent System

In the natural deduction system for classical propositional logic given by G. Gentzen, there are some inference rules with assumptions discharged by the rule. D. Prawitz calls such inference rules improper, and others proper. Improper inference rules are more complicated and are often harder to understand than the proper ones. In the present paper, we distinguish between proper and improper derivations by using sequent systems. Specifically, we introduce a sequent system \(\vdash_{\bf Sc}\) for classical propositional logic with only structural rules, and prove that \(\vdash_{\bf Sc}\) does not allow improper derivations in general. For instance, the sequent \(\Rightarrow p \to q\) cannot be derived from the sequent \(p \Rightarrow q\) in \(\vdash_{\bf Sc}\). In order to prove the failure of improper derivations, we modify the usual notion of truth valuation, and using the modified valuation, we prove the completeness of \(\vdash_{\bf Sc}\). We also consider whether an improper derivation can be described generally by using \(\vdash_{\bf Sc}\).

Which structural rules admit cut elimination? An algebraic criterion

2007 ◽  
Vol 72 (3) ◽  
pp. 738-754 ◽  
Author(s):  
Kazushige Terui
Keyword(s):  
Intuitionistic Logic ◽  
General Class ◽  
Residuated Lattices ◽  
Cut Elimination ◽  
Inference Rules ◽  
Lambek Calculus ◽  
Free Logic ◽  
Algebraic Semantics ◽  
Propagation Property ◽  
Structural Rules

AbstractConsider a general class of structural inference rules such as exchange, weakening, contraction and their generalizations. Among them, some are harmless but others do harm to cut elimination. Hence it is natural to ask under which condition cut elimination is preserved when a set of structural rules is added to a structure-free logic. The aim of this work is to give such a condition by using algebraic semantics.We consider full Lambek calculus (FL), i.e., intuitionistic logic without any structural rules, as our basic framework. Residuated lattices are the algebraic structures corresponding to FL. In this setting, we introduce a criterion, called the propagation property, that can be stated both in syntactic and algebraic terminologies. We then show that, for any set ℛ of structural rules, the cut elimination theorem holds for FL enriched with ℛ if and only if ℛ satisfies the propagation property.As an application, we show that any set ℛ of structural rules can be “completed” into another set ℛ*, so that the cut elimination theorem holds for FL enriched with ℛ*. while the provability remains the same.

scholarly journals A note on cut-elimination for classical propositional logic

2021 ◽  
Author(s):  
Gabriele Pulcini
Keyword(s):  
New York ◽  
Propositional Logic ◽  
Order Logic ◽  
Cut Elimination ◽  
Classical Propositional Logic ◽  
First Order ◽  
Sequent System ◽  
Logical Rules ◽  
Specific Formulation ◽  
Automatic Theorem

AbstractIn Schwichtenberg (Studies in logic and the foundations of mathematics, vol 90, Elsevier, pp 867–895, 1977), Schwichtenberg fine-tuned Tait’s technique (Tait in The syntax and semantics of infinitary languages, Springer, pp 204–236, 1968) so as to provide a simplified version of Gentzen’s original cut-elimination procedure for first-order classical logic (Gallier in Logic for computer science: foundations of automatic theorem proving, Courier Dover Publications, London, 2015). In this note we show that, limited to the case of classical propositional logic, the Tait–Schwichtenberg algorithm allows for a further simplification. The procedure offered here is implemented on Kleene’s sequent system G4 (Kleene in Mathematical logic, Wiley, New York, 1967; Smullyan in First-order logic, Courier corporation, London, 1995). The specific formulation of the logical rules for G4 allows us to provide bounds on the height of cut-free proofs just in terms of the logical complexity of their end-sequent.

Test d’Indépendance au Contexte (TIC) et Structure des Représentations Sociales

2008 ◽  
Vol 67 (2) ◽  
pp. 119-123 ◽  
Author(s):  
Grégory Lo Monaco ◽  
Florent Lheureux ◽  
Séverine Halimi-Falkowicz
Keyword(s):  
Double Negation ◽  
Représentations Sociales ◽  
Representations Sociales

Deux techniques permettent le repérage systématique du système central d’une représentation sociale: la technique de la mise en cause (MEC) et le modèle des schèmes cognitifs de base (SCB). Malgré cet apport, ces techniques présentent des inconvénients: la MEC, de par son principe de double négation, et les SCB, de par la longueur de passation. Une nouvelle technique a été développée: le test d’indépendance au contexte (TIC). Elle vise à rendre compte des caractères trans-situationnel ou contingent des éléments représentationnels, tout en présentant un moindre coût cognitif perçu. Deux objets de représentation ont été étudiés auprès d’une population étudiante. Les résultats révèlent que le TIC paraît, aux participants, cognitivement moins coûteux que la MEC. De plus, le TIC permet un repérage du noyau central identique à celui offert par la MEC.

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