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 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 Possession as Linear Knowledge

10.29007/ntkm ◽  
2018 ◽  
Author(s):  
Frank Pfenning
Keyword(s):  
Distributed System ◽  
Epistemic Logic ◽  
Linear Logic ◽  
Expressive Power ◽  
Cut Elimination ◽  
Multi Agent Systems ◽  
Agent Systems ◽  
Ongoing Effort ◽  
Multi Agent ◽  
Localized Knowledge

Epistemic logic analyzes reasoning governing localized knowledge, and is thus fundamental to multi- agent systems. Linear logic treats hypotheses as consumable resources, allowing us to model evolution of state. Combining principles from these two separate traditions into a single coherent logic allows us to represent localized consumable resources and their flow in a distributed system. The slogan “possession is linear knowledge” summarizes the underlying idea. We walk through the design of a linear epistemic logic and discuss its basic metatheoretic properties such as cut elimination. We illustrate its expressive power with several examples drawn from an ongoing effort to design and implement a linear epistemic logic programming language for multi-agent distributed systems.

scholarly journals Cut Elimination Theorem for Non-Commutative Hypersequent Calculus

2017 ◽  
Vol 46 (1/2) ◽  
Author(s):  
Andrzej Indrzejczak
Keyword(s):  
Temporal Logic ◽  
Cut Elimination ◽  
Temporal Logics ◽  
Hypersequent Calculi ◽  
Flow Of Time

Hypersequent calculi (HC) can formalize various non-classical logics. In [9] we presented a non-commutative variant of HC for the weakest temporal logic of linear frames Kt4.3 and some its extensions for dense and serial flow of time. The system was proved to be cut-free HC formalization of respective temporal logics by means of Schütte/Hintikka-style semantical argument using models built from saturated hypersequents. In this paper we present a variant of this calculus for Kt4.3 with a constructive syntactical proof of cut elimination.

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.

scholarly journals NONASSOCIATIVE SUBSTRUCTURAL LOGICS AND THEIR SEMILINEAR EXTENSIONS: AXIOMATIZATION AND COMPLETENESS PROPERTIES

2013 ◽  
Vol 6 (3) ◽  
pp. 394-423 ◽  
Author(s):  
PETR CINTULA ◽  
ROSTISLAV HORČÍK ◽  
CARLES NOGUERA
Keyword(s):  
Algebraic Approach ◽  
Substructural Logics ◽  
Unit Interval ◽  
Residuated Lattices ◽  
Lambek Calculus ◽  
Deduction Theorem ◽  
The Real ◽  
Special Stress ◽  
Ordered Algebras

AbstractSubstructural logics extending the full Lambek calculus FL have largely benefited from a systematical algebraic approach based on the study of their algebraic counterparts: residuated lattices. Recently, a nonassociative generalization of FL (which we call SL) has been studied by Galatos and Ono as the logic of lattice-ordered residuated unital groupoids. This paper is based on an alternative Hilbert-style presentation for SL which is almost MP-based. This presentation is then used to obtain, in a uniform way applicable to most (both associative and nonassociative) substructural logics, a form of local deduction theorem, description of filter generation, and proper forms of generalized disjunctions. A special stress is put on semilinear substructural logics (i.e., logics complete with respect to linearly ordered algebras). Axiomatizations of the weakest semilinear logic over SL and other prominent substructural logics are provided and their completeness with respect to chains defined over the real unit interval is proved.

scholarly journals Proof theory for lattice-ordered groups

10.29007/3szk ◽  
2018 ◽  
Author(s):  
George Metcalfe
Keyword(s):  
Proof Theory ◽  
Family Resemblance ◽  
Substructural Logics ◽  
Residuated Lattices ◽  
Cut Elimination ◽  
Theoretic Approach ◽  
Gentzen Systems ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Mv Algebras

Proof theory can provide useful tools for tackling problems in algebra. In particular, Gentzen systems admitting cut-eliminationhave been used to establish decidability, complexity, amalgamation, admissibility, and generation results for varieties of residuated lattices corresponding to substructural logics. However, for classes of algebras bearing some family resemblance to groups, such as lattice-ordered groups, MV-algebras, BL-algebras, and cancellative residuated lattices, the proof-theoretic approach has met so far only with limited success.The main aim of this talk will be to introduce proof-theoretic methods for the class of lattice-ordered groups and to explain how these methods can be used to obtain new syntactic proofs of two core theorems: namely, Holland's result that this class is generated as a variety by the lattice-ordered group of order-preserving automorphisms of the real numbers, and the decidability of the word problem for free lattice-ordered groups.

Towards an algorithmic construction of cut-elimination procedures

2008 ◽  
Vol 18 (1) ◽  
pp. 81-105 ◽  
Author(s):  
AGATA CIABATTONI ◽  
ALEXANDER LEITSCH
Keyword(s):  
Substructural Logics ◽  
Cut Elimination ◽  
Sequent Calculi ◽  
Structural Rules ◽  
Algorithmic Construction ◽  
Logical Rules

We investigate cut elimination in propositional substructural logics. The problem is to decide whether a given calculus admits (reductive) cut elimination. We show that for commutative single-conclusion sequent calculi containing generalised knotted structural rules and arbitrary logical rules the problem can be decided by resolution-based methods. A general cut-elimination proof for these calculi is also provided.

scholarly journals Hypersequent calculi for non-normal modal and deontic logics: countermodels and optimal complexity

2020 ◽  
Author(s):  
Tiziano Dalmonte ◽  
Björn Lellmann ◽  
Nicola Olivetti ◽  
Elaine Pimentel
Keyword(s):  
Search Strategy ◽  
Cut Elimination ◽  
Relational Semantics ◽  
Optimal Complexity ◽  
Proof Search ◽  
Polynomial Size ◽  
Hypersequent Calculi

Abstract We present some hypersequent calculi for all systems of the classical cube and their extensions with axioms ${T}$, ${P}$ and ${D}$ and for every $n \geq 1$, rule ${RD}_n^+$. The calculi are internal as they only employ the language of the logic, plus additional structural connectives. We show that the calculi are complete with respect to the corresponding axiomatization by a syntactic proof of cut elimination. Then, we define a terminating proof search strategy in the hypersequent calculi and show that it is optimal for coNP-complete logics. Moreover, we show that from every failed proof of a formula or hypersequent it is possible to directly extract a countermodel of it in the bi-neighbourhood semantics of polynomial size for coNP logics, and for regular logics also in the relational semantics. We finish the paper by giving a translation between hypersequent rule applications and derivations in a labelled system for the classical cube.

scholarly journals S5-Style Non-Standard Modalities in a Hypersequent Framework

2021 ◽  
pp. 1-30
Author(s):  
Yaroslav Petrukhin
Keyword(s):  
Cut Elimination ◽  
Hypersequent Calculi ◽  
Soundness And Completeness ◽  
Completeness Theorems

The aim of the paper is to present some non-standard modalities (such as non-contingency, contingency, essence and accident) based on S5-models in a framework of cut-free hypersequent calculi. We also study negated modalities, i.e. negated necessity and negated possibility, which produce paraconsistent and paracomplete negations respectively. As a basis for our calculi, we use Restall's cut-free hypersequent calculus for S5. We modify its rules for the above-mentioned modalities and prove strong soundness and completeness theorems by a Hintikka-style argument. As a consequence, we obtain a cut admissibility theorem. Finally, we present a constructive syntactic proof of cut elimination theorem.

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