Greg Restall · Logic Journal of IGPL · 1994
Journal Article A Useful Substructural Logic Get access GREG RESTALL GREG RESTALL Automated Reasoning Project, Australian National UniversityCanberra, 0200, Australia E-mail: gar@arp.anu.edu.au Search for other works by this author on: Oxford Academic Google Scholar Logic Journal of the IGPL, Volume 2, Issue 2, September 1994, Pages 137–148, https://doi.org/10.1093/jigpal/2.2.137 Published: 01 September 1994 Article history Revision received: 05 April 1994 Received: 01 September 1994 Published: 01 September 1994
Cited by 13
Didier Galmiche, Pierre Kimmel, David Pym · INRIA a CCSD electronic archive server · 2017
We present a substructural epistemic logic, based on Boolean BI, in which the epistemic modalities are parametrized on agents’ local resources. The new modalities can be seen as generalizations of the usual epistemic modalities. The logic combines Boolean Bl’s resource semantics with epistemic agency. We give a labelled tableaux calculus and establish soundness and completeness with respect to the resource semantics. We illustrate the use of the logic by discussing an example of side-channels in access control using resource tokens.
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James H. Andrews, Verónica Dahl, Fred Popowich · The Journal of Logic Programming · 1996
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Natasha Alechina, Michiel van Lambalgen · Journal of Symbolic Logic · 1996
Abstract We show how sequent calculi for some generalized quantifiers can be obtained by generalizing the Herbrand approach to ordinary first order proof theory. Typical of the Herbrand approach, as compared to plain sequent calculus, is increased control over relations of dependence between variables. In the case of generalized quantifiers, explicit attention to relations of dependence becomes indispensible for setting up proof systems. It is shown that this can be done by turning variables into structured objects, governed by various types of structural rules. These structured variables are interpreted semantically by means of a dependence relation. This relation is an analogue of the accessibility relation in modal logic. We then isolate a class of axioms for generalized quantifiers which correspond to first-order conditions on the dependence relation.
Cited by 7Open access
Dexter Kozen, Jerzy Tiuryn · ACM Transactions on Computational Logic · 2003
We formulate a noncommutative sequent calculus for partial correctness that subsumes propositional Hoare Logic. Partial correctness assertions are represented by intuitionistic linear implication. We prove soundness and completeness over relational and trace models. As a corollary, we obtain a complete sequent calculus for inclusion and equivalence of regular expressions.
Cited by 21
Sara Kalvala, Coventry Cv Al, Valeria De Paiva · 2007
sabelle is that it provides a quick route from a formal logic to a proof environment that can be applied directly in verification tasks. 1 Background We build on results from two active areas of research: formal verification of computer systems, and computational logic. Formal verification, whether of software, hardware, or a hybrid of the two, takes as a basis a logic system and applies it as a rigorous, unbiased tool for reasoning about the correctness of the modelled system. Computational logic is concerned, in part, with providing a mathematical model of computation, and relating the different models available. Both these research communities look at similar issues but from different perspectives. Formal verification of dynamic systems Computational systems are designed to effect a change in the environment. Formal verification of a computational system involves modeling the system mathematically and proving properties of the model through rigorou
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Daniel Lehmann · arXiv (Cornell University) · 2022
This paper presents a substructural logic of sequents with very restricted exchange and weakening rules. It is sound with respect to sequences of measurements of a quantic system. A sound and complete semantics is provided. The semantic structures include a binary relation that expresses orthogonality between elements and enables the definition of an operation that generalizes the projection operation in Hilbert spaces. The language has a unitary connective, a sort of negation, and two dual binary connectives that are neither commutative nor associative, sorts of conjunction and disjunction. This provides a logic for quantum measurements whose proof theory is aesthetically pleasing.
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Manuela Busaniche, Roberto Cignoli · CONICET Digital (CONICET) · 2010
Spinks and Veroff have shown that constructive logic with strong negation (CLSN for short), can be considered as a substructural logic. We use algebraic tools developed to study substructural logics to investigate some axiomatic extensions of CLSN. For instance, we prove that Nilpotent minimum logic is the extension of CLSN by the prelinearity axiom. This generalizes the well-known result by Monteiro and Vakarelov that three-valued ukasiewicz logic is an extension of CLSN. A Glivenko-like theorem relating CLSN and three-valued ukasiewicz logic is proved.
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Manuela Busaniche; IMAL-CONICET-UNL, Roberto Cignoli · CIMEC · 2009
Spinks and Veroff have shown that constructive logic with strong negation (CLSN for short), can be considered as a substructural logic. We use algebraic tools developed to study substructural logics to investigate some axiomatic extensions of CLSN. For instance we prove that Nilpotent Minimum Logic is the extension of CLSN by the prelinearity axiom. This generalizes the well known result by Monteiro and Vakarelov that three-valued \\Lukasiewicz logic is an extension of CLSN. A Glivenko-like theorem relating CLSN and three-valued \\Lukasiewicz logic is proved.\n\nAccepted: Journal of Logic and Computation ; doi: 10.1093/logcom/exn081.
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Pfenning, Frank, Simmons, Robert J. · KiltHub Repository · 2009
We describe a substructural logic with ordered, linear, and persistent propositions and then endow a fragment with a committed choice forward-chaining operational interpretation. Exploiting higher-order terms in this metalanguage, we specify the operational semantics of a number of object language features, such as call-by-value, call-by-name, call-by-need, mutable store, parallelism, communication, exceptions and continuations. The specifications exhibit a high degree of uniformity and modularity that allows us to analyze the structural properties required for each feature in isolation. Our substructural framework thereby provides a new methodology for language specification that synthesizes structural operational semantics, abstract machines, and logical approaches.
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Daisuke Bekki · Lecture notes in computer science · 2011
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Spinks, Matthew, Veroff, R. · Open Access CRIS of the University of Bern · 2018
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Matthew Spinks, Robert Veroff · Studia Logica · 2008
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Pfenning, Frank, Simmons, Robert J. · KiltHub Repository · 2009
We describe a substructural logic with ordered, linear, and persistent propositions and then endow a fragment with a committed choice forward-chaining operational interpretation. Exploiting higher-order terms in this metalanguage, we specify the operational semantics of a number of object language features, such as call-by-value, call-by-name, call-by-need, mutable store, parallelism, communication, exceptions and continuations. The specifications exhibit a high degree of uniformity and modularity that allows us to analyze the structural properties required for each feature in isolation. Our substructural framework thereby provides a new methodology for language specification that synthesizes structural operational semantics, abstract machines, and logical approaches.
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J. Michael Dunn, Katalin Bimbó · 1999
The dissertation deals with problems in logic, more precisely, it deals with particular formal systems aiming at capturing patterns of valid reasoning. Sequent calculi were proposed to characterize logical connectives via introduction rules. These systems customarily also have structural rules which allow one to rearrange the set of premises and conclusions. In the structurally free logic of Dunn and Meyer the structural rules are replaced by combinatory rules which allow the same reshuffling of formulae, and additionally introduce an explicit marker for the application of the rule---the combinator. The dissertation gives syntactic extensions of such systems introducing conjunction and disjunction (both distributively and non-distributively). Cut elimination is proved for these systems.
Syntactic systems as a rule are accompanied by a semantics which interprets them. The extended systems are supplied with semantics using algebraic methods. First, the Lindenbaum algebras are formed, and then these algebras are represented in an algebra of sets. This gives a so called Kripke-style semantics for the systems. The representation for the distributive extension utilizes well-known techniques; it uses sets of prime filters and set theoretical operations on them to deal with conjunction and disjunction, plus a ternary accessibility relation to accommodate the intensional operators as fusion and implication. The representation theorem is proven generally, i.e., for arbitrary sets of combinators using the squeeze lemma of Routley and Meyer.
The non-distributive calculi require a different representation since union and intersection are distributive. Two previous representations for lattices are overviewed, and a modification of the Hartonas-Dunn representation is successfully augmented with the intensional operations. Instead of sets of filters, sets of pairs of minimally inconsistent filters and ideals are used and the representation theorem is proven. The representation results provide soundness and completeness for the systems. The dissertation also investigates properties of dual combinators (introduced by Dunn and Meyer), proving a series of lemmas about dual combinatory systems not being Church-Rosser and being (weakly) inconsistent. Combinatorially (un)definable classes of lambda-terms are considered as well.
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E. Haas, Pieter Adriaans · UvA-DARE (University of Amsterdam) · 2000
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Marcelo Finger · Advances in Modal Logic · 2002
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Frank Pfenning, Robert J. Simmons · 2009
We describe a substructural logic with ordered, linear, and persistent propositions and then endow a fragment with a committed choice forward-chaining operational interpretation. Exploiting higher-order terms in this metalanguage, we specify the operational semantics of a number of object language features, such as call-by-value, call-by-name, call-by-need, mutable store, parallelism, communication, exceptions and continuations. The specifications exhibit a high degree of uniformity and modularity that allows us to analyze the structural properties required for each feature in isolation. Our substructural framework thereby provides a new methodology for language specification that synthesizes structural operational semantics, abstract machines, and logical approaches.
Cited by 34
Shawn Standefer · Journal of Logic Language and Information · 2022
Cited by 3
Ryo Kashima, Norihiro Kamide · Tokyo Tech Research Repository (Tokyo Institute of Technology) · 2006
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