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Formal Theories for Linear Algebra [chapter]

Stephen Cook, Lila Fontes
2010 Lecture Notes in Computer Science  
We introduce two-sorted theories in the style of [CN10] for the complexity classes ⊕L and DET , whose complete problems include determinants over Z2 and Z, respectively.  ...  The result shows equivalences of standard theorems of linear algebra over Z2 and Z can be proved in the corresponding theory, but leaves open the interesting question of whether the theorems themselves  ...  The underlying logic used by LAp is Gentzen's propositional sequent system, in which all axioms are valid sequents and for each rule, the bottom sequent is a logical consequence of the top sequents.  ... 
doi:10.1007/978-3-642-15205-4_21 fatcat:psmpgg5gubgfdmva5qxi74tgyi

Canonical Signed Calculi, Non-deterministic Matrices and Cut-Elimination [chapter]

Arnon Avron, Anna Zamansky
2008 Lecture Notes in Computer Science  
Modular semantics based on nondeterministic matrices are provided for every coherent canonical signed calculus.  ...  Canonical propositional Gentzen-type calculi are a natural class of systems which in addition to the standard axioms and structural rules have only logical rules where exactly one occurrence of a connective  ...  This is generalized to nondeterministic matrices in [3] , where a signed canonical calculus is constructed for every finite Nmatrix.  ... 
doi:10.1007/978-3-540-92687-0_3 fatcat:d5bdhvknjraaxnvs4f4jbkpxou

Rough Sets and 3-Valued Logics

A. Avron, B. Konikowska
2008 Studia Logica: An International Journal for Symbolic Logic  
We give sound and complete, cut-free sequent calculi for both versions of the logic generated by the rough set Nmatrix.  ...  Then we derive from these calculi sequent calculi with the same properties for the various "determinizations" of those two versions of the logic (including Lukasiewicz 3-valued logic).  ...  This research was supported by the THE ISRAEL SCIENCE FOUNDATION (grant No 809-06) and by the grant N N206 399334 of Polish Ministry of Science and Higher Education. Rough Sets and 3-valued Logics  ... 
doi:10.1007/s11225-008-9144-3 fatcat:emspkawupvgk3a3intsw2ggodm

Validity, dialetheism and self-reference

Federico Matias Pailos
2018 Synthese  
We show that LP Val is sound and complete with respect to the three-sided sequent calculus SLP Val . Moreover, LP Val can be safely expanded with a transparent truth predicate.  ...  We will also present an alternative theory LP Val * , which includes a nondeterministic validity predicate.  ...  Matrices for the logic LP Our goal, then, is to internalize the notion of naive validity.  ... 
doi:10.1007/s11229-018-1731-x fatcat:ptcmksy3v5g55gcrsv5ccf2whu

Proof Systems for Reasoning about Computation Errors

Arnon Avron, Beata Konikowska
2009 Studia Logica: An International Journal for Symbolic Logic  
For the resulting logics, we provide sound and complete calculi of ordinary, two-valued sequents.  ...  If we want to be able to deal with both strategies without distinguishing between them, we combine Kleene and McCarthy logics into a logic based on a non-deterministic, 3-valued matrix, incorporating both  ...  For all the three considered logics, we provide sound and complete calculi of ordinary sequents.  ... 
doi:10.1007/s11225-009-9175-4 fatcat:wd4d6ionmjf3botl773vlyx4iy

Two-valued logics for naive truth theory

Lucas Daniel Rosenblatt
2015 The Australasian Journal of Logic  
It is part of the current wisdom that the Liar and similar semanticparadoxes can be taken care of by the use of certain non-classicalmultivalued logics.  ...  In this paper I want to suggest that bivalent logic can do just as well. This is accomplished by using a non-deterministic matrix to define the negation connective.  ...  I am grateful to the audiences of those conferences for their comments and suggestions.  ... 
doi:10.26686/ajl.v12i1.2082 fatcat:b6bdrelzqrfetmp5wvrdfru23y

Focusing on pattern matching

Neelakantan R. Krishnaswami
2008 Proceedings of the 36th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '09  
We show how to extend the Curry-Howard correspondence to pattern matching, by showing how it arises as a natural proof term assignment to a focused sequent calculus for propositional logic.  ...  We demonstrate the value of this calculus by deriving a simple, novel algorithm to check the exhaustiveness of pattern matching, and to reconstruct a program transformation corresponding to compiling patterns  ...  Acknowledgments The author would like to thank Jonathan Aldrich, Robert Harper, Dan Licata, William Lovas, Frank Pfenning, Jason Reed, Kevin Watkins, and Noam Zeilberger for valuable encouragement and  ... 
doi:10.1145/1480881.1480927 dblp:conf/popl/Krishnaswami09 fatcat:olw65mezerf65fwhwgbfhmvs7a

Formal Theories for Linear Algebra

Stephen Cook, Lila Fontes, Anuj Dawar
2012 Logical Methods in Computer Science  
We introduce two-sorted theories in the style of [CN10] for the complexity classes \oplusL and DET, whose complete problems include determinants over Z2 and Z, respectively.  ...  The result shows equivalences of standard theorems of linear algebra over Z2 and Z can be proved in the corresponding theory, but leaves open the interesting question of whether the theorems themselves  ...  Since our translation of formulas and sequents is transparent to the logical connectives, the logical axioms and rule applications for LAp (the propositional part of Gentzen's system LK) translate to logical  ... 
doi:10.2168/lmcs-8(1:25)2012 fatcat:ubvieluiijddxjszuj3pqgskme

Focusing on pattern matching

Neelakantan R. Krishnaswami
2009 SIGPLAN notices  
We show how to extend the Curry-Howard correspondence to pattern matching, by showing how it arises as a natural proof term assignment to a focused sequent calculus for propositional logic.  ...  We demonstrate the value of this calculus by deriving a simple, novel algorithm to check the exhaustiveness of pattern matching, and to reconstruct a program transformation corresponding to compiling patterns  ...  Acknowledgments The author would like to thank Jonathan Aldrich, Robert Harper, Dan Licata, William Lovas, Frank Pfenning, Jason Reed, Kevin Watkins, and Noam Zeilberger for valuable encouragement and  ... 
doi:10.1145/1594834.1480927 fatcat:ddttavcfp5dpnmjjlse73jdq2y

Dynamic Logics of Dynamical Systems [article]

André Platzer
2012 arXiv   pre-print
., induction principles for differential equations. We survey theoretical results, including soundness and completeness and deductive power.  ...  We survey dynamic logics for specifying and verifying properties of dynamical systems, including hybrid systems, distributed hybrid systems, and stochastic hybrid systems.  ...  ACKNOWLEDGMENTS I am grateful to Jan-David Quesel for indispensable help with implementing KeYmaera and to David Renshaw for implementing KeYmaeraD.  ... 
arXiv:1205.4788v1 fatcat:2ahunsfdzfamdiwzd66m5e4wzq

Non-Deterministic Semantics for Logical Systems [chapter]

Arnon Avron, Anna Zamansky
2010 Handbook of Philosophical Logic  
We begin with some preliminaries and a review of many-valued matrices in Section 2. The basic definitions of the framework of Nmatrices are presented in Section 3.  ...  In Section 5 we extend the modular approach to two non-canonical families of Gentzentype calculi: those that are obtained from the positive fragments of classical logic and intuitionistic logic by adding  ...  INTRODUCTION The Key Idea The principle of truth-functionality (or compositionality) is a basic principle in many-valued logic in general, and in classical logic in particular.  ... 
doi:10.1007/978-94-007-0479-4_4 fatcat:wibs2sitz5hnto45c5ctgprpbu

An Epistemic Interpretation of Paraconsistent Weak Kleene Logic

Damian E. Szmuc
2019 Logic and Logical Philosophy  
For a generalization of this notion that also applies to nondeterministic matrices (as defined in [3]) see [51] . 7 Fitting actually denotes the operations ⊓ and ⊔ with the symbols ∧ and ∨, respectively  ...  These considerations motivate a special approach to defining sound and complete Gentzen-style sequent calculi for some of their four-valued generalizations. 1 It should be noted that PWK and K w 3 are  ...  I would like to thank Eduardo Barrio, Federico Pailos, Lucas Rosenblatt, the members of the Buenos Aires Logic Group, and an anonymous reviewer for their comments and suggestions.  ... 
doi:10.12775/llp.2019.014 fatcat:v3kkazb5grcjhduzwsktwy447a

Matrix-Based Constructive Theorem Proving [chapter]

Christoph Kreitz, Jens Otten, Stephan Schmitt, Brigitte Pientka
2000 Applied Logic Series  
Bibel's connection method, originally developed for classical logic, has been extended to a variety of non-classical logics and used to control the creation of sequent proofs for interactive proof assistants  ...  Matrix methods for intuitionistic logic also are the central inference engine for program synthesis and verification.  ...  Proving Theorems in Constructive Logic As program synthesis and verification often relies on constructive arguments proof assistants should be supported by automatic proof search procedures for intuitionistic  ... 
doi:10.1007/978-94-015-9383-0_12 fatcat:psdwy6yocnhtve5oitvdf7lsxy

Defining LFIs and LFUs in extensions of infectious logics

Damian Enrique Szmuc
2016 Journal of Applied Non-Classical Logics  
Here, we extend these logics with classical negations, and we furthermore show that some of these extended systems can be properly regarded as Logics of Formal Inconsistency (LFIs) and Logics of Formal  ...  , such as the ones due to Bochvar and Halldén, among others.  ...  Acknowledgements I am grateful to Eduardo Barrio, Federico Pailos, Lucas Rosenblatt and all the members of BA-LOGIC (Buenos Aires Logic Group) for their comments on earlier versions of this paper.  ... 
doi:10.1080/11663081.2017.1290488 fatcat:czle7u7gxrbili5z42fmvkqzyu

Model checking for the practical verificationist

Lee Pike
2007 Proceedings of the second workshop on Automated formal methods - AFM '07  
and model checker interplay.  ...  In particular, I discuss the use of higher-order functions in model checking, infinitestate bounded model checking, compositional specification and verification, and finally, mechanical theorem prover  ...  I received detailed comments from the workshop's anonymous reviewers and from Levent Erkök at Galois, Inc.  ... 
doi:10.1145/1345169.1345172 fatcat:oqszu4vnnrgeniulyw4o36uula
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