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Semantics for Intuitionistic Arithmetic Based on Tarski Games with Retractable Moves [chapter]

Stefano Berardi
2007 Lecture Notes in Computer Science  
We define an effective, sound and complete game semantics for HA inf , Intuitionistic Arithmetic with ω-rule.  ...  Our semantics is equivalent to the original semantics proposed by Lorentzen [6], but it is based on the more recent notions of "backtracking" ([5], [2] ) and of isomorphism between proofs and strategies  ...  In this paper we adapt Coquand's game semantics of PA inf to game semantics of Intuitionistic Arithmetic HA inf with intuitionistic implication ⇒.  ... 
doi:10.1007/978-3-540-73228-0_4 fatcat:6roel7tou5gshklmqiq4bl3weq

Learning, Realizability and Games in Classical Arithmetic [article]

Federico Aschieri
2011 arXiv   pre-print
In particular, we show there is a one-to-one correspondence between realizers and recursive winning strategies in the 1-Backtracking version of Tarski games.  ...  Secondly, we extend the class of learning based realizers to a classical version PCFclass of PCF and, then, compare the resulting notion of realizability with Coquand game semantics and prove a full soundness  ...  In [14] , Coquand introduced a new game semantics for Peano Arithmetic, centered on the concept of "Backtracking Tarski game": a special Tarski game in which players have the additional possibility of  ... 
arXiv:1012.4992v5 fatcat:4iksfoith5b6zjarfihdtosr3e

A Cook's tour of duality in logic: from quantifiers, through Vietoris, to measures [article]

Mai Gehrke, Tomas Jakl, Luca Reggio
2020 arXiv   pre-print
In particular, we focus on the use of (i) topological duality methods to solve problems in logic and computer science; (ii) category theory and, more particularly, free (and co-free) constructions; (iii  ...  Other examples include large parts of modal and intuitionistic logics, where Jónsson-Tarski duality yields Kripke semantics (Blackburn et al., 2001) .  ...  For one, duality theory often connects syntax and semantics.  ... 
arXiv:2007.15415v1 fatcat:d7fs2d66tffg3m7o5ob6xz6tt4

Proof-search in type-theoretic languages: an introduction

Didier Galmiche, David J. Pym
2000 Theoretical Computer Science  
The strong connections between di erent proof-search methods in logics, type theories and logical frameworks, together with their impact on programming and implementation issues, are central in this context  ...  Acknowledgements We are grateful to Maurice Nivat for encouraging us to produce this article and for acting as a referee. Any remaining errors are our responsibility.  ...  A ÿrst attempt has been made with a games semantics for linear logic [1] in which formulae denote games and proofs denote winning strategies.  ... 
doi:10.1016/s0304-3975(99)00169-3 fatcat:gy4nmmsupfeype23pw3jzbrvwu

Automated deduction for verification

Natarajan Shankar
2009 ACM Computing Surveys  
Truth table semantics for ¬ and ∨. assignment { p → , q → }. On the other hand, p ∧ ¬p is not satisfiable.  ...  We then move on to proof search procedures (Section 4) which typically work in domains where the validity problem is not decidable.  ...  Bill Mc-Cune and Matt Kaufmann responded with clarifications on specific points.  ... 
doi:10.1145/1592434.1592437 fatcat:satgicglyneqvl2nw5pjnr3d3a


1976 Philosophical Books  
to semantic context.  ...  Arguments similar to those given to defend the IC theory show that with one minor alteration, Kripke's formal theory may be used to model the IC notion of truth. ACKNOWLEDGEMENTS  ...  ACKNOWLEDGEMENTS I wish to thank my committee members James Van Cleve, Ernest Sosa, and Nathan Salmon for their help with this dissertation, and for their ferocious sense of justice in doing philosophy  ... 
doi:10.1111/j.1468-0149.1976.tb01645.x fatcat:q45zokr7rneqjnkre443kjtnva

Human Rationality Challenges Universal Logic

Brian R. Gaines
2010 Logica Universalis  
At an abstract level it focuses on consequence operators as the primary unit of analysis in modeling reasoning, and represents them as relations on the powerset generated by an arbitrary set, with no a  ...  Cellucci [46] has discussed the parallels between the basic anticipatory processes of all organisms and the development of formal proofs in mathematics and noted that "analytic proof is based on the procedure  ...  Consolidate research on computer-based inference engines for the logics analyzed in challenges 10 through 13 to design an inference engine for universal logic with a plug-in architecture for various axiom  ... 
doi:10.1007/s11787-010-0019-9 fatcat:xbhnkoblvrhfnjkomkqcowgi5i

Interactive Realizers and Monads [article]

Stefano Berardi, Ugo de'Liguoro
2010 arXiv   pre-print
We propose a realizability interpretation of a system for quantifier free arithmetic which is equivalent to the fragment of classical arithmetic without "nested" quantifiers, called here EM1-arithmetic  ...  We interpret classical proofs as interactive learning strategies, namely as processes going through several stages of knowledge and learning by interacting with the "environment" and with each other.  ...  Coquand's game semantics of classical arithmetic. A primary source of the present research is Coquand's semantics of evidence for classical arithmetic [Coquand 1991 ].  ... 
arXiv:1005.2907v1 fatcat:lkkcc2gqgzetxcdazgaz5uihou

On Full Abstraction for PCF: I, II, and III

J.M.E. Hyland, C.-H.L. Ong
2000 Information and Computation  
Note that with 288 HYLAND AND ONG 2 In [69] Scott considered a version of PCF based on typed S-and K-combinators.  ...  Our model of computation is based on a kind of game in which each play consists of a dialogue of questions and answers between two players who observe the following principles of civil conversation: view-dependent  ...  ACKNOWLEDGMENTS Pierre-Louis Curien has given us helpful comments on a draft of this paper. We thank him for his interest in our work.  ... 
doi:10.1006/inco.2000.2917 fatcat:jktaw5yrsjdxfmh3uvaktomxqq

A Brief History of Truth [chapter]

Stewart Candlish, Nic Damnjanovic
2007 Philosophy of Logic  
Thus, based on his diagnosis of the conditions that allow the liar paradox to be generated for the 'ordinary' truth-predicate, Tarski proposed a number of formal constraints on any adequate definition  ...  Putnam thought that this conclusion is absurd, and that, for reasons that are obscure to us, a semantics based on the idea that truth is ideal justification is the only way to avoid it.  ... 
doi:10.1016/b978-044451541-4/50012-9 fatcat:dywvtt3g5ndcrgxrbh7wkppbgu

On the finitizability problem in algebraic logic; recent results [article]

Tarek Sayed Ahmed
2013 arXiv   pre-print
This is a survey article in algebraic logic, where we take a magical tour from old concepts due to Henkin, Monk and Tarski like neat embeddings, to modern views and perspectives, culminating in the use  ...  It is based on Henkin's ideas of changing the semantics to obtain a completeness theorem for higher order logics. The second term comes from the Amsterdam Group, in fact it is due to Venema [182] .  ...  This is similar to studying non-standard models of arithmetic, that do shed light on the standard model.  ... 
arXiv:1302.1368v1 fatcat:bguykr2afvfajda4qskyr265ge


2012 New Mathematics and Natural Computation  
Finally, I see no evidence for the practical need for new axioms to settle open arithmetical and …nite combinatorial problems."  ...  The Berry Paradox deals with the notion of …nite de…nability and is semantical in nature.  ... 
doi:10.1142/s1793005712400017 fatcat:iuupodfe2jflnjtwdqhejvwtoi

Foundations for Computable Topology [chapter]

Paul Taylor
2011 The Western Ontario Series in Philosophy of Science  
Foundations should be designed for the needs of mathematics and not vice versa.  ...  This method is applied to devising a new paradigm for general topology, called Abstract Stone Duality.  ...  helpful comments on earlier drafts of this paper.  ... 
doi:10.1007/978-94-007-0431-2_14 fatcat:cjdzdpazrrfajduvnd7s3cndoy

On the Semantics of Intensionality and Intensional Recursion [article]

G. A. Kavvos
2017 arXiv   pre-print
This thesis is concerned with the crafting of a logical toolkit through which these phenomena can be studied.  ...  This is a familiar setting for computer scientists, who often study different programs or processes that are interchangeable, i.e. extensionally equal, even though they are not implemented in the same  ...  Tarski's Undefinability Theorem, on the other hand is the result that truth cannot be defined in arithmetic [Smullyan, 1992] . Theorem 47 (Tarski).  ... 
arXiv:1712.09302v1 fatcat:5jfwgc2yujglraaxsuqr7jvmqi

Discrete Mathematics for Computer Science, Some Notes [article]

Jean Gallier
2008 arXiv   pre-print
These are notes on discrete mathematics for computer scientists. The presentation is somewhat unconventional.  ...  Indeed I begin with a discussion of the basic rules of mathematical reasoning and of the notion of proof formalized in a natural deduction system "a la Prawitz".  ...  So far, we have presented a very proof-theoretic view of logic, that is, a view based on the notion of provability as opposed to a more semantic view of based on the notions of truth and models.  ... 
arXiv:0805.0585v1 fatcat:pzc5agtaxvh3hpqo3yvygy2sjq
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