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Naming proofs in classical propositional logic
 IN PAWE̷L URZYCZYN, EDITOR, TYPED LAMBDA CALCULI AND APPLICATIONS, TLCA 2005, VOLUME 3461 OF LECTURE
"... We present a theory of proof denotations in classical propositional logic. The abstract definition is in terms of a semiring of weights, and two concrete instances are explored. With the Boolean semiring we get a theory of classical proof nets, with a geometric correctness criterion, a sequentiali ..."
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Cited by 24 (8 self)
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We present a theory of proof denotations in classical propositional logic. The abstract definition is in terms of a semiring of weights, and two concrete instances are explored. With the Boolean semiring we get a theory of classical proof nets, with a geometric correctness criterion, a sequentialization theorem, and a strongly normalizing cutelimination procedure. This gives us a “Boolean ” category, which is not a poset. With the semiring of natural numbers, we obtain a sound semantics for classical logic, in which fewer proofs are identified. Though a “real” sequentialization theorem is missing, these proof nets have a grip on complexity issues. In both cases the cut elimination procedure is closely related to its equivalent in the calculus of structures.
From proof nets to the free * autonomous category
 Logical Methods in Computer Science, 2(4:3):1–44, 2006. Available from: http://arxiv.org/abs/cs/0605054. [McK05] Richard McKinley. Classical categories and deep inference. In Structures and Deduction 2005 (Satellite Workshop of ICALP’05
, 2005
"... Vol. 2 (4:3) 2006, pp. 1–44 www.lmcsonline.org ..."
L.: Constructing free Boolean categories
, 2005
"... By Boolean category we mean something which is to a Boolean algebra what a category is to a poset. We propose an axiomatic system for Boolean categories, which is different in several respects from the ones proposed recently. In particular everything is done from the start in a *autonomous category ..."
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Cited by 21 (6 self)
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By Boolean category we mean something which is to a Boolean algebra what a category is to a poset. We propose an axiomatic system for Boolean categories, which is different in several respects from the ones proposed recently. In particular everything is done from the start in a *autonomous category and not in a weakly distributive one, which simplifies issues like the Mix rule. An important axiom, which is introduced later, is a “graphical ” condition, which is closely related to denotational semantics and the Geometry of Interaction. Then we show that a previously
Categorical Proof Theory of Classical Propositional Calculus
, 2005
"... We investigate semantics for classical proof based on the sequent calculus. We show that the propositional connectives are not quite wellbehaved from a traditional categorical perspective, and give a more refined, but necessarily complex, analysis of how connectives may be characterised abstractly. ..."
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Cited by 9 (1 self)
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We investigate semantics for classical proof based on the sequent calculus. We show that the propositional connectives are not quite wellbehaved from a traditional categorical perspective, and give a more refined, but necessarily complex, analysis of how connectives may be characterised abstractly. Finally we explain the consequences of insisting on more familiar categorical behaviour.
Deep Inference Proof Theory Equals Categorical Proof Theory Minus Coherence
, 2004
"... This paper links deep inference proof theory, as studied by Guglielmi et al., to categorical proof theory in the sense of Lambek et al.. It observes how deep inference proof theory is categorical proof theory, minus the coherence diagrams/laws. Coherence yields a readymade and well studied notion o ..."
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Cited by 5 (1 self)
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This paper links deep inference proof theory, as studied by Guglielmi et al., to categorical proof theory in the sense of Lambek et al.. It observes how deep inference proof theory is categorical proof theory, minus the coherence diagrams/laws. Coherence yields a readymade and well studied notion of equality on deep inference proofs. The paper notes a precise correspondence between the symmetric deep inference system for multiplicative linear logic (the linear fragment of SKSg) and the presentation of #autonomous categories as symmetric linearly distributive categories with negation. Contraction and weakening in SKSg corresponds precisely to the presence of (co)monoids.
ON THE AXIOMATISATION OF BOOLEAN CATEGORIES WITH AND WITHOUT MEDIAL
"... should be used for describing an object that ..."
Proofs as Polynomials
"... Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be ..."
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Cited by 2 (1 self)
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Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be