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31
Orderenriched categorical models of the classical sequent calculus
 LECTURE AT INTERNATIONAL CENTRE FOR MATHEMATICAL SCIENCES, WORKSHOP ON PROOF THEORY AND ALGORITHMS
, 2003
"... It is wellknown that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. Starting from a convenient formulation of the wellknown categorical semantics of linear classical sequent proofs, we give models of weakening and contra ..."
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It is wellknown that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. Starting from a convenient formulation of the wellknown categorical semantics of linear classical sequent proofs, we give models of weakening and contraction that do not collapse. Cutreduction is interpreted by a partial order between morphisms. Our models make no commitment to any translation of classical logic into intuitionistic logic and distinguish nondeterministic choices of cutelimination. We show soundness and completeness via initial models built from proof nets, and describe models built from sets and relations.
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 21 (7 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.
On the axiomatisation of boolean categories with and without medial
 THEORY APPL. CATEG
, 2007
"... ..."
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.
Expansion nets: proofnets for propositional classical logic
 IN PROCEEDINGS OF THE 17TH INTERNATIONAL CONFERENCE ON LOGIC FOR PROGRAMMING, ARTIFICIAL INTELLIGENCE, AND REASONING, LPAR’10
, 2010
"... We give a calculus of proofnets for classical propositional logic. These nets improve on a proposal due to Robinson by validating the associativity and commutativity of contraction, and provide canonical representants for classical sequent proofs modulo natural equivalences. We present the relation ..."
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Cited by 4 (1 self)
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We give a calculus of proofnets for classical propositional logic. These nets improve on a proposal due to Robinson by validating the associativity and commutativity of contraction, and provide canonical representants for classical sequent proofs modulo natural equivalences. We present the relationship between sequent proofs and proofnets as an annotated sequent calculus, deriving formulae decorated with expansion/deletion trees. We then see a subcalculus, expansion nets, which in addition to these good properties has a polynomialtime correctness criterion.
Breaking Paths in Atomic Flows for Classic Logic
, 2010
"... This work belongs to a wider effort aimed at eliminating syntactic bureaucracy from proof systems. In this paper, we present a novel cut elimination procedure for classical propositional logic. It is based on the recently introduced away from much of the typical bureaucracy of proofs. We make cruci ..."
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Cited by 4 (2 self)
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This work belongs to a wider effort aimed at eliminating syntactic bureaucracy from proof systems. In this paper, we present a novel cut elimination procedure for classical propositional logic. It is based on the recently introduced away from much of the typical bureaucracy of proofs. We make crucial use of the path breaker, an atomicflow construction that avoids some nasty termination problems, and that can be used in any proof system with sufficient symmetry. This paper contains an original 2dimensionaldiagram exposition of atomic flows, which helps us to connect atomic flows with other known formalisms.
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 4 (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 categorical models of classical logic and the geometry of interaction
, 2005
"... It is wellknown that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. In previous work, summarized briefly herein, we have provided a class of models called classical categories which is sound and complete and avoids this co ..."
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It is wellknown that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. In previous work, summarized briefly herein, we have provided a class of models called classical categories which is sound and complete and avoids this collapse by interpreting cutreduction by a posetenrichment. Examples of classical categories include boolean lattices and the category of sets and relations, where both conjunction and disjunction are modelled by the settheoretic product. In this article, which is selfcontained, we present an improved axiomatization of classical categories, together with a deep exploration of their structural theory. Observing that the collapse already happens in the absence of negation, we start with negationfree models called Dummett categories. Examples include, besides the classical categories above, the category of sets and relations, where both conjunction and disjunction are modelled by the disjoint union. We prove that Dummett categories are MIX, and that the partial order can be derived from homsemilattices which have a straightforward prooftheoretic definition. Moreover, we show that the GeometryofInteraction construction can be extended from multiplicative linear logic to classical logic, by applying it to obtain a classical
Classical categories and deep inference
"... Deep inference is a prooftheoretic notion in which proof rules apply arbitrarily deeply inside a formula. We show that the essense of deep inference is the bifunctorality of the connectives. We demonstrate that, when given an inequational theory that models cutreduction, a deep inference calculus ..."
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Cited by 4 (0 self)
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Deep inference is a prooftheoretic notion in which proof rules apply arbitrarily deeply inside a formula. We show that the essense of deep inference is the bifunctorality of the connectives. We demonstrate that, when given an inequational theory that models cutreduction, a deep inference calculus for classical logic (SKSg) is a categorical model of the classical sequent calculus LK in the sense of Führmann and Pym. We uncover a mismatch between this notion of cutreduction and the usual notion of cut in SKSg. Viewing SKSg as a model of the sequent calculus uncovers new insights into the Craig interpolation lemma and intuitionistic provablility.
What is the Problem with Proof Nets for Classical Logic?
"... Abstract. This paper is an informal (and nonexhaustive) overview over some existing notions of proof nets for classical logic, and gives some hints why they might be considered to be unsatisfactory. 1 ..."
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Abstract. This paper is an informal (and nonexhaustive) overview over some existing notions of proof nets for classical logic, and gives some hints why they might be considered to be unsatisfactory. 1