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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.
Polycategories via pseudodistributive laws
"... In this paper, we give a novel abstract description of Szabo’s polycategories. We use the theory of double clubs – a generalisation of Kelly’s theory of clubs to ‘pseudo ’ (or ‘weak’) double categories – to construct a pseudodistributive law of the free symmetric strict monoidal category pseudocomo ..."
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In this paper, we give a novel abstract description of Szabo’s polycategories. We use the theory of double clubs – a generalisation of Kelly’s theory of clubs to ‘pseudo ’ (or ‘weak’) double categories – to construct a pseudodistributive law of the free symmetric strict monoidal category pseudocomonad on Mod over itself qua pseudomonad, and show that monads in the ‘twosided Kleisli bicategory’ of this pseudodistributive law are precisely symmetric polycategories. 1
Rescorla, Quarterly
 Journal of Experimental Psychology
, 2003
"... Abstract. We show that Shoenfield’s functional interpretation of Peano arithmetic can be factorized as a negative translation due to J. L. Krivine followed by Gödel’s Dialectica interpretation. Mathematics Subject Classification: 03F03, 03F10. ..."
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Abstract. We show that Shoenfield’s functional interpretation of Peano arithmetic can be factorized as a negative translation due to J. L. Krivine followed by Gödel’s Dialectica interpretation. Mathematics Subject Classification: 03F03, 03F10.
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
Modified realizability interpretation of classical linear logic
 In Proceedings of Logic in Computer Science (LiCS
, 2007
"... This paper presents a modified realizability interpretation of classical linear logic. The interpretation is based on work of de Paiva (1989), Blass (1995), and Shirahata (2006) on categorical models of classical linear logic using Gödel’s Dialectica interpretation. Whereas the Dialectica categories ..."
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This paper presents a modified realizability interpretation of classical linear logic. The interpretation is based on work of de Paiva (1989), Blass (1995), and Shirahata (2006) on categorical models of classical linear logic using Gödel’s Dialectica interpretation. Whereas the Dialectica categories provide models of linear logic, our interpretation is presented as an endointerpretation of proofs, which does not leave the realm of classical linear logic. The advantage is that we obtain stronger versions of the disjunction and existence properties, and new conservation results for certain choice principles. Of particular interest is the simple branching quantifier used in order to obtain a completeness result for the modified realizability interpretation. 1
A categorical semantics for polarized mall
 Ann. Pure Appl. Logic
"... In this paper, we present a categorical model for Multiplicative Additive Polarized Linear Logic MALLP, which is the linear fragment (without structural rules) of Olivier Laurent’s Polarized Linear Logic. Our model is based on an adjunction between reflective/coreflective full subcategories C−/C+ of ..."
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In this paper, we present a categorical model for Multiplicative Additive Polarized Linear Logic MALLP, which is the linear fragment (without structural rules) of Olivier Laurent’s Polarized Linear Logic. Our model is based on an adjunction between reflective/coreflective full subcategories C−/C+ of an ambient ∗autonomous category C (with products). Similar structures were first introduced by M. Barr in the late 1970’s in abstract duality theory and more recently in work on game semantics for linear logic. The paper has two goals: to discuss concrete models and to present various completeness theorems. As concrete examples, we present (i) a hypercoherence model, using Ehrhard’s hereditary/antihereditary objects, (ii) a Chuspace model, (iii) a double gluing model over our categorical framework, and (iv) a model based on iterated double gluing over a ∗autonomous category. For the multiplicative fragment MLLP of MALLP, we present both weakly full (Läuchlistyle) as well as full completeness theorems, using a polarized version of functorial
Towards Hilbert's 24th Problem: Combinatorial Proof Invariants
, 2006
"... Proofs Without Syntax [37] introduced polynomialtime checkable combinatorial proofs for classical propositional logic. This sequel approaches Hilbert’s 24 th Problem with combinatorial proofs as abstract invariants for sequent calculus proofs, analogous to homotopy groups as abstract invariants for ..."
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Proofs Without Syntax [37] introduced polynomialtime checkable combinatorial proofs for classical propositional logic. This sequel approaches Hilbert’s 24 th Problem with combinatorial proofs as abstract invariants for sequent calculus proofs, analogous to homotopy groups as abstract invariants for topological spaces. The paper lifts a simple, strongly normalising cut elimination from combinatorial proofs to sequent calculus, factorising away the mechanical commutations of structural rules which litter traditional syntactic cut elimination. Sequent calculus fails to be surjective onto combinatorial proofs: the paper extracts a semantically motivated closure of sequent calculus from which there is a surjection, pointing towards an abstract combinatorial refinement of Herbrand’s theorem.
Cartesian Closed Dialectica Categories
"... When Gödel developed his functional interpretation, also known as the Dialectica interpretation, his aim was to prove (relative) consistency of first order arithmetic by reducing it to a quantifierfree theory with finite types. Like other functional interpretations (e.g. Kleene’s realizability in ..."
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When Gödel developed his functional interpretation, also known as the Dialectica interpretation, his aim was to prove (relative) consistency of first order arithmetic by reducing it to a quantifierfree theory with finite types. Like other functional interpretations (e.g. Kleene’s realizability interpretation and Kreisel’s modified realizability) Gödel’s Dialectica interpretation gives rise to category theoretic constructions that serve both as new models for logic and semantics and as tools for analysing and understanding various aspects of the Dialectica interpretation itself. Gödel’s Dialectica interpretation gives rise to the Dialectica categories (described by V. de Paiva in [dP89] and J.M.E. Hyland in [Hyl02]). These categories are symmetric monoidal closed and have finite products and weak coproducts, but they are not Cartesian closed in general. We give an analysis of how to obtain weakly Cartesian closed and Cartesian closed Dialectica categories, and we also reflect on what the analysis might tell us about the Dialectica interpretation. 1