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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 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.
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 17 (5 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
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.
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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Cited by 4 (0 self)
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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
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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Cited by 3 (0 self)
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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.
V VIContents Invited Lecture
, 2009
"... The topic of this workshop is the application of algebraic, geometric, and combinatorial methods in proof theory. In recent years many researchers have proposed approaches to understand and reduce ”syntactic beaucracy ” in the presentation of proofs. Examples are proof nets, atomic flows, new deduct ..."
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The topic of this workshop is the application of algebraic, geometric, and combinatorial methods in proof theory. In recent years many researchers have proposed approaches to understand and reduce ”syntactic beaucracy ” in the presentation of proofs. Examples are proof nets, atomic flows, new deductive systems based on deep inference, and new algebraic semantics for proofs. These efforts have also led to new methods of proof normalisation and new results in proof complexity. The workshop is relevant to a wide range of people. The list of topics includes among others: algebraic semantics of proofs, game semantics, proof
Frobenius Algebras and Classical Proof Nets
"... The semantics of proofs for classical logic is a very recent discipline; the construction of proofs semantics that are completely faithful to the natural symmetries of classical logic is even more recent. In this paper we present a theory of proof nets which is related to those in [LS05,Hyl04,FP05], ..."
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The semantics of proofs for classical logic is a very recent discipline; the construction of proofs semantics that are completely faithful to the natural symmetries of classical logic is even more recent. In this paper we present a theory of proof nets which is related to those in [LS05,Hyl04,FP05], but which differs from them in its ability to take account of resources, in the sense of linear logic. It also has the interesting property (like [Hyl04]) of being based on a topological foundation. This work originated as an investigation in the denotational semantics of classical logic [LN09], furthering the work in [Lam07]. As it often happens here, it involved the construction of bialgebras, in this particular case in the category of posets and bimodules. The fact that these bialgebras were actually Frobenius algebras was noticed, but it took some time for the extreme interest of this property to sink in. Definition 1 (Frobenius algebra). Let (C, ⊗,1) be a symmetric monoidal category (SMC), and A an object of it. A Frobenius algebra is a sextuple (A,∆,Π, ∇, ∐) where (A, ∇, ∐) is a commutative monoid, (A,∆,Π) a cocommutative comonoid, where the following diagram commutes:
Ordinals in Frobenius Monads
, 809
"... This paper provides geometrical descriptions of the Frobenius monad freely generated by a single object. These descriptions are related to results connecting Frobenius algebras and topological quantum field theories. In these descriptions, which are based on coherence results for selfadjunctions (a ..."
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This paper provides geometrical descriptions of the Frobenius monad freely generated by a single object. These descriptions are related to results connecting Frobenius algebras and topological quantum field theories. In these descriptions, which are based on coherence results for selfadjunctions (adjunctions where an endofunctor is adjoint to itself), ordinals in ε0 play a prominent role. The paper ends by considering how the notion of Frobenius algebra induces the collapse of the hierarchy of ordinals in ε0, and by raising the question of the exact categorial abstraction of the notion of Frobenius algebra.