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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 ..."
Understanding the small object argument
 Applied Categorical Structures
, 2008
"... The small object argument is a transfinite construction which, starting from a set of maps in a category, generates a weak factorisation system on that ..."
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The small object argument is a transfinite construction which, starting from a set of maps in a category, generates a weak factorisation system on that
On the axiomatisation of boolean categories with and without medial
 THEORY APPL. CATEG
, 2007
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System BV is NPcomplete
, 2005
"... System BV is an extension of multiplicative linear logic (MLL) with the rules mix, nullary mix, and a selfdual, noncommutative logical operator, called seq. While the rules mix and nullary mix extend the deductive system, the operator seq extends the language of MLL. Due to the operator seq, syste ..."
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Cited by 9 (4 self)
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System BV is an extension of multiplicative linear logic (MLL) with the rules mix, nullary mix, and a selfdual, noncommutative logical operator, called seq. While the rules mix and nullary mix extend the deductive system, the operator seq extends the language of MLL. Due to the operator seq, system BV extends the applications of MLL to those where sequential composition is crucial, e.g., concurrency theory. System FBV is an extension of MLL with the rules mix and nullary mix. In this paper, by relying on the fact that system BV is a conservative extension of system FBV, I show that system BV is NPcomplete by encoding the 3Partition problem in FBV. I provide a simple completeness proof of this encoding by resorting to a novel proof theoretical method for reducing the nondeterminism in proof search, which is also of independent interest.
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
Extension without Cut
, 2008
"... In proof theory one distinguishes sequent proofs with cut and cutfree sequent proofs, while for proof complexity one distinguishes Fregesystems and extended Fregesystems. In this paper we show how deep inference can provide a uniform treatment for both classifications, such that we can define cut ..."
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In proof theory one distinguishes sequent proofs with cut and cutfree sequent proofs, while for proof complexity one distinguishes Fregesystems and extended Fregesystems. In this paper we show how deep inference can provide a uniform treatment for both classifications, such that we can define cutfree systems with extension, which is neither possible with Fregesystems, nor with the sequent calculus. We show that the propositional pidgeonhole principle admits polynomialsize proofs in a cutfree system with extension. We also define cutfree systems with substitution and show that the system with extension psimulates the system with substitution. This yields a new (and simpler) proof that extended Fregesystems psimulate Fregesystems with substitution. Finally, we propose a new class of tautologies that have short proofs in extended systems, but might not in Frege systems without extension.
Deep Inference and Probabilistic Coherence Spaces
, 2009
"... This paper proposes a definition of categorical model of the deep inference system BV, introduced by Guglielmi. Our definition is based on the notion of a linear functor, due to Cockett and Seely. A BVcategory is a linearly distributive category, possibly with negation, with an additional tensor pr ..."
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This paper proposes a definition of categorical model of the deep inference system BV, introduced by Guglielmi. Our definition is based on the notion of a linear functor, due to Cockett and Seely. A BVcategory is a linearly distributive category, possibly with negation, with an additional tensor product which, when viewed as a bivariant functor, is linear with a degeneracy condition. We show that this simple definition implies all of the key isomorphisms of the theory. We show that coherence spaces, with Retoré’s noncommutative tensor, is a model.We then consider Girard’s category of probabilistic coherence spaces and show that it contains a selfdual monoidal structure in addition to the ∗autonomous structure exhibited by Girard. This
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
DOI: 10.1016/j.apal.2012.07.004 Extension without Cut
, 2012
"... In proof theory one distinguishes sequent proofs with cut and cutfree sequent proofs, while for proof complexity one distinguishes Fregesystems and extended Fregesystems. In this paper we show how deep inference can provide a uniform treatment for both classifications, such that we can define cut ..."
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In proof theory one distinguishes sequent proofs with cut and cutfree sequent proofs, while for proof complexity one distinguishes Fregesystems and extended Fregesystems. In this paper we show how deep inference can provide a uniform treatment for both classifications, such that we can define cutfree systems with extension, which is neither possible with Fregesystems, nor with the sequent calculus. We show that the propositional pigeonhole principle admits polynomialsize proofs in a cutfree system with extension. We also define cutfree systems with substitution and show that the cutfree system with extension psimulates the cutfree system with substitution. 1.
3.2. Proof Nets, Sequent Calculus and Typed Lambda Calculi 3
"... c t i v it y e p o r t 2008 Table of contents 1. Team.................................................................................... 1 ..."
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c t i v it y e p o r t 2008 Table of contents 1. Team.................................................................................... 1