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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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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.
Sufficient conditions for cut elimination with complexity analysis
 Annals of Pure and Applied Logic
, 2007
"... Sufficient conditions for first order based sequent calculi to admit cut elimination by a SchütteTait style cut elimination proof are established. The worst case complexity of the cut elimination is analysed. The obtained upper bound is parameterized by a quantity related with the calculus. The con ..."
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Cited by 9 (4 self)
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Sufficient conditions for first order based sequent calculi to admit cut elimination by a SchütteTait style cut elimination proof are established. The worst case complexity of the cut elimination is analysed. The obtained upper bound is parameterized by a quantity related with the calculus. The conditions are general enough to be satisfied by a wide class of sequent calculi encompassing, among others, some sequent calculi presentations for the first order and the propositional versions of classical and intuitionistic logic, classical and intuitionistic modal logic S4, and classical and intuitionistic linear logic and some of its fragments. Moreover the conditions are such that there is an algorithm for checking if they are satisfied by a sequent calculus.
Modal Sequent Calculi Labelled with Truth Values: Completeness, Duality and Analyticity
 LOGIC JOURNAL OF THE IGPL
, 2003
"... Labelled sequent calculi are provided for a wide class of normal modal systems using truth values as labels. The rules for formula constructors are common to all modal systems. For each modal system, specific rules for truth values are provided that reflect the envisaged properties of the accessi ..."
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Cited by 8 (5 self)
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Labelled sequent calculi are provided for a wide class of normal modal systems using truth values as labels. The rules for formula constructors are common to all modal systems. For each modal system, specific rules for truth values are provided that reflect the envisaged properties of the accessibility relation. Both local and global reasoning are supported. Strong completeness is proved for a natural twosorted algebraic semantics. As a corollary, strong completeness is also obtained over general Kripke semantics. A duality result
Asymptotic cyclic expansion and bridge groups of formal proofs
 JOURNAL OF ALGEBRA
, 2001
"... Formal proofs, even simple ones, may hide an unexpected intricate combinatorics. We define a new combinatorial invariant, the bridge group of a proof, which encodes the cyclic structure of proofs in the sequent calculus. We compute the bridge groups of two infinite families of proofs and identify th ..."
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Cited by 5 (1 self)
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Formal proofs, even simple ones, may hide an unexpected intricate combinatorics. We define a new combinatorial invariant, the bridge group of a proof, which encodes the cyclic structure of proofs in the sequent calculus. We compute the bridge groups of two infinite families of proofs and identify them with the Baumslag–Solitar and Gersten groups. We observe that the distortion of cyclic subgroups in these groups equals the asymptotic growth of the procedure of elimination of lemmas from the proofs.
The Cost of a Cycle is a Square
, 1999
"... The logical flow graphs of sequent calculus proofs might contain oriented cycles. For the predicate calculus the elimination of cycles might be nonelementary and this was shown in [Car96]. For the propositional calculus, we prove that if a proof of k lines contains n cycles then there exists an ..."
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Cited by 3 (2 self)
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The logical flow graphs of sequent calculus proofs might contain oriented cycles. For the predicate calculus the elimination of cycles might be nonelementary and this was shown in [Car96]. For the propositional calculus, we prove that if a proof of k lines contains n cycles then there exists an acyclic proof with O(k n+1 ) lines. In particular, there is a quadratic time algorithm which eliminates a single cycle from a proof. These results are motivated by the search for general methods on proving lower bounds on proof size and by the design of more efficient heuristic algorithms for proof search.
Streams and Strings in Formal Proofs
"... Streams are acyclic directed subgraphs of the logical flow graph of a proof and represent bundles of paths with the same origin and the same end. Streams can be described with a natural algebraic formalism which allows to explain in algebraic terms the evolution of proofs during cutelimination. ..."
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Streams are acyclic directed subgraphs of the logical flow graph of a proof and represent bundles of paths with the same origin and the same end. Streams can be described with a natural algebraic formalism which allows to explain in algebraic terms the evolution of proofs during cutelimination. In our approach, "logic" is often forgotten and combinatorial properties of graphs are taken into account to explain logical phenomena.
The Geometry of NonDistributive Logics
, 2005
"... Abstract: In this paper we introduce a new natural deduction system for the logic of lattices, and a number of extensions of lattice logic with different negation connectives. We provide the class of natural deduction proofs with both a standard inductive definition and a global graphtheoretical cr ..."
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Abstract: In this paper we introduce a new natural deduction system for the logic of lattices, and a number of extensions of lattice logic with different negation connectives. We provide the class of natural deduction proofs with both a standard inductive definition and a global graphtheoretical criterion for correctness, and we show how normalisation in this system corresponds to cut elimination in the sequent calculus for lattice logic. This natural deduction system is inspired both by Shoesmith and Smiley’s multiple conclusion systems for classical logic and Girard’s proofnets for linear logic. In this paper we present a new kind of natural deduction system for lattice logic and extensions of lattice logic with negation. This natural deduction system has a natural symmetry. The rules for lattice disjunction are exactly the reverse of the rules for lattice conjunction. In this way, the natural deduction system shares some features with Girard’s proofnets for multiplicative linear logic [12]. Our proofs differ from proofnets, however, in that they are oriented graphs in which the flow of information from premise to conclusion in inference
Complexity Analysis of Cut Elimination in First Order Based Logics
, 2005
"... The worstcase complexity of cut elimination in sequent calculi for first order based logics is investigated in terms of the increase in logical depth of the deduction. It is shown that given a calculus satisfying a general collection of sufficient conditions for cut elimination and given a deductio ..."
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The worstcase complexity of cut elimination in sequent calculi for first order based logics is investigated in terms of the increase in logical depth of the deduction. It is shown that given a calculus satisfying a general collection of sufficient conditions for cut elimination and given a deduction with cuts, there exists a cut free deduction with a logical depth, in the worst case, hyperexponentially greater than the logical depth of the original deduction. Moreover an interesting relation, as far as we know not yet reported in the literature, between this complexity and the greatest cut length of a pair of introduction rules in the calculus for a constructor, is established. By the cut length of a pair of introduction rules of a constructor it is meant the minimal length of a cut sequence for that pair, where a cut sequence for a pair of introduction rules is a sequence formed by premises of that rules, without contexts, that when combined in a deduction using a generic cut rule, respecting the sequence order, lead to the empty sequent. The nonelementary bound in the worst case complexity of the cut elimination