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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 6 (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.
Proof nets for Herbrand’s Theorem
"... This paper explores Herbrand’s theorem as the source of a natural notion of abstract proof object for classical logic, embodying the “essence ” of a sequent calculus proof. We we see how to view a calculus of abstract Herbrand proofs (“Herbrand nets”) as an analytic proof system with syntactic cute ..."
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Cited by 2 (1 self)
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This paper explores Herbrand’s theorem as the source of a natural notion of abstract proof object for classical logic, embodying the “essence ” of a sequent calculus proof. We we see how to view a calculus of abstract Herbrand proofs (“Herbrand nets”) as an analytic proof system with syntactic cutelimination. Herbrand nets can also be seen as a natural generalization of Miller’s expansion tree proofs to a setting including cut. We demonstrate sequentialization of Herbrand nets into a sequent calculus LKH; each net corresponds to an equivalence class of LKH proofs under natural proof transformations. A surprising property of our cutreduction algorithm is that it is nonconfluent, despite not supporting the usual examples of nonconfluent reduction in classical logic.
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 ..."
Abstract
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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. 1.
Appl Categor Struct DOI 10.1007/s1048501092410 Deep Inference and Probabilistic Coherence Spaces
, 2009
"... Abstract This paper proposes a definition of categorical model of the deep inference system BV, defined by Guglielmi. Deep inference introduces the idea of performing a deduction in the interior of a formula, at any depth. Traditional sequent calculus rules only see the roots of formulae. However in ..."
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Abstract This paper proposes a definition of categorical model of the deep inference system BV, defined by Guglielmi. Deep inference introduces the idea of performing a deduction in the interior of a formula, at any depth. Traditional sequent calculus rules only see the roots of formulae. However in these new systems, one can rewrite at any position in the formula tree. Deep inference in particular allows the syntactic description of logics for which there is no sequent calculus. One such system is BV, which extends linear logic to include a noncommutative selfdual connective. This is the logic our paper proposes to model. Our definition is based on the notion of a linear functor, due to Cockett and Seely. A BVcategory is a linearly distributive
Categories and Subject Descriptors: F4.1 [Mathematical logic and formal languages]: Mathematical
"... This paper explores Herbrand’s theorem as the source of a natural notion of abstract proof object for classical logic, embodying the “essence ” of a sequent calculus proof. We see how to view a calculus of abstract Herbrand proofs (“Herbrand nets”) as an analytic proof system with syntactic cutelim ..."
Abstract
 Add to MetaCart
This paper explores Herbrand’s theorem as the source of a natural notion of abstract proof object for classical logic, embodying the “essence ” of a sequent calculus proof. We see how to view a calculus of abstract Herbrand proofs (“Herbrand nets”) as an analytic proof system with syntactic cutelimination. Herbrand nets can also be seen as a natural generalization of Miller’s expansion tree proofs to a setting including cut. We demonstrate sequentialization of Herbrand nets into a sequent calculus LKH; each net corresponds to an equivalence class of LKH proofs under natural proof transformations. A surprising property of our cutreduction algorithm is that it is nonconfluent, despite not supporting the usual examples of nonconfluent reduction in classical logic.