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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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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
 Add to MetaCart
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.
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.