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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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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.
Expansion nets: proofnets for propositional classical logic
 IN PROCEEDINGS OF THE 17TH INTERNATIONAL CONFERENCE ON LOGIC FOR PROGRAMMING, ARTIFICIAL INTELLIGENCE, AND REASONING, LPAR’10
, 2010
"... We give a calculus of proofnets for classical propositional logic. These nets improve on a proposal due to Robinson by validating the associativity and commutativity of contraction, and provide canonical representants for classical sequent proofs modulo natural equivalences. We present the relation ..."
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We give a calculus of proofnets for classical propositional logic. These nets improve on a proposal due to Robinson by validating the associativity and commutativity of contraction, and provide canonical representants for classical sequent proofs modulo natural equivalences. We present the relationship between sequent proofs and proofnets as an annotated sequent calculus, deriving formulae decorated with expansion/deletion trees. We then see a subcalculus, expansion nets, which in addition to these good properties has a polynomialtime correctness criterion.
A sequent calculus demonstration of Herbrand’s Theorem
, 2010
"... Herbrand’s theorem is often presented as a corollary of Gentzen’s sharpened Hauptsatz for the classical sequent calculus. However, the midsequent gives Herbrand’s theorem directly only for formulae in prenex normal form. In the Handbook of Proof Theory, Buss claims to give a proof of the full statem ..."
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Herbrand’s theorem is often presented as a corollary of Gentzen’s sharpened Hauptsatz for the classical sequent calculus. However, the midsequent gives Herbrand’s theorem directly only for formulae in prenex normal form. In the Handbook of Proof Theory, Buss claims to give a proof of the full statement of the theorem, using sequent calculus methods to show completeness of a calculus of Herbrand proofs, but as we demonstrate there is a flaw in the proof. In this note we give a correct demonstration of Herbrand’s theorem in its full generality, as a corollary of the full cutelimination theorem for LK. The major difficulty is to show that, if there is an Herbrand proof of the premiss of a contraction rule, there is an Herbrand proof of its conclusion. We solve this problem by showing the admissibility of a deep contraction rule. 1
Canonical proof nets for classical logic
"... Abstract. Proof nets provide abstract counterparts to sequent proofs modulo rule permutations; the idea being that if two proofs have the same underlying proofnet, they are in essence the same proof. Providing a convincing proofnet counterpart to proofs in the classical sequent calculus is thus an ..."
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Abstract. Proof nets provide abstract counterparts to sequent proofs modulo rule permutations; the idea being that if two proofs have the same underlying proofnet, they are in essence the same proof. Providing a convincing proofnet counterpart to proofs in the classical sequent calculus is thus an important step in understanding classical sequent calculus proofs. By convincing, we mean that (a) there should be a canonical function from sequent proofs to proof nets, (b) it should be possible to check the correctness of a net in polynomial time, (c) every correct net should be obtainable from a sequent calculus proof, and (d) there should be a cutelimination procedure which preserves correctness. Previous attempts to give proofnetlike objects for propositional classical logic have failed at least one of the above conditions. In [23], the author presented a calculus of proof nets (expansion nets) satisfying (a) and (b); the paper defined a sequent calculus corresponding to expansion nets but gave no explicit demonstration of (c). That sequent calculus, called LK ∗ in this paper, is a novel onesided sequent calculus with both additively and multiplicatively formulated disjunction rules. In this paper (a selfcontained extended version of [23]) , we give a full proof of (c) for expansion nets with respect to LK ∗, and in addition give a cutelimination procedure internal to expansion nets – this makes expansion nets the first notion of proofnet for classical logic satisfying all four criteria. 1