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This article 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 cut-elimination. Herbrand nets can also be seen as… (More)

We give a calculus of proof-nets 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 proof-nets as an annotated… (More)

We set out to find something that corresponds to deep inference in the same way that the lambda-calculus corresponds to natural deduction. Starting from natural deduction for the conjunction-implication fragment of intuitionistic logic we design a corresponding deep inference system together with reduction rules on proofs that allow a fine-grained… (More)

Proof nets provide abstract counterparts to sequent proofs modulo rule permutations; the idea being that if two proofs have the same underlying proof-net, they are in essence the same proof. Providing a convincing proof-net counterpart to proofs in the classical sequent calculus is thus an important step in understanding classical sequent calculus proofs.… (More)

MRSI grids frequently show spectra with poor quality, mainly because of the high sensitivity of MRS to field inhomogeneities. These poor quality spectra are prone to quantification and/or interpretation errors that can have a significant impact on the clinical use of spectroscopic data. Therefore, quality control of the spectra should always precede their… (More)

A formulation of naive set theory is given in Lafont's Soft Linear Logic, a logic with polynomial time cut-elimination. We demonstrate that the provably total functions of this set theory are precisely the PTIME functions. A novelty of this approach is the representation of the unary/binary natural numbers by two distinct sets (the safe naturals and the… (More)

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