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The Focused Calculus of Structures

Kaustuv Chaudhuri 1 Nicolas Guenot 1 Lutz Straßburger 1
1 PARSIFAL - Proof search and reasoning with logic specifications
LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau], Inria Saclay - Ile de France
Abstract : The focusing theorem identifies a complete class of sequent proofs that have no inessential non-deterministic choices and restrict the essential choices to a particular normal form. Focused proofs are therefore well suited both for the search and for the representation of sequent proofs. The calculus of structures is a proof formalism that allows rules to be applied deep inside a formula. Through this freedom it can be used to give analytic proof systems for a wider variety of logics than the sequent calculus, but standard presentations of this calculus are too permissive, allowing too many proofs. In order to make it more amenable to proof search, we transplant the focusing theorem from the sequent calculus to the calculus of structures. The key technical contribution is an incremental treatment of focusing that avoids trivializing the calculus of structures. We give a direct inductive proof of the completeness of the focused calculus of structures with respect to a more standard unfocused form. We also show that any focused sequent proof can be represented in the focused calculus of structures, and, conversely, any proof in the focused calculus of structures corresponds to a focused sequent proof.
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Kaustuv Chaudhuri, Nicolas Guenot, Lutz Straßburger. The Focused Calculus of Structures. 20th EACSL Annual Conference on Computer Science Logic, Sep 2011, Bergen, Norway. pp.159-173, ⟨10.4230/LIPIcs.CSL.2011.159⟩. ⟨hal-00772420⟩

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