Section: New Results

The Focused Calculus of Structures

Participants : Kaustuv Chaudhuri, Nicolas Guenot, Lutz Straßburger.

The sequent calculus is a proof system for logic that has many nice properties from a proof search perspective, the most famous being the subformula property that is essential to tame the search space. In recent years, the focusing property of sequent systems has become another useful property, both for shrinking the search space and to improve the representation of proofs. However, the sequent calculus does have some limitations. Primarily, not all logics have analytic (i.e., cut-free) proof systems, which are the sine qua non of proof search. A less obvious but equally bothersome limitation is that cut-free sequent proofs tend to contain large repeated sub-proofs.

To remedy these deficiencies, one can use the calculus of structures, a proof system that allows inferences anywhere inside a formula. This system can represent many more logics than the sequent calculus and can produce better (i.e., usually smaller) proofs because it can avoid sharing large subformulas. Nevertheless, because the rules of the calculus of structures have finer granularity than sequent rules, it has more non-determinism during search.

In this work, we show how to transplant the focusing result from the sequent calculus to the calculus of structures [19] . We thus improve the search capabilities of the calculus of structures, including the ability to go back and forth between focused sequent proofs and focused calculus of structures proofs, but we retain all the distinguishing features of the calculus of structures.

In particular, we preserve the ability to permute contractions below all other rules (first observed for the calculus of structures in  [55] , [31] ). This permutation enables a two-stage normal form of proofs. The first stage contains only contractions, which increases the complexity of the formulas and is therefore a potential source of unbounded search; this phase needs to be recorded in order to reconstruct proofs by bounded search. The second stage that contains the remaining logical rules (except contraction) is strictly bounded and finite—hence decidable—and can be reconstructed if omitted from the proof object. Thus, we have the potential of obtaining very simple proof objects, recording only the first phase, for focused proofs; moreover, because of the bidirectional link, we can reconstruct focused sequent proofs from such proof objects.

Both the search and the representational aspects of focused calculus of structures proofs are being investigated in the Profound tool (see section  5.1 ).