Nicolas Guenot

The Focused Calculus of Structures.

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.

Computation in focused intuitionistic logic

We investigate the control of evaluation strategies in a variant of the Λ-calculus derived through the Curry-Howard correspondence from LJF, a sequent calculus for intuitionistic logic implementing the focusing technique. The proof theory of focused intuitionistic logic yields a single calculus in which a number of known Λ-calculi appear as subsystems obtained by restricting types to a certain fragment of LJF. In particular, standard Λ-calculi as well as the call-by-push-value calculus are analysed using this framework, and we relate cut elimination for LJF to a new abstract machine subsuming well-known machines for these different strategies.

Nested proof search as reduction in the Lambda-calculus

We present a system for propositional implicative intuitionistic logic in the calculus of structures, which is a generalisation of the sequent calculus to the deep inference methodology. We show that it is sound and complete with respect to the usual sequent calculus, and consider a restricted system for a smaller class of formulas. Then, we encode lambda-terms with explicit substitutions in these formulas and exhibit a correspondence between proof search in this system and reduction in a lambda-calculus with explicit substitutions. Finally, we present a further restriction to allow a correspondence with the standard lambda-calculus, and show that we can prove results on lambda-calculi by proving results on derivations in the proof systems.

FOCUSED PROOF SEARCH FOR LINEAR LOGIC IN THE CALCULUS OF STRUCTURES

The proof-theoretic approach to logic programming has benefited from the introduction of focused proof systems, through the non-determinism reduction and control they provide when searching for proofs in the sequent calculus. However, this technique was not available in the calculus of structures, known for inducing even more non-determinism than other logical formalisms. This work in progress aims at translating the notion of focusing into the presentation of linear logic in this setting, and use some of its specific features, such as deep application of rules and fine granularity, in order to improve proof search procedures. The starting point for this research line is the multiplicative fragment of linear logic, for which a simple focused proof system can be built.

Symmetric normalisation for intuitionistic logic

We present two proof systems for implication-only intuitionistic logic in the calculus of structures. The first is a direct adaptation of the standard sequent calculus to the deep inference setting, and we describe a procedure for cut elimination, similar to the one from the sequent calculus, but using a non-local rewriting. The second system is the symmetric completion of the first, as normally given in deep inference for logics with a DeMorgan duality: all inference rules have duals, as cut is dual to the identity axiom. We prove a generalisation of cut elimination, that we call symmetric normalisation, where all rules dual to standard ones are permuted up in the derivation. The result is a decomposition theorem having cut elimination and interpolation as corollaries.

Cut Elimination in Multifocused Linear Logic

We study cut elimination for a multifocused variant of full linear logic in the sequent calculus. The multifocused normal form of proofs yields problems that do not appear in a standard focused system, related to the constraints in grouping rule instances in focusing phases. We show that cut elimination can be performed in a sensible way even though the proof requires some specific lemmas to deal with multifocusing phases, and discuss the difficulties arising with cut elimination when considering normal forms of proofs in linear logic. The two most important results in the proof theory of linear logic [Gir87] are the admissibility of the cut rule, and the completeness of the focused normal form of proofs. The notion of focusing was originally developped by Andreoli [And92] with the purpose of improving proof search, but recently it has been considered more often as a normal form that can be obtained by reorganising inference steps in a given proof [MS07]: permutations can be used to group positive rule instances — and to move negative rule instances down. This viewpoint is particularly useful in the natural extension of focusing tomultifocusing [CMS08], where several positive formulas are selected to be decomposed in parallel. This stronger normal form is difficult to use for proof search, since not all positive formulas can be selected in a given sequent: there are complex dependencies. In the multiplicative fragment without units, when the selection of positives is done maximally, proofs are canonical in the sense that they are in bijection with proof-nets [Gir96]. For this reason, investigating the proof theory of multifocused linear logic is necessary to understand the notion of canonicity in sequent calculi, and possibly design normal forms of proofs that could be used as proof-nets, while retaining the usual syntax based on trees of rule instances.

Sequent Calculus and Equational Programming.

Proof assistants and programming languages based on type theories usually come in two flavours: one is based on the standard natural deduction presentation of type theory and involves eliminators, while the other provides a syntax in equational style. We show here that the equational approach corresponds to the use of a focused presentation of a type theory expressed as a sequent calculus. A typed functional language is presented, based on a sequent calculus, that we relate to the syntax and internal language of Agda. In particular, we discuss the use of patterns and case splittings, as well as rules implementing inductive reasoning and dependent products and sums.

Focused Linear Logic and the λ-calculus

Linear logic enjoys strong symmetries inherited from classical logic while providing a constructive framework comparable to intuitionistic logic. However, the computational interpretation of sequent calculus presentations of linear logic remains problematic, mostly because of the many rule permutations allowed in the sequent calculus. We address this problem by providing a simple interpretation of focused proofs, a complete subclass of linear sequent proofs known to have a much stronger structure than the standard sequent calculus for linear logic. Despite the classical setting, the interpretation relates proofs to a refined linear λ-calculus, and we investigate its properties and relation to other calculi, such as the usual λ-calculus, the λµ-calculus, and their variants based on sequent calculi.

Equality and fixpoints in the calculus of structures

The standard proof theory for logics with equality and fixpoints suffers from limitations of the sequent calculus, where reasoning is separated from computational tasks such as unification or rewriting. We propose in this paper an extension of the calculus of structures, a deep inference formalism, that supports incremental and contextual reasoning with equality and fixpoints in the setting of linear logic. This system allows deductive and computational steps to mix freely in a continuum which integrates smoothly into the usual versatile rules of multiplicative-additive linear logic in deep inference.

Hybrid Extensions in a Logical Framework

We discuss the extension of the LF logical framework with operators for manipulating worlds, as found in hybrid logics or in the HLF framework. To overcome the restrictions of HLF, we present a more general approach to worlds in LF, where the structure of worlds can be described in an explicit way. We give a canonical presentation for this system and discuss the encoding of logical systems, beyond the limited scope of linear logic that formed the main goal of HLF.