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.Comment: In Proceedings LFMTP 2015, arXiv:1507.0759

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

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

Focused Linear Logic and the λ-calculus

International audienceLinear 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

A Climate for Change in the UN Security Council? Member States' Approaches to the Climate-Security Nexus

This research report is the first to systematically engage with the growing political agenda of the climate-security nexus and to place a particular focus on the relationship between the state and the only international organ with a mandate to maintain international peace and security: the United Nations Security Council (UNSC). Discussions that have been ongoing since 2007, scattered governmental positions and the difficulty of achieving an overview of the various understandings, topics, concerns and responses of the UNSC member states in relation to the climate-security nexus all indicate a need to address this topic. This report therefore assesses and maps if and how the UNSC members acknowledge the linkages between climate change and security and how they position themselves with respect to these debates in the UNSC. With a large international network of interdisciplinary and country-specialized partner scientists, the analysis relies on an extensive spectrum of official primary sources from member state governments, various ministry strategies (such as those addressing security and climate change), UNSC documents and interdisciplinary academic literature on the climate-security nexus. It is located in the context of substantiated planetary climate emergencies and existential threats as well as urgent calls for action from the UN and member state representatives, scientific networks in Earth System Sciences and youth protests. Based on broad empirical research findings, this report concludes that all 15 current UNSC member states acknowledge the climate-security nexus in complex, changing and partly country-dependent ways. The report formulates an outlook and recommendations for decision-makers and scholars with a particular focus on strengthening the science-policy interface and dialogue and emphasizing the urgent need for institutional, multilateral and scientifically informed change. It also illustrates how essential it is for the UNSC to recognize and adapt institutional working methods to the interrelations of climate change and security and their effects as a cross-cutting issue

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.Comment: In Proceedings LINEARITY 2014, arXiv:1502.0441

Deduction Imbriquée et Fondements Logiques du Calcul

This thesis investigates the use of deep inference formalisms as basis for a computational interpretation of proof systems, following the two main approaches: proofs-as-programs and proof-search-as-computation. The first contribution is the development of a family of proof systems for intuitionistic logic in the calculus of structures and in nested sequents, for which internal normalisation procedures are given. One of these procedure is then interpreted in terms of computation, as a refinement of the Curry-Howard correspondence allowing to introduce a form of sharing and communication operators in a lambda-calculus with explicit substitution. On the side of proof-search, the notion of focused proof in linear logic is transferred from the sequent calculus into the calculus of structures, where it yields an incremental form of focusing, with a simple proof of completeness. Finally, another interpretation of proof-search is given through the encoding of reduction in a lambda-calculus with explicit substitution into the inference rules of a subsystem of intuitionistic logic in the calculus of structures.Cette thèse s'intéresse à l'usage des formalismes d'inférence profonde comme fondement des interprétations calculatoires des systèmes de preuve, en suivant les deux approches principales: celle des preuves comme programmes et celle de la recherche de preuve comme calcul. La première contribution est le développement d'une famille de systèmes de preuve pour la logique intuitionniste dans le calcul des structures et dans les séquents imbriqués. pour lesquels des procédures de normalisation internes sont fournies. L'une de ces procédures est alors interprétée en termes calculatoires, comme un raffinement de la correspondance de Curry-Howard permettant d'introduire une forme de partage ainsi que des opérateurs de communication dans un lambda-calcul avec substitution explicite. Du coté de la recherche de preuve, la notion de preuve focalisée en logique linéaire est transférée du calcul des séquents au calcul des structures, où elle induit une forme incrémentale de focalisation, dotée d'une preuve de complétude très simple. Enfin, une autre interprétation de la recherche de preuve est donnée par l'encodage de la réduction d'un lambda-calcul avec substitution explicite dans les règles d'inférence d'un sous-système de la logique intuitionniste dans le calcul des structures

Deduction Imbriquée et Fondements Logiques du Calcul

This thesis investigates the use of deep inference formalisms as basis for a computational interpretation of proof systems, following the two main approaches: proofs-as-programs and proof-search-as-computation. The first contribution is the development of a family of proof systems for intuitionistic logic in the calculus of structures and in nested sequents, for which internal normalisation procedures are given. One of these procedure is then interpreted in terms of computation, as a refinement of the Curry-Howard correspondence allowing to introduce a form of sharing and communication operators in a lambda-calculus with explicit substitution. On the side of proof-search, the notion of focused proof in linear logic is transferred from the sequent calculus into the calculus of structures, where it yields an incremental form of focusing, with a simple proof of completeness. Finally, another interpretation of proof-search is given through the encoding of reduction in a lambda-calculus with explicit substitution into the inference rules of a subsystem of intuitionistic logic in the calculus of structures.Cette thèse s'intéresse à l'usage des formalismes d'inférence profonde comme fondement des interprétations calculatoires des systèmes de preuve, en suivant les deux approches principales: celle des preuves comme programmes et celle de la recherche de preuve comme calcul. La première contribution est le développement d'une famille de systèmes de preuve pour la logique intuitionniste dans le calcul des structures et dans les séquents imbriqués. pour lesquels des procédures de normalisation internes sont fournies. L'une de ces procédures est alors interprétée en termes calculatoires, comme un raffinement de la correspondance de Curry-Howard permettant d'introduire une forme de partage ainsi que des opérateurs de communication dans un lambda-calcul avec substitution explicite. Du coté de la recherche de preuve, la notion de preuve focalisée en logique linéaire est transférée du calcul des séquents au calcul des structures, où elle induit une forme incrémentale de focalisation, dotée d'une preuve de complétude très simple. Enfin, une autre interprétation de la recherche de preuve est donnée par l'encodage de la réduction d'un lambda-calcul avec substitution explicite dans les règles d'inférence d'un sous-système de la logique intuitionniste dans le calcul des structures

Symmetric normalisation for intuitionistic logic

International audienceWe 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