Cut Elimination for a Logic with Induction and Co-induction

Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and co-induction. These proof principles are based on a proof theoretic (rather than set-theoretic) notion of definition. Definitions are akin to logic programs, where the left and right rules for defined atoms allow one to view theories as "closed" or defining fixed points. The use of definitions and free equality makes it possible to reason intentionally about syntax. We add in a consistent way rules for pre and post fixed points, thus allowing the user to reason inductively and co-inductively about properties of computational system making full use of higher-order abstract syntax. Consistency is guaranteed via cut-elimination, where we give the first, to our knowledge, cut-elimination procedure in the presence of general inductive and co-inductive definitions.Comment: 42 pages, submitted to the Journal of Applied Logi

Proving termination of evaluation for System F with control operators

We present new proofs of termination of evaluation in reduction semantics (i.e., a small-step operational semantics with explicit representation of evaluation contexts) for System F with control operators. We introduce a modified version of Girard's proof method based on reducibility candidates, where the reducibility predicates are defined on values and on evaluation contexts as prescribed by the reduction semantics format. We address both abortive control operators (callcc) and delimited-control operators (shift and reset) for which we introduce novel polymorphic type systems, and we consider both the call-by-value and call-by-name evaluation strategies.Comment: In Proceedings COS 2013, arXiv:1309.092

Uniform Proofs of Normalisation and Approximation for Intersection Types

We present intersection type systems in the style of sequent calculus, modifying the systems that Valentini introduced to prove normalisation properties without using the reducibility method. Our systems are more natural than Valentini's ones and equivalent to the usual natural deduction style systems. We prove the characterisation theorems of strong and weak normalisation through the proposed systems, and, moreover, the approximation theorem by means of direct inductive arguments. This provides in a uniform way proofs of the normalisation and approximation theorems via type systems in sequent calculus style.Comment: In Proceedings ITRS 2014, arXiv:1503.0437

Strong Normalization for HA + EM1 by Non-Deterministic Choice

We study the strong normalization of a new Curry-Howard correspondence for HA + EM1, constructive Heyting Arithmetic with the excluded middle on Sigma01-formulas. The proof-term language of HA + EM1 consists in the lambda calculus plus an operator ||_a which represents, from the viewpoint of programming, an exception operator with a delimited scope, and from the viewpoint of logic, a restricted version of the excluded middle. We give a strong normalization proof for the system based on a technique of "non-deterministic immersion".Comment: In Proceedings COS 2013, arXiv:1309.092

Covariant theory of asymptotic symmetries, conservation laws and central charges

Under suitable assumptions on the boundary conditions, it is shown that there is a bijective correspondence between equivalence classes of asymptotic reducibility parameters and asymptotically conserved n-2 forms in the context of Lagrangian gauge theories. The asymptotic reducibility parameters can be interpreted as asymptotic Killing vector fields of the background, with asymptotic behaviour determined by a new dynamical condition. A universal formula for asymptotically conserved n-2 forms in terms of the reducibility parameters is derived. Sufficient conditions for finiteness of the charges built out of the asymptotically conserved n-2 forms and for the existence of a Lie algebra g among equivalence classes of asymptotic reducibility parameters are given. The representation of g in terms of the charges may be centrally extended. An explicit and covariant formula for the central charges is constructed. They are shown to be 2-cocycles on the Lie algebra g. The general considerations and formulas are applied to electrodynamics, Yang-Mills theory and Einstein gravity.Comment: 86 pages Latex file; minor correction