Sense, denotation and semantics natural deduction the Curry-Howard isomorphism the normalisation theorem Godel's system T coherence spaces denotational semantics of T sums in natural deduction system F coherence semantics of the sum cut elimination (Hauptsatz) strong normalisation for F representation theorem semantics of System F what is linear logic?

Proofs and types

Publisher Summary This chapter focuses on finite automata on infinite sequences and infinite trees. The chapter discusses the complexity of the complementation process and the equivalence test. Deterministic Muller automata and nondeterministic Buchi automata are equivalent in recognition power. Any nonempty Rabin recognizable set contains a regular tree and shows that the emptiness problem for Rabin tree automata is decidable. The chapter discusses the formulation of two interesting generalizations of Rabin's Tree Theorem and presents some remarks on the undecidable extensions of the monadic theory of the binary tree. A short overview of the work that studies the fine structure of the class of Rabin recognizable sets of trees is also presented in the chapter. Depending on the formalism in which tree properties are classified, the results fall in three categories: monadic second-order logic, tree automata, and fixed-point calculi.

Automata on infinite objects

Coq is a proof assistant based on a higher-order logic allowing powerful definitions of functions. Coq V6.1 is available by anonymous ftp at ftp.inria.fr:/INRIA/Projects/coq/V6.1 and ftp.ens-lyon.fr:/pub/LIP/COQ/V6.1

https://hal.inria.fr/inria-00069968/document

The Coq proof assistant : reference manual, version 6.1

1 INTRODUCTION This paper presents a way of extending the paradigm "proofs as programs" to classical proofs. The system we use is derived from the general Free Deduction system presented in [31. Usually when considering proofs as programs, one has only in mind some kind of intuitionistic proofs. There is an obvious reason for that restriction: only intuitionistic proofs are contructive, in the sense that from the proof of an existential statement, one can get a witness of this existential statement. But from the programming point of view, constructivity is only needed for E~-statements, for which classical and intuitionistic provability coincide. This means that, classical proofs are also candidates for being programs. In order to use them as programs, one has two tasks to achieve: (i) to find a system in which one can extract directly a program from a classical proof (and not by means of a translation to intuitionistic logic), and (ii) to understand the algorithmic meaning of classical constructions. The system we will consider is a natural deduction system with multiple conclusions, we will call it Classical Natural Deduction (the one with the absurdity rule being called Usual Natural Deduction). It is a particular subsystem of Free Deduction (FD) with inputs fixed to the left, chosen for its simplicity: it can be seen as a simple extension of intuitionistic natural deduction, whose algorithmic interpretation is very well known. In this context, the contribution of classical constructs to programming appears clearly: they correspond to control operators added to functional languages, like call/ce in Scheme. In both contexts, the role of the classical constructs is the same: they allow to take shorter routes in the construction of a proof~program. The link between control operators and classical constructs has first been made by T. Griffin in [1], where he proposes to type the C operator of Felleisen, with the type-~'-,A-* A. The system he obtains is not satisfactory from the logical point of view: the reduction is in fact a reduction strategy and the type assigned to C doesn't fit in general the reduction rule for C. C. Murthy further analysed the connections t91 between control operators, classical constructs and translations from classical logic to intuitionistic logic (see [4]). The difficulties met in trying to use-~-A ~ A (or the classical absurdity rule) as a type for control operators is not really due to classical logic, but much nore to …

/pdf/lambda-my-calculus-an-algorithmic-interpretation-of-4xuctj4qr7.pdf

Lambda-My-Calculus: An Algorithmic Interpretation of Classical Natural Deduction

In the pure Calculus of Constructions, it is possible to represent data structures and predicates using higher-order quantification. However, this representation is not satisfactory, from the point of view of both the efficiency of the underlying programs and the power of the logical system. For these reasons, the calculus was extended with a primitive notion of inductive definitions [8]. This paper describes the rules for inductive definitions in the system Coq. They are general enough to be seen as one formulation of adding inductive definitions to a typed lambda-calculus. We prove strong normalization for a subsystem of Coq corresponding to the pure Calculus of Constructions plus Inductive Definitions with only weak eliminations.

Inductive Definitions in the system Coq - Rules and Properties

We give two proofs of strong normalisation for second order classical natural deduction. The first one is an adaptation of the method of reducibility candidates introduced in [9] for second order intuitionistic natural deduction; the extension to the classical case requires in particular a simplification of the notion of reducibility candidate. The second one is a reduction to the intuitionistic case, using a Kolmogorov translation.

Proofs of strong normalisation for second order classical natural deduction

The strong normalization theorem for second-order classical natural deduction is proved. The method used is an adaptation of the one of reducibility candidates introduced in a thesis by J.Y. Girard (Univ. Paris 7, 1972) for second-order intuitionistic natural deduction. The extension to the classical case requires, in particular, a simplification of the notion of reducibility candidates. >

Strong normalization for second order classical natural deduction

We present an extension of the correspondence between intuitionistic proofs and functional programs to classical proofs, and more precisely to second order classical proofs. The advantage of classical logic in this context is that it allows to model imperative features of programming languages too (cf [5]). But there is an intrinsic difficulty with classical logic which lies in certain non-determinism of its computational interpretations. The use of a natural deduction system removes a part of this non determinism by fixing the inputs to the left of the sequents (cf [10] and [11]). However a conflict remains between the confluence of the computation mechanism and the uniqueness of the representation of data (for instance the uniqueness of the representation of the natural number 1). In this paper we develop the solution to this problem proposed in [11]: we show how to extract the intuitionistic representation of a data from a classical one using an “output” operator, while keeping a confluent computation mechanism. This result allows to extend in a sound way the proofs-as-programs paradigm to classical proofs in a framework where all the usual theoretical properties of intuitionistic proofs still hold.

Michel Parigot

Papers

Lambda-My-Calculus: An Algorithmic Interpretation of Classical Natural Deduction

Proofs of strong normalisation for second order classical natural deduction

Strong normalization for second order classical natural deduction

Classical Proofs as Programs

Recursive programming with proofs