Natural deduction systems for some non-commutative logics

Varieties of natural deduction systems are introduced for Wansing’s paraconsistent non-commutative substructural logic, called a constructive sequential propositional logic (COSPL), and its fragments. Normalization, strong normalization and Church-Rosser theorems are proved for these systems. These results include some new results on full Lambek logic (FL) and its fragments, because FL is a fragment of COSPL

Lecture notes on the lambda calculus

This is a set of lecture notes that developed out of courses on the lambda calculus that I taught at the University of Ottawa in 2001 and at Dalhousie University in 2007 and 2013. Topics covered in these notes include the untyped lambda calculus, the Church-Rosser theorem, combinatory algebras, the simply-typed lambda calculus, the Curry-Howard isomorphism, weak and strong normalization, polymorphism, type inference, denotational semantics, complete partial orders, and the language PCF.Comment: 120 pages. Added in v2: section on polymorphis

Pure Type System conversion is always typable

International audiencePure Type Systems are usually described in two different ways, one that uses an external notion of computation like beta-reduction, and one that relies on a typed judgment of equality, directly in the typing system. For a long time, the question was open to know whether both presentations described the same theory. A first step toward this equivalence has been made by Adams for a particular class of \emph{Pure Type Systems} (PTS) called functional. Then, his result has been relaxed to all semi-full PTS in previous work. In this paper, we finally give a positive answer to the general issue, and prove that equivalence holds for any Pure Type System.Les Systèmes de Types Purs (PTS) sont habituellement présentés de deux manières différentes, une qui utilise une notion de calcul indépendante du typage, comme la béta-reduction, et une qui défini un jugement d'égalité typée au sein du système de types. La question de savoir si ces deux présentations représentaient la même théorie est restée ouverte pendant de nombreuses années. Une première réponse partielle à cette question a été apportée par Adams pour une classe particulière de PTS dit "fonctionnels". Nous avons récement étendu ce résultat à tous les PTS "semi-complets" . Dans cet article, nous pouvons finalement donner une réponse positive à la question dans toute sa généralité: l'équivalence entre les deux présentations est prouvée correcte pour n'importe quel Système de Types Purs

The computational content of atomic polymorphism

We show that the number-theoretic functions de nable in the atomic polymorphic system (Fat) are exactly the extended polynomials. Two proofs of the above result are presented: one reducing the functions' de n- ability problem in Fat to de nability in the simply typed lambda-calculus and other directly adapting Helmut Schwichtenberg's strategy for de nability in the simply typed lambda-calculus to the atomic polymorphic setting. The uniformity granted in the polymorphic system, when compared with the simply typed lambda-calculus, is emphasized.This work was supported by Fundação para a Ciência e a Tecnologia [UID/MAT/ 04561/2013, UID/CEC/00408/2013 and grant SFRH/BPD/93278/2013 to G.F.]. The first author is also grateful to Centro de Matemática, Aplicações Fundamentais e Investigação Operacional and to Large-Scale Informatics Systems Laboratory (Universidade de Lisboa).info:eu-repo/semantics/publishedVersio

A Computational Interpretation of Parametricity

Reynolds' abstraction theorem has recently been extended to lambda-calculi with dependent types. In this paper, we show how this theorem can be internalized. More precisely, we describe an extension of Pure Type Systems with a special parametricity rule (with computational content), and prove fundamental properties such as Church-Rosser's and strong normalization. All instances of the abstraction theorem can be both expressed and proved in the calculus itself. Moreover, one can apply parametricity to the parametricity rule: parametricity is itself parametric

On Equivalence and Canonical Forms in the LF Type Theory

Decidability of definitional equality and conversion of terms into canonical form play a central role in the meta-theory of a type-theoretic logical framework. Most studies of definitional equality are based on a confluent, strongly-normalizing notion of reduction. Coquand has considered a different approach, directly proving the correctness of a practical equivalance algorithm based on the shape of terms. Neither approach appears to scale well to richer languages with unit types or subtyping, and neither directly addresses the problem of conversion to canonical. In this paper we present a new, type-directed equivalence algorithm for the LF type theory that overcomes the weaknesses of previous approaches. The algorithm is practical, scales to richer languages, and yields a new notion of canonical form sufficient for adequate encodings of logical systems. The algorithm is proved complete by a Kripke-style logical relations argument similar to that suggested by Coquand. Crucially, both the algorithm itself and the logical relations rely only on the shapes of types, ignoring dependencies on terms.Comment: 41 page

Some Lambda Calculus and Type Theory Formalized

"This paper is about our hobby." That is the first sentence of [MP93], the first report on our formal development of lambda calculus and type theory, written in autumn 1992. We have continued to pursue this hobby on and off ever since, and have developed a substantial body of formal knowledge, including Church-Rosser and standardizationtheorems for beta reduction, and the basic theory ofPure Type Systems (PTS) leading to the strengthening theorem and type checking algorithms for PTS. Some of this work is reported in [MP93, vBJMP94, Pol94b, Pol95]. In the present paper we survey this work, including some new proofs, and point out what we feel has been learned about the general issues of formalizing mathematics. On the technical side, we describe an abstract, and simplified, proof of standardization for beta reduction, not previously published, that doesnot mention redex positions or residuals. On the general issues, we emphasize the search for formal definitions that are convenient for formal proof and convincingly represent the intended informal concepts. The LEGO Proof Development System [LP92] was used to check the work in an implementation of the Extended Calculus of Constructions(ECC) with inductive types [Luo94]. LEGO is a refinement styleproof checker, publicly available by ftp and WWW, with a User's Manual [LP92] and a large collection of examples. Section 1.3 contains information on accessing the formal development described in this paper. Other interesting examples formalized in LEGO include program specification and data refinement [Luo91], strong normalization of System F [Alt93], synthetic domain theory [Reu95, Reu96], and operational semantics for imperative programs [Sch97]

A Theory of Explicit Substitutions with Safe and Full Composition

Many different systems with explicit substitutions have been proposed to implement a large class of higher-order languages. Motivations and challenges that guided the development of such calculi in functional frameworks are surveyed in the first part of this paper. Then, very simple technology in named variable-style notation is used to establish a theory of explicit substitutions for the lambda-calculus which enjoys a whole set of useful properties such as full composition, simulation of one-step beta-reduction, preservation of beta-strong normalisation, strong normalisation of typed terms and confluence on metaterms. Normalisation of related calculi is also discussed.Comment: 29 pages Special Issue: Selected Papers of the Conference "International Colloquium on Automata, Languages and Programming 2008" edited by Giuseppe Castagna and Igor Walukiewic