Realizability, Covers, and Sheaves I. Application to the Simply-Typed Lambda-Calculus

We present a general method for proving properties of typed λ-terms. This method is obtained by introducing a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory, a cover algebra being a Grothendieck topology in the case of a preorder). For this, we introduce a new class of semantic structures equipped with preorders, called pre-applicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Kleene\u27s recursive realizability and a variant of Kreisel\u27s modified realizability both fit into this framework. Applying this theorem to the special case of the term model, yields a general theorem for proving properties of typed λ-terms, in particular, strong normalization and confluence. This approach clarifies the reducibility method by showing that the closure conditions on candidates of reducibility can be viewed as sheaf conditions. Part I of this paper applies the above approach to the simply-typed λ-calculus (with types →, ×, +, and ⊥). Part II of this paper deals with the second-order (polymorphic) λ-calculus (with types → and ∀)

Kripke Models for the Second-Order Lambda-Calculus

We define a new class of Kripke structures for the second-order λ-calculus, and investigate the soundness and completeness of some proof systems for proving inequalities (rewrite rules) or equations. The Kripke structures under consideration are equipped with preorders that correspond to an abstract form of reduction, and they are not necessarily extensional. A novelty of our approach is that we define these structures directly as functors A:W→ Preor equipped with certain natural transformations corresponding to application and abstraction (where is a preorder, the set of worlds, and Preor is the category of preorders). We make use of an explicit construction of the exponential of functors in the Cartesian-closed category PreorW, and we also define a kind of exponential ∏Φ(As)s∈Τ to take care of type abstraction. We obtain soundness and completeness theorems that generalize some results of Mitchell and Moggi to the second-order λ-calculus, and to sets of inequalities (rewrite rules)

On Girard\u27s Candidats de Reductibilité

We attempt to elucidate the conditions required on Girard\u27s candidates of reducibility (in French, candidats de reductibilité ) in order to establish certain properties of various typed lambda calculi, such as strong normalization and Church-Rosser property. We present two generalizations of the candidates of reducibility, an untyped version in the line of Tait and Mitchell, and a typed version which is an adaptation of Girard\u27s original method. As an application of this general result, we give two proofs of strong normalization for the second-order polymorphic lambda calculus under βη-reduction (and thus under β-reduction). We present two sets of conditions for the typed version of the candidates. The first set consists of conditions similar to those used by Stenlund (basically the typed version of Tait\u27s conditions), and the second set consists of Girard\u27s original conditions. We also compare these conditions, and prove that Girard\u27s conditions are stronger than Tait\u27s conditions. We give a new proof of the Church-Rosser theorem for both β-reduction and βη-reduction, using the modified version of Girard\u27s method. We also compare various proofs that have appeared in the literature (see section 11). We conclude by sketching the extension of the above results to Girard\u27s higher-order polymorphic calculus Fω, and in appendix 1, to Fω with product types

Unification Procedures in Automated Deduction Methods Based on Matings: A Survey

Unification procedures arising in methods for automated theorem proving based on matings are surveyed. We begin by reviewing some fundamentals of automated deduction, including the Skolem form and the Skolem-Herbrand-Gödel theorem. Next, the method of matings for first-order languages without equality due to Andrews and Bibel is presented. Standard unification is described in terms of transformations on systems (following the approach of Martelli and Montanari, anticipated by Herbrand). Some fast unification algorithms are also sketched, in particular, a unification closure algorithm inspired by Paterson and Wegman\u27s method. The method of matings is then extended to languages with equality. This extention leads naturally to a generalization of standard unification called rigid E-unification (due to Gallier, Narendran, Plaisted, and Snyder). The main properties of rigid E-unification, decidability, NP-completeness, and finiteness of complete sets, are discussed

Realizability, Covers, and Sheaves II. Applications to the Second-Order Lambda-Calculus

We present a general method for proving properties of typed λ-terms. This method is obtained by introducing a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory, a cover algebra being a Grothendieck topology in the case of a preorder). For this, we introduce a new class of semantic structures equipped with preorders, called pre-applicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Applying this theorem to the special case of the term model, yields a general theorem for proving properties of typed λ-terms, in particular, strong normalization and confluence. This approach clarifies the reducibility method by showing that the closure conditions on candidates of reducibility can be viewed as sheaf conditions. Part II of this paper applies the above approach to the second-order (polymorphic) λ-calculus λ→,∀2 (with types → and ∀)

Constructive Logics Part I: A Tutorial on Proof Systems and Typed Lambda-Calculi

The purpose of this paper is to give an exposition of material dealing with constructive logic, typed λ-calculi, and linear logic. The emergence in the past ten years of a coherent field of research often named logic and computation has had two major (and related) effects: firstly, it has rocked vigorously the world of mathematical logic; secondly, it has created a new computer science discipline, which spans from what is traditionally called theory of computation, to programming language design. Remarkably, this new body of work relies heavily on some old concepts found in mathematical logic, like natural deduction, sequent calculus, and λ-calculus (but often viewed in a different light), and also on some newer concepts. Thus, it may be quite a challenge to become initiated to this new body of work (but the situation is improving, there are now some excellent texts on this subject matter). This paper attempts to provide a coherent and hopefully gentle initiation to this new body of work. We have attempted to cover the basic material on natural deduction, sequent calculus, and typed λ-calculus, but also to provide an introduction to Girard\u27s linear logic, one of the most exciting developments in logic these past five years. The first part of these notes gives an exposition of background material (with the exception of the Girard-translation of classical logic into intuitionistic logic, which is new). The second part is devoted to linear logic and proof nets

On the Correspondence Between Proofs and Lambda-Terms

The correspondence between natural deduction proofs and λ-terms is presented and discussed. A variant of the reducibility method is presented, and a general theorem for establishing properties of typed (first-order) λ-terms is proved. As a corollary, we obtain a simple proof of the Church-Rosser property, and of the strong normalization property, for the typed λ-calculus associated with the system of (intuitionistic) first-order natural deduction, including all the connectors →, ×, +, ∀,∃ and ⊥ (falsity) (with or without η-like rules)

Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations

Some basic mathematical tools such as convex sets, polytopes and combinatorial topology are used quite heavily in applied fields such as geometric modeling, meshing, computer vision, medical imaging and robotics. This report may be viewed as a tutorial and a set of notes on convex sets, polytopes, polyhedra, combinatorial topology, Voronoi Diagrams and Delaunay Triangulations. It is intended for a broad audience of mathematically inclined readers. One of my (selfish!) motivations in writing these notes was to understand the concept of shelling and how it is used to prove the famous Euler-Poincare formula(Poincare, 1899) and the more recent Upper Bound Theorem (McMullen, 1970) for polytopes. Another of my motivations was to give a correct account of Delaunay triangulations and Voronoi diagrams in terms of (direct and inverse) stereographic projections onto a sphere and prove rigorously that the projective map that sends the (projective) sphere to the (projective) paraboloid works correctly, that is, maps the Delaunay triangulation and Voronoi diagram w.r.t. the lifting onto the sphere to the Delaunay diagram and Voronoi diagrams w.r.t. the traditional lifting onto the paraboloid. Here, the problem is that this map is only well defined (total) in projective space and we are forced to define the notion of convex polyhedron in projective space. It turns out that in order to achieve (even partially) the above goals, I found that it was necessary to include quite a bit of background material on convex sets, polytopes, polyhedra and projective spaces. I have included a rather thorough treatment of the equivalence of V-polytopes and H-polytopes and also of the equivalence of V-polyhedra and H-polyhedra, which is a bit harder. In particular, the Fourier-Motzkin elimination method (a version of Gaussian elimination for inequalities) is discussed in some detail. I also had to include some material on projective spaces, projective maps and polar duality w.r.t. a nondegenerate quadric in order to define a suitable notion of \projective polyhedron based on cones. To the best of our knowledge, this notion of projective polyhedron is new. We also believe that some of our proofs establishing the equivalence of V-polyhedra and H-polyhedra are new