Gallier, Jean H
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Publication Typing Untyped Lambda-Terms, or Reducibility Strikes Again!(1994-12-07) Gallier, Jean HIt was observed by Curry that when (untyped) λ-terms can be assigned types, for example, simple types, these terms have nice properties (for example, they are strongly normalizing). Coppo, Dezani, and Veneri, introduced type systems using conjunctive types, and showed that several important classes of (untyped) terms can be characterized according to the shape of the types that can be assigned to these terms. For example, the strongly normalizable terms, the normalizable terms, and the terms having head-normal forms, can be characterized in some systems D and DΩ. The proofs use variants of the method of reducibility. In this paper, we present a uniform approach for proving several meta-theorems relating properties of λ-terms and their typability in the systems D and DΩ. Our proofs use a new and more modular version of the reducibility method. As an application of our metatheorems, we show how the characterizations obtained by Coppo, Dezani, Veneri, and Pottinger, can be easily rederived. We also characterize the terms that have weak head-normal forms, which appears to be new. We conclude by stating a number of challenging open problems regarding possible generalizations of the realizability method.Publication On the Correspondence Between Proofs and Lambda-Terms(1993-05-27) Gallier, Jean HThe 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).Publication Proving Properties of Typed Lambda-Terms Using Realizability, Covers, and Sheaves (Preliminary Version)(1993-11-10) Gallier, Jean HWe 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). 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's recursive realizability and a variant of Kreisel's 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. The above approach is applied to the simply-typed λ-calculus (with types →, x, +, and ⊥), and to the second-order (polymorphic λ-calculus (with types → and ∀2), for which it yields a new theorem.Publication On Some Quadratic Optimization Problems Arising in Computer Vision(2011-11-18) Gallier, Jean HPublication On Girard's "Candidats de Reductibilité"(1989-11-01) Gallier, Jean HWe attempt to elucidate the conditions required on Girard's 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's 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's conditions), and the second set consists of Girard's original conditions. We also compare these conditions, and prove that Girard's conditions are stronger than Tait's conditions. We give a new proof of the Church-Rosser theorem for both β-reduction and βη-reduction, using the modified version of Girard's 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's higher-order polymorphic calculus Fω, and in appendix 1, to Fω with product types.Publication Structure From Motion With Directional Correspondence for Visual Odometry(2011-01-01) Naroditsky, Oleg; Zhou, Xun S.; Gallier, Jean H; Roumeliotis, Stergios I.; Daniilidis, KostasThis report presents two efficient solutions to the two-view, relative pose problem from three image point correspondences and one common reference direction. This three-plus-one problem can be used either as a substitute for the classic five-point algorithm using a vanishing point for the reference direction, or to make use of an inertial measurement unit commonly available on robots and mobile devices, where the gravity vector becomes the reference direction. We provide a simple closed-form solution and a solution based on techniques from algebraic geometry and investigate numerical and computational advantages of each approach. In a set of real experiments, we demonstrate the power of our approach by comparing it to the five-point method in a hypothesize-and-test visual odometry setting.Publication Logarithms and Square Roots of Real Matrices(2008-05-02) Gallier, Jean HThe need for computing logarithms or square roots of real matrices arises in a number of applied problems. A significant class of problems comes from medical imaging. One of these problems is to interpolate and to perform statistics on data represented by certain kinds of matrices (such as symmetric positive definite matrices in DTI). Another important and difficult problem is the registration of medical images. For both of these problems, the ability to compute logarithms of real matrices turns out to be crucial. However, not all real matrices have a real logarithm and thus, it is important to have sufficient conditions for the existence (and possibly the uniqueness) of a real logarithm for a real matrix. Such conditions (involving the eigenvalues of a matrix) are known, both for the logarithm and the square root. As far as I know, with the exception of Higham's recent book [18], proofs of the results involving these conditions are scattered in the literature and it is not easy to locate them. Moreover, Higham's excellent book assumes a certain level of background in linear algebra that readers interested in applications to medical imaging may not possess so we feel that a more elementary presentation might be a valuable supplement to Higham [18]. In this paper, I present a unified exposition of these results, including a proof of the existence of the Real Jordan Form, and give more direct proofs of some of these results using the Real Jordan Form.Publication A Proof of Strong Normalization for the Theory of Constructions Using a Kripke-Like Interpretation(1990-07-01) Coquand, Thierry; Gallier, Jean HWe give a proof that all terms that type-check in the theory of contructions are strongly normalizing (under ß-reduction). The main novelty of this proof is that it uses a "Kripke-like" interpretation of the types and kinds, and that it does not use infinite contexts. We explore some consequences of strong normalization, consistency and decidability of typechecking. We also show that our proof yields another proof of strong normalization for LF (under ß-reduction), using the reducibility method.Publication Complete Sets of Transformations for General E-Unification(1989-02-01) Gallier, Jean H; Snyder, WayneThis paper is concerned with E-unification in arbitrary equational theories. We extend the method of transformations on systems of terms, developed by Martelli-Montanari for standard unification, to E-unification by giving two sets of transformations, BT and T, which are proved to be sound and complete in the sense that a complete set of E-unifiers for any equational theory E can be enumerated by either of these sets. The set T is an improvement of BT, in that many E-unifiers produced by BT will be weeded out by T. In addition, we show that a generalization of surreduction (also called narrowing) combined with the computation of critical pairs is complete. A new representation of equational proofs as certain kinds of trees is used to prove the completeness of the set BT in a rather direct fashion that parallels the completeness of the transformations in the case of (standard) unification. The completeness of T and the generalization of surreduction is proved by a method inspired by the concept of unfailing completion, using an abstract (and simpler) notion of the completion of a set of equations.Publication Kripke Models and the (In)equational Logic of the Second-Order Lambda-Calculus(1995-09-01) Gallier, Jean HWe 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) as well as 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 W 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) sЄТ to take care of type abstraction. However, we strive for simplicity, and we only use very elementary categorical concepts. Consequently, we believe that the models described in this paper are more palatable than abstract categorical models which require much more sophisticated machinery,(and are not models of rewrite rules anyway). 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).