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21
Comparing Cubes
"... We study the cube of type assignment systems, as introduced in [10]. This cube is obtained from Barendregt's typed cube [1] via a natural type erasing function E, that erases type information from terms. We prove that the systems in the former cube enjoy good computational properties, like ..."
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We study the cube of type assignment systems, as introduced in [10]. This cube is obtained from Barendregt's typed cube [1] via a natural type erasing function E, that erases type information from terms. We prove that the systems in the former cube enjoy good computational properties, like subject reduction and strong normalization. We study the relationship between the two cubes, which leads to some unexpected results in the eld of systems with dependent types.
CurryStyle Types for Nominal Terms ⋆
"... Abstract. We define a rank 1 polymorphic type system for nominal terms, where typing environments type atoms, variables and function symbols. The interaction between type assumptions for atoms and substitution for variables is subtle: substitution does not avoid capture and so can move an atom into ..."
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Abstract. We define a rank 1 polymorphic type system for nominal terms, where typing environments type atoms, variables and function symbols. The interaction between type assumptions for atoms and substitution for variables is subtle: substitution does not avoid capture and so can move an atom into multiple different typing contexts. We give typing rules such that principal types exist and are decidable for a fixed typing environment. αequivalent nominal terms have the same types; a nontrivial result because nominal terms include explicit constructs for renaming atoms. We investigate rule formats to guarantee subject reduction. Our system is in a convenient Currystyle, so the user has no need to explicitly type abstracted atoms.
Coq and Hardware Verification: a Case Study
 TPHOLs'96, LCNS 1125
, 1996
"... . We present, on the example of a lefttoright comparator, several approaches for verifying a class of circuits with the Coq proofassistant. The great expressiveness of the Calculus of Inductive Constructions allows us to give precise and general specifications. Thanks to Coq's higherorde ..."
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. We present, on the example of a lefttoright comparator, several approaches for verifying a class of circuits with the Coq proofassistant. The great expressiveness of the Calculus of Inductive Constructions allows us to give precise and general specifications. Thanks to Coq's higherorder logic, we state general results for establishing the correctness of such circuits. Finally, exploiting the constructive aspect of the logic, we show how to synthezise automatically a certified circuit from its specification. 1 Introduction During the past decade, intensive and dynamic research has developed in the field of mechanized theorem prover design, resulting in a great deal of new proof assistants. Hardware verification has been one of the original motivations and main applications of this area. Among the earliest and most significant achievements, let us mentionned the works of Gordon's group using HOL [14, 6] and the proof of the FM8501 [?] with Nqthm [5]. On the one hand, using ...
A short article for the Encyclopedia of Artificial Intelligence: Second Edition “Logic, Higherorder”
, 1991
"... While firstorder logic has syntactic categories for individuals, functions, and predicates, only quantification over individuals is permitted. Many concepts when translated into logic are, however, naturally expressed using quantifiers over functions and predicates. Leibniz’s principle of equality, ..."
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While firstorder logic has syntactic categories for individuals, functions, and predicates, only quantification over individuals is permitted. Many concepts when translated into logic are, however, naturally expressed using quantifiers over functions and predicates. Leibniz’s principle of equality, for example, states that two objects are to be taken as equal if they share the same properties; that is, a = b can be defined as ∀P [P (a) ≡ P (b)]. Of course, firstorder logic is very strong and it is possible to encode such a statement into it. For example, let app be a firstorder predicate symbol of arity two that is used to stand for the application of a predicate to an individual. Semantically, app(P, x) would mean P satisfies x or that the extension of the predicate P contains x. In this case, the quantified expression could be rewritten as the firstorder expression ∀P [app(P, a) ≡ app(P, b)] (appropriate axioms for describing app are required). Such an encoding is often done in a multisorted logic setting, where one sort is for individuals and another sort is for predicates over individuals. Settheory is another firstorder language that encodes such higherorder concepts
October 1990 A short article for the
"... tax of a higherorder logic is to introduce some kind of typing scheme. One approach types firstorder individuals with #, sets of individuals with ###, sets of pairs of individuals with ####, sets of sets of individuals with #####, etc. Such a typing scheme does not provide types for function ..."
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tax of a higherorder logic is to introduce some kind of typing scheme. One approach types firstorder individuals with #, sets of individuals with ###, sets of pairs of individuals with ####, sets of sets of individuals with #####, etc. Such a typing scheme does not provide types for function symbols. Since in some treatments of higherorder logic, functions can be represented by their graphs, i.e. certain kinds of sets of ordered pairs, this lack is not a serious restriction. Identifying functions up to their graphs does, of course, treat functions extensionally, something that might be 1 too strong in some applications. (A logic is extensional if whenever two predicates or two functions are equal on all their arguments, they themselves are equal.) A more general approach to typing is that used in the Simple Theory of Types (Church, 1940). Here again, the type # is used to denote the set of firstorder individuals, and the type o is used to denote the sort of booleans, false
Abstract
"... We study the cube of type assignment systems, as introduced in [10]. This cube is obtained from Barendregt’s typedcube [1] via a natural type erasing function E, that erases type information from terms. We prove that the systems in the former cube enjoy good computational properties, like subject r ..."
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We study the cube of type assignment systems, as introduced in [10]. This cube is obtained from Barendregt’s typedcube [1] via a natural type erasing function E, that erases type information from terms. We prove that the systems in the former cube enjoy good computational properties, like subject reduction and strong normalization. We study the relationship between the two cubes, which leads to some unexpected results in the field of systems with dependent types.
Note Coherence spaces are untopological
, 1999
"... The purpose of this note is to show that stability is not a topological property. That is, that coherence spaces  except for the socalled flat ones  cannot be equipped with a topology such that the notions of continuity and stability coincide. 1 ..."
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The purpose of this note is to show that stability is not a topological property. That is, that coherence spaces  except for the socalled flat ones  cannot be equipped with a topology such that the notions of continuity and stability coincide. 1
This is a preprint of a paper that has been submitted to Information and Computation. On Functors Expressible in the Polymorphic Typed Lambda Calculus
, 1991
"... Given a model of the polymorphic typed lambda calculus based upon a Cartesian closed category K, there will be functors from K to K whose action on objects can be expressed by type expressions and whose action on morphisms can be expressed by ordinary expressions. We show that if T is such a functor ..."
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Given a model of the polymorphic typed lambda calculus based upon a Cartesian closed category K, there will be functors from K to K whose action on objects can be expressed by type expressions and whose action on morphisms can be expressed by ordinary expressions. We show that if T is such a functor then there is a weak initial Talgebra and if, in addition, K possesses equalizers of all subsets of its morphism sets, then there is an initial Talgebra. These results are used to establish the impossibility of certain models, including those in which types denote sets and S → S ′ denotes the set of all functions from S to S ′.