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MartinLöf Complexes
, 2009
"... In this paper we define MartinLöf complexes to be algebras for monads on the category of (reflexive) globular sets which freely add cells in accordance with the rules of intensional MartinLöf type theory. We then study the resulting categories of algebras for several theories. Our principal resu ..."
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In this paper we define MartinLöf complexes to be algebras for monads on the category of (reflexive) globular sets which freely add cells in accordance with the rules of intensional MartinLöf type theory. We then study the resulting categories of algebras for several theories. Our principal result is that there exists a cofibrantly generated Quillen model structure on
Fractional Types
"... Abstract. In previous work, we developed a firstorder, informationpreserving, and reversible programming language Π founded on type isomorphisms. Being restricted to firstorder types limits the expressiveness of the language: it is not possible, for example, to abstract common program fragments in ..."
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Abstract. In previous work, we developed a firstorder, informationpreserving, and reversible programming language Π founded on type isomorphisms. Being restricted to firstorder types limits the expressiveness of the language: it is not possible, for example, to abstract common program fragments into a higherlevel combinator. In this paper, we introduce a higherorder extension of Π based on the novel concept of fractional types 1/b. Intuitively, a value of a fractional type 1/v represents negative information. A function is modeled by a pair (1/v1, v2) with 1/v1 representing the needed argument and v2 representing the result. Fractional values are firstclass: they can be freely propagated and transformed but must ultimately — in a complete program — be offset by the corresponding amount of positive information. 1
Program Verification in Synthetic Domain Theory
, 1995
"... Synthetic Domain Theory provides a setting to consider domains as sets with certain closure properties for computing suprema of ascending chains. As a consequence the notion of domain can be internalized which allows one to construct and reason about solutions of recursive domain equations. Moreover ..."
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Synthetic Domain Theory provides a setting to consider domains as sets with certain closure properties for computing suprema of ascending chains. As a consequence the notion of domain can be internalized which allows one to construct and reason about solutions of recursive domain equations. Moreover, one can derive that all functions are continuous. In this thesis such a synthetic theory of domains (#domains) is developed based on a few axioms formulated in an adequate intuitionistic higherorder logic. This leads to an elegant theory of domains. It integrates the positive features of several approaches in the literature. In contrast to those, however, it is model independent and can therefore be formalized. A complete formalization of the whole theory of #domains has been coded into a proofchecker (Lego) for impredicative type theory. There one can exploit dependent types in order to express program modules and modular specifications. As an application of this theory an entirely fo...
Automating Inversion of Inductive Predicates in Coq
 In BRA Workshop on Types for Proofs and Programs
, 1995
"... . An inductive definition of a set is often informally presented by giving some rules that explain how to build the elements of the set. The closure property states that any object is in the set if and only if it has been generated according to the formation rules. This is enough to justify case ..."
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. An inductive definition of a set is often informally presented by giving some rules that explain how to build the elements of the set. The closure property states that any object is in the set if and only if it has been generated according to the formation rules. This is enough to justify case analysis reasoning: we can read the formation rules backwards to derive the necessary conditions for a given instance to hold. The problem of inversion consists in finding out these conditions. In this paper we address the problem of deriving inversion lemmas in logical frameworks based on Type Theory that have been extended with inductive definitions at the primitive level. These frameworks associate to each inductive definition a case analysis principle corresponding to the closure property. In this formal context, inversion lemmas can be seen as derived case analysis principles. Though they are intuitively simple they are curiously hard to formalize. We relate first inversion to co...
LogiCal Project
, 2004
"... This document is the Reference Manual of version 8.0 of the COQ proof assistant. A companion volume, the COQ Tutorial, is provided for the beginners. It is advised to read the Tutorial first. A new book [13] on practical uses of the COQ system will be published in 2004 and is a good support for both ..."
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This document is the Reference Manual of version 8.0 of the COQ proof assistant. A companion volume, the COQ Tutorial, is provided for the beginners. It is advised to read the Tutorial first. A new book [13] on practical uses of the COQ system will be published in 2004 and is a good support for both the beginner and the advanced user.
A Framework for Internalizing Relations into Type Theory
"... Abstract. This paper introduces the concept of internalization structure, which can be used to incorporate certain relations into F Π, a variant of system F, while maintaining termination of the new system. We will call this process of incorporation internalization, F Π the base system and the new s ..."
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Abstract. This paper introduces the concept of internalization structure, which can be used to incorporate certain relations into F Π, a variant of system F, while maintaining termination of the new system. We will call this process of incorporation internalization, F Π the base system and the new system after the incorporation the internalized system. We first specify the syntax, and then the semantics of F Π via the TaitGirard reducibility method. We then define internalization structure. We show that we can obtain a terminating internalized system from an internalization structure. Finally, as motivating examples, we demonstrate how our framework can be applied to internalize subtyping, fullbeta term equality and termtype inhabitation relations. 1
General Terms
"... In some substructural logics, the memory used by proofs is treated as a firstclass multiplicative resource, but the choices made by those proofs are not. Since we can convert between space and time complexity, these “resource conscious ” logics are therefore not actually guaranteed to preserve memo ..."
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In some substructural logics, the memory used by proofs is treated as a firstclass multiplicative resource, but the choices made by those proofs are not. Since we can convert between space and time complexity, these “resource conscious ” logics are therefore not actually guaranteed to preserve memory—for example, linear logic allows the erasure and duplication of natural numbers with time complexity proportional to their size. In order to fully account for spacetime tradeoffs, we augment contexts to track all information, not just multiplicative resources. This creates a reversible, fully resourcepreserving logic which allows us to examine the hidden information effects in linear logic and study reversible computation from a prooftheoretic perspective.