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Reasoning with inductively defined relations in the HOL theorem prover
, 1992
"... Abstract: Inductively defined relations are among the basic mathematical tools of computer science. Examples include evaluation and computation relations in structural operational semantics, labelled transition relations in process algebra semantics, inductivelydefined typing judgements, and proof ..."
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Abstract: Inductively defined relations are among the basic mathematical tools of computer science. Examples include evaluation and computation relations in structural operational semantics, labelled transition relations in process algebra semantics, inductivelydefined typing judgements, and proof systems in general. This paper describes a set of HOL theoremproving tools for reasoning about such inductively defined relations. We also describe a suite of worked examples using these tools. First printed: August 1992
Deliverables: A Categorical Approach to Program Development in Type Theory
, 1992
"... This thesis considers the problem of program correctness within a rich theory of dependent types, the Extended Calculus of Constructions (ECC). This system contains a powerful programming language of higherorder primitive recursion and higherorder intuitionistic logic. It is supported by Pollack's ..."
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Cited by 24 (1 self)
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This thesis considers the problem of program correctness within a rich theory of dependent types, the Extended Calculus of Constructions (ECC). This system contains a powerful programming language of higherorder primitive recursion and higherorder intuitionistic logic. It is supported by Pollack's versatile LEGO implementation, which I use extensively to develop the mathematical constructions studied here. I systematically investigate Burstall's notion of deliverable, that is, a program paired with a proof of correctness. This approach separates the concerns of programming and logic, since I want a simple program extraction mechanism. The \Sigmatypes of the calculus enable us to achieve this. There are many similarities with the subset interpretation of MartinLof type theory. I show that deliverables have a rich categorical structure, so that correctness proofs may be decomposed in a principled way. The categorical combinators which I define in the system package up much logical bo...
Complete sequent calculi for induction and infinite descent
 Proceedings of LICS22
, 2007
"... This paper compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system supports traditional proof by induction, with induction rules formulated as sequent rules for introducing induct ..."
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Cited by 18 (6 self)
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This paper compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system supports traditional proof by induction, with induction rules formulated as sequent rules for introducing inductively defined predicates on the left of sequents. We show this system to be cutfree complete with respect to a natural class of Henkin models; the eliminability of cut follows as a corollary. The second system uses infinite (nonwellfounded) proofs to represent arguments by infinite descent. In this system, the left rules for inductively defined predicates are simple casesplit rules, and an infinitary, global condition on proof trees is required to ensure soundness. We show this system to be cutfree complete with respect to standard models, and again infer the eliminability of cut. The second infinitary system is unsuitable for formal reasoning. However, it has a natural restriction to proofs given by regular trees, i.e. to those proofs representable by finite graphs. This restricted “cyclic ” system subsumes the first system for proof by induction. We conjecture that the two systems are in fact equivalent, i.e., that proof by induction is equivalent to regular proof by infinite descent.
Programming interfaces and basic topology
 Annals of Pure and Applied Logic
, 2005
"... A pattern of interaction that arises again and again in programming, is a “handshake”, in which two agents exchange data. The exchange is thought of as provision of a service. Each interaction is initiated by a specific agent —the client or Angel, and concluded by the other —the server or Demon. We ..."
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A pattern of interaction that arises again and again in programming, is a “handshake”, in which two agents exchange data. The exchange is thought of as provision of a service. Each interaction is initiated by a specific agent —the client or Angel, and concluded by the other —the server or Demon. We present a category in which the objects —called interaction structures in the paper — serve as descriptions of services provided across such handshaken interfaces. The morphisms —called (general) simulations— model components that provide one such service, relying on another. The morphisms are relations between the underlying sets of the interaction structures. The proof that a relation is a simulation can serve (in principle) as an executable program, whose specification is that it provides the service described by its domain, given an implementation of the service described by its codomain.
The Mathematical Import Of Zermelo's WellOrdering Theorem
 Bull. Symbolic Logic
, 1997
"... this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs ..."
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Cited by 5 (1 self)
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this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his wellknown paradox, an early expression of our motif. The motif becomes fully manifest through the study of functions f :
Reasoning with Inductively Defined Relations in the HOL Theorem Prover
, 1992
"... : Inductively defined relations are among the basic mathematical tools of computer science. Examples include evaluation and computation relations in structural operational semantics, labelled transition relations in process algebra semantics, inductivelydefined typing judgements, and proof systems ..."
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: Inductively defined relations are among the basic mathematical tools of computer science. Examples include evaluation and computation relations in structural operational semantics, labelled transition relations in process algebra semantics, inductivelydefined typing judgements, and proof systems in general. This paper describes a set of HOL theoremproving tools for reasoning about such inductively defined relations. We also describe a suite of worked examples using these tools. First printed: August 1992 Parts of this report have previously appeared as: T. Melham, `A Package for Inductive Relation Definitions in HOL', in Proceedings of the 1991 International Workshop on the HOL Theorem Proving System and its Applications, Davis, August 1991, edited by M. Archer, J. J. Joyce, K. N. Levitt, and P. J. Windley (IEEE Computer Society Press, 1992), pp. 350357. Contents Introduction 4 1 Inductive definitions 5 1.1 Rule induction : : : : : : : : : : : : : : : : : : : : : : : : :...
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...
The Bulletin of Symbolic Logic
 Bulletin of Symbolic Logic
, 2002
"... We consider fixed point logics, i.e., extensions of first order predicate logic with operators defining fixed points. A number of such operators, generalizing inductive definitions, have been studied in the context of finite model theory, including nondeterministic and alternating operators. We r ..."
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We consider fixed point logics, i.e., extensions of first order predicate logic with operators defining fixed points. A number of such operators, generalizing inductive definitions, have been studied in the context of finite model theory, including nondeterministic and alternating operators. We review results established in finite model theory, and also consider the expressive power of the resulting logics on infinite structures. In particular, we establish the relationship between inflationary and nondeterministic fixed point logics and second order logic, and we consider questions related to the determinacy of games associated with alternating fixed points.
One Useful Logic That Defines Its Own Truth
"... Abstract. Existential fixed point logic (EFPL) is a natural fit for some applications, and the purpose of this talk is to attract attention to EFPL. The logic is also interesting in its own right as it has attractive properties. One of those properties is rather unusual: truth of formulas can be def ..."
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Abstract. Existential fixed point logic (EFPL) is a natural fit for some applications, and the purpose of this talk is to attract attention to EFPL. The logic is also interesting in its own right as it has attractive properties. One of those properties is rather unusual: truth of formulas can be defined (given appropriate syntactic apparatus) in the logic. We mentioned that property elsewhere, and we use this opportunity to provide the proof. Believe those who are seeking the truth. Doubt those who find it. —André Gide 1
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"... Abstract. This is the first of a series of three articles devoted to the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers) and establishing the relationships between these notions. In the present paper, we undertake an extended survey o ..."
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Abstract. This is the first of a series of three articles devoted to the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers) and establishing the relationships between these notions. In the present paper, we undertake an extended survey of the different strands of research to date on higher type computability, bringing together material from recursion theory, constructive logic and computer science, and emphasizing the historical development of the ideas. The paper thus serves as a reasonably comprehensive survey of the literature on higher type computability.