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105
Coinductive Axiomatization of Recursive Type Equality and Subtyping
, 1998
"... e present new sound and complete axiomatizations of type equality and subtype inequality for a firstorder type language with regular recursive types. The rules are motivated by coinductive characterizations of type containment and type equality via simulation and bisimulation, respectively. The mai ..."
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Cited by 76 (2 self)
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e present new sound and complete axiomatizations of type equality and subtype inequality for a firstorder type language with regular recursive types. The rules are motivated by coinductive characterizations of type containment and type equality via simulation and bisimulation, respectively. The main novelty of the axiomatization is the fixpoint rule (or coinduction principle), which has the form A; P ` P A ` P (Fix) where P is either a type equality = 0 or type containment 0 and the proof of the premise must be contractive in a formal sense. In particular, a proof of A; P ` P using the assumption axiom is not contractive. The fixpoint rule embodies a finitary coinduction principle and thus allows us to capture a coinductive relation in the fundamentally inductive framework of inference systems. The new axiomatizations are more concise than previous axiomatizations, particularly so for type containment since no separate axiomatization of type equality is required, as in A...
On the Origins of Bisimulation and Coinduction
"... The origins of bisimulation and bisimilarity are examined, in the three fields where they have been ..."
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Cited by 61 (0 self)
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The origins of bisimulation and bisimilarity are examined, in the three fields where they have been
TypeBased Termination of Recursive Definitions
, 2002
"... This article The purpose of this paper is to introduce b, a simply typed calculus that supports typebased recursive definitions. Although heavily inspired from previous work by Giménez (Giménez 1998) and closely related to recent work by Amadio and Coupet (Amadio and CoupetGrimal 1998), the techn ..."
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Cited by 53 (4 self)
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This article The purpose of this paper is to introduce b, a simply typed calculus that supports typebased recursive definitions. Although heavily inspired from previous work by Giménez (Giménez 1998) and closely related to recent work by Amadio and Coupet (Amadio and CoupetGrimal 1998), the technical machinery behind our system puts a slightly different emphasis on the interpretation of types. More precisely, we formalize the notion of typebased termination using a restricted form of type dependency (a.k.a. indexed types), as popularized by (Xi and Pfenning 1998; Xi and Pfenning 1999). This leads to a simple and intuitive system which is robust under several extensions, such as mutually inductive datatypes and mutually recursive function definitions; however, such extensions are not treated in the paper
Structural Recursive Definitions in Type Theory
 Automata, Languages and Programming, 25th International Colloquium, ICALP’98
, 1998
"... We introduce an extension of the Calculus of Construction with inductive and coinductive types that preserves strong normalisation for a lazy computation relation. This extension considerably enlarges the expressiveness of the language, enabling a direct translation of recursive programs, while kee ..."
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Cited by 44 (0 self)
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We introduce an extension of the Calculus of Construction with inductive and coinductive types that preserves strong normalisation for a lazy computation relation. This extension considerably enlarges the expressiveness of the language, enabling a direct translation of recursive programs, while keeping a relatively simple collection of typing rules. 1 Introduction The last twenty five years have seen an increasing development of different proof environments based on type theory. Several type theories have been proposed as a foundation of such proof environments [15, 6, 16], trying to find an accurate compromise between two criteria. On the one hand, we search for extensions of type theory that preserve its conceptual simplicity of type theory (a few primitive constructions, a small number of typing rules) and metatheoretical properties ensuring its soundness and a direct mechanisation (strong normalisation, decidability of typechecking, etc). On the other hand, we would like to pro...
General recursion via coinductive types
 Logical Methods in Computer Science
"... Vol. 1 (2:1) 2005, pp. 1–28 ..."
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Termination Checking with Types
, 1999
"... The paradigm of typebased termination is explored for functional programming with recursive data types. The article introduces , a lambdacalculus with recursion, inductive types, subtyping and bounded quanti cation. Decorated type variables representing approximations of inductive types ..."
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Cited by 36 (6 self)
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The paradigm of typebased termination is explored for functional programming with recursive data types. The article introduces , a lambdacalculus with recursion, inductive types, subtyping and bounded quanti cation. Decorated type variables representing approximations of inductive types are used to track the size of function arguments and return values. The system is shown to be type safe and strongly normalizing. The main novelty is a bidirectional type checking algorithm whose soundness is established formally.
Filters on coinductive streams, an application to eratosthenes’ sieve
 Typed Lambda Calculi and Applications, 7th International Conference, TLCA 2005
, 2005
"... Our objective is to describe a formal proof of correctness for the following Haskell [13] program in a type theorybased proof verification system, such as the Coq system [10, 1]. sieve (p:rest) = p:sieve [r  r < rest, r ‘rem ‘ p / = 0] primes = sieve [2..] This program is a functional impleme ..."
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Cited by 27 (6 self)
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Our objective is to describe a formal proof of correctness for the following Haskell [13] program in a type theorybased proof verification system, such as the Coq system [10, 1]. sieve (p:rest) = p:sieve [r  r < rest, r ‘rem ‘ p / = 0] primes = sieve [2..] This program is a functional implementation of Eratosthenes ’ sieve that consists in removing all multiples of previously found primes from the sequence of natural numbers. We want to prove that the expression primes is the stream containing all the prime numbers in increasing order. This work relies on coinductive types [5, 11, 12] because the program manipulates infinite lists, also known as streams. It first uses the infinite list of natural numbers larger than 2, then the infinite list of numbers larger than 3 and containing no multiples of 2, then the infinite list of numbers larger than 4 and containing no multiples of prime numbers smaller than 4, and so on. This example was initially proposed as a challenge by G. Kahn and used as an illustration of a program and its proof of correctness in a
Calculus in Coinductive Form
, 1998
"... artment of Computer Science, University of Edinburgh, Edinburgh. Email: mhe@dcs.ed.ac.uk tions (2) and (1) normalize the integral with respect to the subintegral function and the interval of integration. 1 Reapplying equation (3) yields the Taylor (Maclaurin) expansion f = f(0) :: f 0 = f(0) ..."
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Cited by 25 (2 self)
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artment of Computer Science, University of Edinburgh, Edinburgh. Email: mhe@dcs.ed.ac.uk tions (2) and (1) normalize the integral with respect to the subintegral function and the interval of integration. 1 Reapplying equation (3) yields the Taylor (Maclaurin) expansion f = f(0) :: f 0 = f(0) :: f 0 (0) :: f 00 . . . = f(0) :: f 0 (0) :: \Delta \Delta \Delta :: f (n) (0) :: \Delta \Delta \Delta Unfolding the above definition of ::, which amounts to iterated integration, one finally gets f(x) = f(0) + f 0 (0)x + \Delta \Delta \Delta + f (n) (0)x n<F2
Cyclic proofs for firstorder logic with inductive definitions
 In TABLEAUX’05, volume 3702 of LNCS
, 2005
"... Abstract. We consider a cyclic approach to inductive reasoning in the setting of firstorder logic with inductive definitions. We present a proof system for this language in which proofs are represented as finite, locally sound derivation trees with a “repeat function ” identifying cyclic proof sect ..."
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Cited by 23 (7 self)
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Abstract. We consider a cyclic approach to inductive reasoning in the setting of firstorder logic with inductive definitions. We present a proof system for this language in which proofs are represented as finite, locally sound derivation trees with a “repeat function ” identifying cyclic proof sections. Soundness is guaranteed by a wellfoundedness condition formulated globally in terms of traces over the proof tree, following an idea due to Sprenger and Dam. However, in contrast to their work, our proof system does not require an extension of logical syntax by ordinal variables. A fundamental question in our setting is the strength of the cyclic proof system compared to the more familiar use of a noncyclic proof system using explicit induction rules. We show that the cyclic proof system subsumes the use of explicit induction rules. In addition, we provide machinery for manipulating and analysing the structure of cyclic proofs, based primarily on viewing them as generating regular infinite trees, and also formulate a finitary trace condition sufficient (but not necessary) for soundness, that is computationally and combinatorially simpler than the general trace condition. 1
Ensuring Streams Flow
 Proc. 6 th AMAST
, 1997
"... . It is our aim to develop an elementary strong functional programming (ESFP) system. To be useful, ESFP should include structures such as streams which can be computationally unwound infinitely often. We describe a syntactic analysis to ensure that infinitely proceeding structures, which we shall t ..."
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Cited by 23 (2 self)
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. It is our aim to develop an elementary strong functional programming (ESFP) system. To be useful, ESFP should include structures such as streams which can be computationally unwound infinitely often. We describe a syntactic analysis to ensure that infinitely proceeding structures, which we shall term codata, are productive. This analysis is an extension of the check for guardedness that has been used with definitions over coinductive types in MartinLof's type theory and in the calculus of constructions. Our analysis is presented as a form of abstract interpretation that allows a wider syntactic class of corecursive definitions to be recognised as productive than in previous work. Thus programmers will have fewer restrictions on their use of infinite streams within a strongly normalizing functional language. 1 Introduction We aim to develop an Elementary Strong Functional Programming (ESFP) system. That is, we wish to exhibit a language that has the strong normalization (every progr...