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A Variable Typed Logic of Effects
 Information and Computation
, 1993
"... In this paper we introduce a variable typed logic of effects inspired by the variable type systems of Feferman for purely functional languages. VTLoE (Variable Typed Logic of Effects) is introduced in two stages. The first stage is the firstorder theory of individuals built on assertions of equalit ..."
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In this paper we introduce a variable typed logic of effects inspired by the variable type systems of Feferman for purely functional languages. VTLoE (Variable Typed Logic of Effects) is introduced in two stages. The first stage is the firstorder theory of individuals built on assertions of equality (operational equivalence `a la Plotkin), and contextual assertions. The second stage extends the logic to include classes and class membership. The logic we present provides an expressive language for defining and studying properties of programs including program equivalences, in a uniform framework. The logic combines the features and benefits of equational calculi as well as program and specification logics. In addition to the usual firstorder formula constructions, we add contextual assertions. Contextual assertions generalize Hoare's triples in that they can be nested, used as assumptions, and their free variables may be quantified. They are similar in spirit to program modalities in ...
Improvement in a Lazy Context: An Operational Theory for CallByNeed
 Proc. POPL'99, ACM
, 1999
"... Machine The semantics presented in this section is essentially Sestoft's \mark 1" abstract machine for laziness [Sestoft 1997]. In that paper, he proves his abstract machine 6 A. K. Moran and D. Sands h fx = Mg; x; S i ! h ; M; #x : S i (Lookup) h ; V; #x : S i ! h fx = V g; V; S i (Up ..."
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Cited by 44 (8 self)
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Machine The semantics presented in this section is essentially Sestoft's \mark 1" abstract machine for laziness [Sestoft 1997]. In that paper, he proves his abstract machine 6 A. K. Moran and D. Sands h fx = Mg; x; S i ! h ; M; #x : S i (Lookup) h ; V; #x : S i ! h fx = V g; V; S i (Update) h ; M x; S i ! h ; M; x : S i (Unwind) h ; x:M; y : S i ! h ; M [ y = x ]; S i (Subst) h ; case M of alts ; S i ! h ; M; alts : S i (Case) h ; c j ~y; fc i ~x i N i g : S i ! h ; N j [ ~y = ~x j ]; S i (Branch) h ; let f~x = ~ Mg in N; S i ! h f~x = ~ Mg; N; S i ~x dom(;S) (Letrec) Fig. 1. The abstract machine semantics for callbyneed. semantics sound and complete with respect to Launchbury's natural semantics, and we will not repeat those proofs here. Transitions are over congurations consisting of a heap, containing bindings, the expression currently being evaluated, and a stack. The heap is a partial function from variables to terms, and denoted in an identical manner to a coll...
Reasoning about local variables with operationallybased logical relations
 In LICS
, 1996
"... A parametric logical relation between the phrases of an Algollike language is presented. Its definition involves the structural operational semantics of the language, but was inspired by recent denotationallybased work of O’Hearn and Reynolds on translating Algol into a predicatively polymorphic l ..."
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Cited by 34 (3 self)
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A parametric logical relation between the phrases of an Algollike language is presented. Its definition involves the structural operational semantics of the language, but was inspired by recent denotationallybased work of O’Hearn and Reynolds on translating Algol into a predicatively polymorphic linear lambda calculus. The logical relation yields an applicative characterisation of contextual equivalence for the language and provides a useful (and complete) method for proving equivalences. Its utility is illustrated by giving simple and direct proofs of some contextual equivalences, including an interesting equivalence due to O’Hearn which hinges upon the undefinability of ‘snapback ’ operations (and which goes beyond the standard suite of ‘MeyerSieber ’ examples). Whilst some of the mathematical intricacies of denotational semantics are avoided, the hard work in this operational approach lies in establishing the ‘fundamental property’ for the logical relation—the proof of which makes use of a compactness property of fixpoint recursion with respect to evaluation of phrases. But once this property has been established, the logical relation provides a verification method with an attractively low mathematical overhead. 1.
A Tutorial on Coinduction and Functional Programming
 IN GLASGOW FUNCTIONAL PROGRAMMING WORKSHOP
, 1994
"... Coinduction is an important tool for reasoning about unbounded structures. This tutorial explains the foundations of coinduction, and shows how it justifies intuitive arguments about lazy streams, of central importance to lazy functional programmers. We explain from first principles a theory based ..."
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Cited by 30 (1 self)
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Coinduction is an important tool for reasoning about unbounded structures. This tutorial explains the foundations of coinduction, and shows how it justifies intuitive arguments about lazy streams, of central importance to lazy functional programmers. We explain from first principles a theory based on a new formulation of bisimilarity for functional programs, which coincides exactly with Morrisstyle contextual equivalence. We show how to prove properties of lazy streams by coinduction and derive Bird and Wadler's Take Lemma, a wellknown proof technique for lazy streams.
From SOS Rules to Proof Principles: An Operational Metatheory for Functional Languages
 In Proc. POPL'97, the 24 th ACM SIGPLANSIGACT Symposium on Principles of Programming Languages
, 1997
"... Structural Operational Semantics (SOS) is a widely used formalism for specifying the computational meaning of programs, and is commonly used in specifying the semantics of functional languages. Despite this widespread use there has been relatively little work on the imetatheoryj for such semantics. ..."
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Structural Operational Semantics (SOS) is a widely used formalism for specifying the computational meaning of programs, and is commonly used in specifying the semantics of functional languages. Despite this widespread use there has been relatively little work on the imetatheoryj for such semantics. As a consequence the operational approach to reasoning is considered ad hoc since the same basic proof techniques and reasoning tools are reestablished over and over, once for each operational semantics speciøcation. This paper develops some metatheory for a certain class of SOS language speciøcations for functional languages. We deøne a rule format, Globally Deterministic SOS (gdsos), and establish some proof principles for reasoning about equivalence which are sound for all languages which can be expressed in this format. More speciøcally, if the SOS rules for the operators of a language conform to the syntax of the gdsos format, then ffl a syntactic analogy of continuity holds, which rel...
Programming Metalogics with a Fixpoint Type
, 1992
"... A programming metalogic is a formal system into which programming languages can be translated and given meaning. The translation should both reflect the structure of the language and make it easy to prove properties of programs. This thesis develops certain metalogics using techniques of category th ..."
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Cited by 12 (6 self)
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A programming metalogic is a formal system into which programming languages can be translated and given meaning. The translation should both reflect the structure of the language and make it easy to prove properties of programs. This thesis develops certain metalogics using techniques of category theory and treats recursion in a new way. The notion of a category with fixpoint object is defined. Corresponding to this categorical structure there are type theoretic equational rules which will be present in all of the metalogics considered. These rules define the fixpoint type which will allow the interpretation of recursive declarations. With these core notions FIX categories are defined. These are the categorical equivalent of an equational logic which can be viewed as a very basic programming metalogic. Recursion is treated both syntactically and categorically. The expressive power of the equational logic is increased by embedding it in an intuitionistic predicate calculus, giving rise to the FIX logic. This contains propositions about the evaluation of computations to values and an induction principle which is derived from the definition of a fixpoint object as an initial algebra. The categorical structure which accompanies the FIX logic is defined, called a FIX hyperdoctrine, and certain existence and disjunction properties of FIX are stated. A particular FIX hyperdoctrine is constructed and used in the proof of the same properties. PCFstyle languages are translated into the FIX logic and computational adequacy reaulta are proved. Two languages are studied: Both are similar to PCF except one has call by value recursive function declararations and the other higher order conditionals. ...
FUNDIO: A LambdaCalculus with a letrec, case, Constructors, and an IOInterface: Approaching a Theory of unsafePerformIO. Frank report 16, Institut für
 Informatik, J.W. GoetheUniversität Frankfurt, September 2003. SSSS04. Manfred SchmidtSchauß, Marko Schütz, and
"... Abstract. A nondeterministic callbyneed lambdacalculus λndlr with case, constructors, letrec and a (nondeterministic) erratic choice, based on rewriting rules is investigated. A standard reduction is defined as a variant of leftmost outermost reduction. The semantics is defined by contextual e ..."
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Cited by 8 (0 self)
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Abstract. A nondeterministic callbyneed lambdacalculus λndlr with case, constructors, letrec and a (nondeterministic) erratic choice, based on rewriting rules is investigated. A standard reduction is defined as a variant of leftmost outermost reduction. The semantics is defined by contextual equivalence of expressions instead of using αβ(η)equivalence. It is shown that several program transformations are correct, for example all (deterministic) rules of the calculus, and in addition the rules for garbage collection, removing indirections and unique copy. This shows that the combination of a context lemma and a metarewriting on reductions using complete sets of commuting (forking, resp.) diagrams is a useful and successful method for providing a semantics of a functional programming language and proving correctness of program transformations. 1
Partial computations in constructive type theory
 JOURNAL OF LOGIC AND COMPUTATION
, 1991
"... Constructive type theory as conceived by Per MartinLöf has a very rich type system, but partial functions cannot be typed. This also makes it impossible to directly write recursive programs. In this paper a constructive type theory Red is defined which includes a partial type constructor A; objects ..."
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Cited by 7 (5 self)
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Constructive type theory as conceived by Per MartinLöf has a very rich type system, but partial functions cannot be typed. This also makes it impossible to directly write recursive programs. In this paper a constructive type theory Red is defined which includes a partial type constructor A; objects in the type A may diverge, but if they converge, they must be members of A. A fixed point typing principle is given to allow typing of recursive functions. The extraction paradigm of type theory, whereby programs are automatically extracted from constructive proofs, is extended to allow extraction of fixed points. There is a Scott fixed point induction principle for reasoning about these functions. Soundness of the theory is proven. Type theory becomes a more expressive programming logic as a result.
Possibilities and limitations of callbyneed space improvement
 In Proceedings of the sixth ACM SIGPLAN International Conference on Functional Programming
, 2001
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Hybrid PartialTotal Type Theory
, 1995
"... In this paper a hybrid type theory HTT is defined which combines the programming language notion of partial type with the logical notion of total type into a single theory. A new partial type constructor A is added to the type theory: objects in A may diverge, but if they converge, they must be memb ..."
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Cited by 5 (0 self)
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In this paper a hybrid type theory HTT is defined which combines the programming language notion of partial type with the logical notion of total type into a single theory. A new partial type constructor A is added to the type theory: objects in A may diverge, but if they converge, they must be members of A. A fixed point typing rule is given to allow for typing of fixed points. The underlying theory is based on ideas from Feferman's Class Theory and Martin Lof's Intuitionistic Type Theory. The extraction paradigm of constructive type theory is extended to allow direct extraction of arbitrary fixed points. Important features of general programming logics such as LCF are preserved, including the typing of all partial functions, a partial ordering ! ¸ on computations, and a fixed point induction principle. The resulting theory is thus intended as a generalpurpose programming logic. Rules are presented and soundness of the theory established. Keywords: Constructive Type Theory, Logics...