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What is a Purely Functional Language?
 Journal of Functional Programming
, 1998
"... Functional programming languages are informally classified into pure and impure languages. The precise meaning of this distinction has been a matter of controversy. We therefore investigate a formal definition of purity. We begin by showing that some proposed definitions that rely on confluence, sou ..."
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Cited by 14 (6 self)
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Functional programming languages are informally classified into pure and impure languages. The precise meaning of this distinction has been a matter of controversy. We therefore investigate a formal definition of purity. We begin by showing that some proposed definitions that rely on confluence, soundness of the beta axiom, preservation of pure observational equivalences, and independence of the order of evaluation, do not withstand close scrutiny. We propose instead a definition based on parameterpassing independence. Intuitively, the definition implies that functions are pure mappings from arguments to results; the operational decision of how to pass the arguments is irrelevant. In the context of Haskell, our definition is consistent with the fact that the traditional callbyname denotational semantics coincides with the traditional callbyneed implementation. Furthermore, our definition is compatible with the streambased, continuationbased, and monadbased integration of computa...
Notions of computability at higher types I
 In Logic Colloquium 2000
, 2005
"... We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a ..."
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Cited by 12 (5 self)
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We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.
An Operational Semantics for I/O in a Lazy Functional Language
 in Proc Functional Programming Languages and Computer Architecture
, 1993
"... I/O mechanisms are needed if functional languages are to be suitable for general purpose programming and several implementations exist. But little is known about semantic methods for specifying and proving properties of lazy functional programs engaged in I/O. As a step towards formal methods of rea ..."
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I/O mechanisms are needed if functional languages are to be suitable for general purpose programming and several implementations exist. But little is known about semantic methods for specifying and proving properties of lazy functional programs engaged in I/O. As a step towards formal methods of reasoning about realistic I/O we investigate three widely implemented mechanisms in the setting of teletype I/O: synchronisedstream (primitive in Haskell), continuationpassing (derived in Haskell) and Landinstream I/O (where programs map an input stream to an output stream of characters) . Using methods from Milner's CCS we give a labelled transition semantics for the three mechanisms. We adopt bisimulation equivalence as equality on programs engaged in I/O and give functions to map between the three kinds of I/O. The main result is the first formal proof of semantic equivalence of the three mechanisms, generalising an informal argument of the Haskell committee. 1 Introduction and motivation...
Bidomains and full abstraction for countable nondeterminism
 In Proceedings of FoSSaCS’06, number 3921 in LNCS
, 2006
"... Abstract. We describe a denotational semantics for a sequential functional language with random number generation over a countably infinite set (the natural numbers), and prove that it is fully abstract with respect to mayandmust testing. Our model is based on biordered sets similar to Berry’s bid ..."
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Abstract. We describe a denotational semantics for a sequential functional language with random number generation over a countably infinite set (the natural numbers), and prove that it is fully abstract with respect to mayandmust testing. Our model is based on biordered sets similar to Berry’s bidomains, and stable, monotone functions. However, (as in prior models of unbounded nondeterminism) these functions may not be continuous. Working in a biordered setting allows us to exploit the different properties of both extensional and stable orders to construct a Cartesian closed category of sequential, discontinuous functions, with least and greatest fixpoints having strong enough properties to prove computational adequacy. We establish full abstraction of the semantics by showing that it contains a simple, firstorder “universal typeobject ” within which all types may be embedded using functions defined by (countable) ordinal induction. 1
An investigation of contracts as projections
, 2004
"... Software contracts help programmers enforce program properties that the language’s type system cannot express. Unlike types, contracts are (usually) enforced at runtime. When a contract fails, the contract system signals an error. Beyond such errors, contracts should have no other observable (funct ..."
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Cited by 7 (4 self)
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Software contracts help programmers enforce program properties that the language’s type system cannot express. Unlike types, contracts are (usually) enforced at runtime. When a contract fails, the contract system signals an error. Beyond such errors, contracts should have no other observable (functional) effect on the program’s results. In most implementations, however, the language of contracts is the fullfledged programming language, which means that programmers may (intentionally or unintentionally) introduce visible effects into their contracts. Here we present the results of investigating the nature of contracts from a denotational perspective. Specifically, we use SPCF and the category of observably sequential functions to show that contracts are best understood as projections. Thus far, the investigation has produced a significantly faster contract implementation and the insight that our contract language cannot express all projections, which in turn has produced a new contract combinator.
What is a Universal HigherOrder Programming Language?
 In Proc. International Conference on Automata, Languages, and Programming. Lecture Notes in Computer Science
, 1993
"... . In this paper, we develop a theory of higherorder computability suitable for comparing the expressiveness of sequential, deterministic programming languages. The theory is based on the construction of a new universal domain T and corresponding universal language KL. The domain T is universal for ..."
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. In this paper, we develop a theory of higherorder computability suitable for comparing the expressiveness of sequential, deterministic programming languages. The theory is based on the construction of a new universal domain T and corresponding universal language KL. The domain T is universal for observably sequential domains; KL can define all the computable elements of T, including the elements corresponding to computable observably sequential functions. In addition, domain embeddings in T preserve the maximality of finite elementspreserving the termination behavior of programs over the embedded domains. 1 Background and Motivation Classic recursion theory [7, 13, 18] asserts that all conventional programming languages are equally expressive because they can define all partial recursive functions over the natural numbers. This statement, however, is misleading because real programming languages support and enforce a more abstract view of data than bitstrings. In particular, mo...
Definitions in Nonstrict Positive Free Logic
 Modern Logic
, 1997
"... Every "practical" programming language supplies the programmer with at least one nonstrict construct, such as the ALGOL60 arithmetic `ifthen else' and the LISP `cond'. Many programming languages also enable the user to define nonstrict functions. In some languages, this is accom ..."
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Cited by 4 (2 self)
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Every "practical" programming language supplies the programmer with at least one nonstrict construct, such as the ALGOL60 arithmetic `ifthen else' and the LISP `cond'. Many programming languages also enable the user to define nonstrict functions. In some languages, this is accomplished through the lazy evaluation of procedure parameters, as realized, for example, by the callbyname devices of ALGOL60 and SIMULA67 and the callbyneed mechanism of Haskell. In other languages, such as Common LISP, a macro definition facility can serve a similar purpose. Programming languages that provide a mechanism for the user to define nonstrict functions are nonstrict languages, and we call the natural underlying logic of these languages nonstrict positive free logic. In this paper, we present the definition theory of nonstrict positive free logic. Suitable transformations of sentences in standard logic into sentences in nonstrict positive free logic preserve many properties of definitions in stand...
Games and Sequential Algorithms
, 2001
"... The relationship between HylandOngstyle games and BerryCurien sequential algorithms is investigated, with the object of describing semantic solutions to two problems  to characterise eectively the \minimal models" of the simplytyped calculus and the fully abstract model of PCF with co ..."
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The relationship between HylandOngstyle games and BerryCurien sequential algorithms is investigated, with the object of describing semantic solutions to two problems  to characterise eectively the \minimal models" of the simplytyped calculus and the fully abstract model of PCF with control operators  which are shown to be equivalent.
Decidability in Syntactic Control of Interference
 Proceedings of ICALP ’05, number 3580 in LNCS
, 2005
"... Abstract. We investigate the decidability of observational equivalence and approximation in “Syntactic Control of Interference ” (SCI). By associating denotations of terms in an inequationally fully abstract model of finitary basic SCI with multitape finite state automata, we show that observational ..."
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Abstract. We investigate the decidability of observational equivalence and approximation in “Syntactic Control of Interference ” (SCI). By associating denotations of terms in an inequationally fully abstract model of finitary basic SCI with multitape finite state automata, we show that observational approximation is not decidable (even at first order), but that observational equivalence is decidable for all terms. We then consider the same problems for basic SCI extended with nonlocal control in the form of backwards jumps. We show that both observational approximation and observational equivalence are decidable in this language by describing a fully abstract games model in which strategies are regular languages. 1
Bistable biorders: A sequential domain theory
 Logical Methods in Computer Science
"... Abstract. We give a simple ordertheoretic construction of a Cartesian closed category of sequential functions. It is based on bistable biorders, which are sets with a partial order — the extensional order — and a bistable coherence, which captures equivalence of program behaviour, up to permutation ..."
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Abstract. We give a simple ordertheoretic construction of a Cartesian closed category of sequential functions. It is based on bistable biorders, which are sets with a partial order — the extensional order — and a bistable coherence, which captures equivalence of program behaviour, up to permutation of top (error) and bottom (divergence). We show that monotone and bistable functions (which are required to preserve bistably bounded meets and joins) are strongly sequential, and use this fact to prove universality results for the bistable biorder semantics of the simplytyped lambdacalculus (with atomic constants), and an extension with arithmetic and recursion. We also construct a bistable model of SPCF, a higherorder functional programming language with nonlocal control. We use our universality result for the lambdacalculus to show that the semantics of SPCF is fully abstract. We then establish a direct correspondence between bistable functions and sequential algorithms by showing that sequential data structures give rise to bistable biorders, and that each bistable function between such biorders is computed by a sequential algorithm. 1.