Dana S. Scott. A type theoretical alternative to CUCH, ISWIM, OWHY. Unpublished manuscript, 1969.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....PLLC. More interesting are settings which model PLLC where the symmetric monodial structure is not Cartesian. 7. 2 Domain Models Domains have played an important role in programming semantics for the past 30 years, dating back to Scott s modeling of typed and untyped lambda calculi with domains [Sco69, Sco70] Ordered sets arise naturally in situations where non termination is possible, where a larger element terminates (is de ned) more often than a smaller element. Having such an ordering of the elements is crucial for modeling recursion. A xed point operator is typically handled by making ....

D. S. Scott. A type theoretical alternative to CUCH, ISWIM and OWHY. Unpublished manuscript, Oxford University, 1969. Reprinted as [Sco93].

....definition of f. In Section 5, we shall prove the two programs equivalent by showing that their defunctionalized versions are equivalent. The correctness then follows from the correctness of defunctionalization (Section 4) 2 The Source Language The source language, PCFA, is a version of PCF [31, 36] with control flow types [15] Each function definition in a program has a unique label; we use to range over these labels. Each call site has a set of labels representing the functions that may flow to that site; we use L to range over finite sets of labels. Function types include these label ....

D. Scott. A type theoretical alternative to CUCH, ISWIM, OWHY. Theoretical Computer Science, 121:411--440. Published version of unpublished manuscript, Oxford University, 1969.

....: A:M j n j succ j pred j ifzero M then M else M j Y(M) j l j M : M j M j mkloc MM j skip j M ; M j new x : n in M where x ranges over a countable set of variables, l ranges over a countable set of locations, and n over natural numbers. The rst two lines de ne the standard PCF constructs [34,32]. We use in x : for variable assignment, pre x for dereferencing, in x ; for command sequencing, and write skip for the null command. Block allocated local variables are introduced by the construct new x : n in M (the local variable x is initialized to n as control enters the block; note ....

D. S. Scott. A type-theoretical alternative to cuch, iswim and owhy. Theoretical Computer Science, 121:411-440, 1993.

....of FPC are algebraically compact partial cartesian closed categories with binary coproducts. Impact of axiomatic domain theory In relating operational and denotational semantics. The investigation of the relation between operational and denotational semantics started with a question of Scott in [Sco69] An answer for PCF (a higher order functional programming language with fixed point operators, and base type and arithmetic operations for natural numbers) was given by Plotkin in [Plo77] there he proved the soundness and adequacy of the standard semantics of PCF for a call by name evaluator. A ....

D.S. Scott. A type-theoretical alternative to CUCH, ISWIM, OWHY. Manuscript (available as [Sco93]), 1969.

....as a satisfiability question. This observation lets us develop a simple algorithm for constructivity that extends easily to non Boolean types. 3. 1 Background To formulate simplification as a fixpoint process, we need some well known concepts from Scott s theory of Complete Partial Orders (CPO s) [10]. The type T is extended with a new element (read as bottom ) to form the type T . T is equipped with the partial order , defined by a b iff a = b or a = Note that is a CPO (every sequence of elements that is monotonically increasing w.r.t. has a least upper bound) The greatest lower ....

D. S. Scott. A type-theoretical alternative to CUCH, ISWIM, OWHY. Unpublished notes, Oxford,

....and the so called Calculus of Constructions and its extensions [12, 45] on the other hand. A number of other foundational systems for computer science has been proposed. Scott s PCF, introduced at the beginning of the 70 s as a manuscript (unpublished until its appearance in the B ohm Festschrift [70]) is notable not only for the large impact it and the related work of nding 3 adequate mathematical models had on computer science, but also for the implementation of a Logic of Computable Functions inspired by it, giving rise to the LCF system [62] The basic ideas for designing an interactive ....

Dana S. Scott. A type-theoretical alternative to CUCH, ISWIM and OWHY. Theoretical Computer Science, 121(411-440), 1993.

....is PCF extended with two constants, por : o o o for parallel or, and 9 : o) o for (continuous) existential quantification. The language PCF is adapted to sequential functional computation; PCF is adapted to parallel functional computation. Scott s model C of continuous functions [Sco93] and Milner s model S of sequential continuous functions [Mil77] are the standard fully abstract models of PCF and PCF, respectively. They have natural submodels C o and S o consisting of those elements definable in, respectively, PCF and PCF; equivalently, these are the submodels of ....

D. Scott, A Type-Theoretical Alternative to CUCH, ISWIM and OWHY, in Logic, Semantics and Theory of Programming, Theoretical Comp. Sci., Vol. 121, Nos. 1,2, pp. 411--440, 1993.

....as i sequential processes, of respective lengths n 1 ; n i , running concurrently. Complete cuboidal sets are cuboidal sets equipped with a formal lub operator satisfying three algebraic laws, which are exactly those needed of the lub operator in order to prove the fixed point theorem [35]. Computationally, the passage from cuboidal sets to complete cuboidal sets corresponds to allowing infinite processes. In fact, the formal lub operator assigns paths of shape C to paths of shape C Theta , for every C. Here the set of paths of shape C Theta is the colimit of the paths of ....

D.S. Scott. A type-theoretical alternative to CUCH, ISWIM, OWHY. Manuscript (available as [37]), 1969.

....of easy corollaries for particular algebras. 1 Introduction The (fi) and (j) axioms form the basis for proving equations in call by name functional languages. In these languages, fi) and (j) yield sound program optimizations. For example, consider a version of the call by name language PCF [11, 15] which is described in Appendix A. Our version of PCF includes simplytyped calculus, numerals 0; 1; 2; successor and predecessor, addition, multiplication, a conditional test for 0, and a fixpoint operator for recursion. In any PCF program, the term ( x Pi 3) Q) may be replaced by 3 ....

....first order functions condition to fail. Here we amend Theorems 11 and 13 to account for . In most models over algebraic signatures, is treated uniformly: algebraic functional constants are strict, i.e. return when any argument is . For instance, in the full continuous model for PCF (cf. [11, 15]) the operations succ, pred, and are strict in all of their arguments. Let A , the strict algebra, denote the result of enhancing the carrier set of A with and extending all the operations of A to their strict versions. We say that a model has enough strict first order functions if for ....

Dana Scott. A type theoretical alternative to CUCH, ISWIM, OWHY. Unpublished manuscript, Oxford University, 1969.

....The denotational semantics most appropriate for a programming language depends crucially upon the observations one makes about computations. In general, an observation is some important behavior of the interpreter [8] For example, in the arithmetic, higher order programming language PCF [11, 13], one usually chooses to observe the results of arithmetic expressions that a term of integer type reduces to a numeral. One may also extend the notion of observation to arbitrary terms, saying that two terms are observationally congruent if they produce the same observable outcomes in any ....

....for the model A Y . Adequacy, full abstraction and universality mark an intimate connection between PCF and the model A Y . The point of obtaining such a model is, in part, to develop techniques for proving properties about code. We give some preliminary results in defining a logic (based on LCF [6, 13]) for a fragment of PCF with up . The logic is shown to be sound for the model A Y . 1 These results were obtained independently from Abramsky [1] and Ong [9, 10] who have proven similar adequacy and full abstraction results for an untyped calculus. Cosmadakis [4] has extended our results to a ....

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D. Scott. A type theoretical alternative to CUCH, ISWIM, OWHY. 1969. Manuscript, Oxford Univ.

....In order to give a precise definition of observational equivalence, we need some information about the programming language in question. pcf was first introduced by Scott in a famous paper [ Scott, 1969 ] which remained an unpublished manuscript until its recent appearance in the Bohm Festschrift [ Scott, 1993 ] Scott intended pcf to be a logical calculus (or algebra) for the purpose of studying computability and logical properties of programs by using type theory. Plotkin presented pcf explicitly as a programming language in a seminal paper entitled PCF as a Programming Language [ Plotkin, 1977 ] ....

....the key to making Hilbert s programme (in the modified sense) viable. Correspondence between semantics 283 The formal system pcf was introduced by Scott in a famous paper [ Scott, 1969 ] which remained an unpublished manuscript for a long time until its recent appearance in the Bohm Festschrift [ Scott, 1993 ] The syntax 7 of pcf is simple enough: it is essentially the simply typed calculus augmented by general recursion in the form of a fixed point operator at every type, and further augmented by basic arithmetic. Scott intended pcf to be a logical calculus (or algebra) for studying ....

D. S. Scott. A type-theoretical alternative to cuch, iswim and owhy. Theoretical Computer Science (Bohm Festschrift, Special Issue), 121:411--440, 1993. Correspondence between semantics 355

....they are programs with typed holes into which type compatible program fragments may fit. Overview of pcf In order to give a precise definition of observational equivalence, we need some information about the programming language in question. pcf was first introduced by Scott in a famous paper [ Scott, 1969 ] which remained an unpublished manuscript until its recent appearance in the Bohm Festschrift [ Scott, 1993 ] Scott intended pcf to be a logical calculus (or algebra) for the purpose of studying computability and logical properties of programs by using type theory. Plotkin presented pcf ....

....M t N , if for every typecompatible program context C[X] such that both C[M ] and C[N ] are programs, and for any value V , C[M ] V if and only if C[N ] V . As usual, C[M ] means the program which is obtained from the context C[X] by substituting M for every occurrence of X in C[X] In [ Scott, 1969 ] Scott introduced what has come to be known as the Scott continuous function space model for pcf programs. He used the standard flat cpos to interpret booleans and the natural numbers respectively, and Scott continuous function space to interpret function types. We shall call this model the ....

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D. S. Scott. A type theoretical alternative to CUCH, ISWIM and OWHY. Unpublished handwritten manuscript, 1969.

....One might also seek results tying the three settings of continuations together. Finally, an Appendix is included which contains proofs of standard theorems for v . Chapter 2 The Language and its Continuation Transform This chapter defines v , a call by value version of the language PCF [20, 25], including an interpreter for v . A call by value continuation transform for the language is then given, along with theorems that show the correctness of the transform. 2.1 Syntax The familiar syntax of the simply typed calculus forms the basis of v . Each term in v has a type of the form o ....

Dana Scott. A type theoretical alternative to CUCH, ISWIM, OWHY. 1969. Manuscript, Oxford Univ.

....discussion on a language with the grammar M : x j (x: M) j (M M) j n j true j false j (succ M) j (pred M) j (zero M) j (if M then M else M) j (fix M) j (store M) j (fetch M) where x and n are from primitive syntax classes of variables and numerals respectively. This is a variant of PCF [Sco, Plo77, BGS90] augmented by primitive operations for forcing and delaying evaluations. The expression (fix M) is used for recursive definitions. The key to providing a semantics for this language is to represent the memoization used in computing the fetch primitive so that certain recomputation is avioded. We ....

D. S. Scott. A type theoretical alternative to CUCH, ISWIM, OWHY. Unpublished manuscript, 1969.

....by a scheme amounting to the First Recursion Theorem. The essentials of the theory are definition by cases and least fixed points, and Platek does introduce a calculus formulation which can be regarded as a precursor to PCF. The formal system PCF was introduced by Scott in a famous paper [26] which has remained an unpublished manuscript for a long time until its recent appearance in the Bohm Festschrift [26] The syntax 2 of PCF is simple enough: it is essentially the simply typed calculus augmented by general recursion in the form of a fixed point operator at every type, and ....

....least fixed points, and Platek does introduce a calculus formulation which can be regarded as a precursor to PCF. The formal system PCF was introduced by Scott in a famous paper [26] which has remained an unpublished manuscript for a long time until its recent appearance in the Bohm Festschrift [26]. The syntax 2 of PCF is simple enough: it is essentially the simply typed calculus augmented by general recursion in the form of a fixed point operator at every type, and further augmented by basic arithmetic. Scott intended PCF to be a logical calculus (or algebra) for studying ....

[Article contains additional citation context not shown here]

D. S. Scott. A type-theoretical alternative to cuch, iswim and owhy. Theoretical Computer Science, 121:411--440, 1993.

....fun ) and (i arg ) in the sense of Parigot [28] and which satisfies the uniqueness of data representation property (i.e. the only normal forms of the type of natural numbers are the Church numerals) This property does not hold for the call by name calculus. It is folklore that Scott s pcf [37, 33], that mother of all toy languages after Pitts memorable turn of phrase, is a prototypical functional language. Another goal of our research is to develop a theory of functional computation with control based on a simple but expressive language; and then to give a precise and systematic account ....

....programming language In this section we introduce a deterministic, sequential, cbv programming language pcf v . Syntactically pcf v is obtained from v by augmenting it by basic arithmetic, conditionals, and fixed point operators. So pcf v is to v what Scott s programming language pcf [37, 33] is to pure simply typed calculus. We distinguish two strengths of the language. First the more expressive version. Programming language pcf v Types of pcf v are generated by the function space constructor ) from three program types, namely, natural number , boolean o, and absurdity . ....

D. S. Scott. A type-theoretical alternative to cuch, iswim and owhy. Theoretical Computer Science, 121:411--440, 1993.

....by the observation that our categorical model induces an order theoretic model for PCF in a canonical way. 1 Introduction The programming language PCF (an acronym for Programming language for Computable F unctions) was in 1969 introduced by Scott in the until recently unpublished paper [Sco93]. The formal system of PCF consists of the # calculus augmented with booleans and numerals together with recursion where the usual reduction rules are replaced by an operational semantics. Scott s pioneering idea was to interpret types as cpos and terms as continuous functions. The possibility of ....

D. S. Scott. A type theoretical alternative to CUCH, ISWIM, OWHY. In Bohm Festscrift, Theoretical Computer Science, volume 121. Elsevier, 1993.

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Dana S. Scott. A type theoretical alternative to CUCH, ISWIM, OWHY. Unpublished manuscript, 1969.

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Dana S. Scott. A type theoretical alternative to CUCH, ISWIM, OWHY. Unpublished manuscript, 1969.

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D. S. Scott. A type-theoretical alternative to CUCH, ISWIM and OWHY. Theoretical Computer Science, 121:411--440, 1993.

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D. S. Scott. A type theoretical alternative to CUCH, ISWIM and OWHY. Theoretical Comput. Sci., 121:411-440, 1993. 264

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D.S. Scott. A type-theoretical alternative to CUCH, ISWIM, OWHY. Theoretical Computer Science, 121:411--440, 1993. (Reprint of a manuscript).

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D.S. Scott. A type-theoretical alternative to CUCH, ISWIM, OWHY. Theoretical Computer Science, 121:411--440, 1993. (Reprint of a manuscript).

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D.S.Scott. A type-theoretical alternative to CUCH, ISWIN and OHWY. Unpublished manuscript. Oxford, 1969.

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D. Scott. A type-theoretical alternative to CUCH, ISWIM, OWHY. Typed script -- Oxford, October 1969.

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