P.L. Curien, Observable Algorithms on Concrete Data Structures, Proc. Seventh Annual IEEE Symposium on Logic in Computer Science, 1992  Home/Search   Document Not in Database   Summary   Related Articles   Check

....algorithms. Recently, Cartwright and Felleisen in [4] discovered that a natural extension of PCF with a construction called catch allows for a naturally defined fully abstract model, which seems to be strongly related to concrete data structures and sequential algorithms; indeed, Curien shows in [6] that concrete data structures and sequential algorithms are a fully abstract model of PCF catch. This report is structured in three chapters. In chapter 2, we define the notions of concrete data structures and concrete domains, and state the relation between them. In chapter 3, we define ....

.... right addition: Add l = Y (f:xy:if i (zero x)y( 1(f( Gamma1x)y) Add r = Y (f:xy:if i (zero y)x( 1(fx( Gamma1y) Although Add l j op Add r (because for a standard model [ Add l ] Add r ] we don t have in the algorithm model A, Add l ] Add r ] However, recent work by [4] and [6] has shown that the ability to distinguish the order of argument evaluation in PCF is enough to make this model fully abstract. We just mention the result from [6] Theorem 7 Let PCF catch be the extension of PCF by the constants catch oe : oe i. The operational semantics of PCF catch is such ....

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P.L. Curien, Observable Algorithms on Concrete Data Structures, Proc. Seventh Annual IEEE Symposium on Logic in Computer Science, 1992

....are not even functions 4 . Recently, drawing on their intuitions as programmers, Cartwright and Felleisen [ Cartwright and Felleisen, 1992; Cartwright et al. 1994 ] introduced a continuous, order extensional model for pcf which is based on what they call observably sequential functions. Curien [ Curien, 1992 ] immediately realized that the observably sequential functions were a natural extensional refinement of sequential algorithms. This is remarkable because the sequential algorithms being considered in the extended setting, which are called observable algorithms, are still very much intensional in ....

....with error handling, Bucciarelli and Ehrhard s recent work on sequential structure and strong stability. 1. 5 A selected bibliography To the best of our knowledge, there are only two survey papers (excluding this one) covering grounds similar to this chapter, namely [ Berry et al. 1986 ] and [ Curien, 1992 ] The first gives an exposition of the state of the art up to Berry and Curien s work on sequential algorithms. The second, based on an invited talk given at the 1991 Durham Symposium, complements the first and brings it more or less up to date. A short paper of Meyer and Cosmadakis [ Meyer ....

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P.-L. Curien. Observable algorithms on concrete data structures. In Proc. 7th IEEE Symp. Logic in Computer Science, pages 432-- 443. IEEE Computer Society Press, 1992.

.... sequential languages such as PCF [Plo77, BCL85] The known constructions of fully abstract models for PCF [Mil77, Ber78, Mul87] are not natural, yet there are natural fully abstract models for an extension of PCF with parallel facilities [Plo77] and, more recently, with control facilities [CF92, Cur92] There is currently no definition of sequential functions suitable for defining a natural extensional semantic model for PCF. The first definitions of sequential functions, given by Milner [Mil77] and Vuillemin [Vui73] were limited to functions on products of flat domains. Kahn and Plotkin ....

P.-L. Curien. Observable algorithms on concrete data structures. In Seventh Annual IEEE Symposium on Logic in Computer Science, pages 432--443. IEEE Computer Society Press, June 1992.

....) d = d 87 The classical model defined in [Sto81] is an extensional model. The stable function models as in [Ber79] based on the stable ordering is also extensional. However, the sequential algorithms model is not extensional. By adding the error elements, it can be made order extensional [Cur92] Cartwright et. al [CCF94] consider various categories that can be obtained by adding 0, 1, and 2 errors to the domain. Without errors, the model is not extensional. By adding a single error, the model can be made into an extensional one. With two errors, the model becomes order extensional. ....

....(i) programmers can observe the order of evaluation in procedures through the propagation of errors, and (ii) with suitable control operators, programs can determine 146 the order of evaluation of procedure arguments. The full details of the language are available in [CF92, CCF94, KCF93] In [Cur92] Curien shows that Cartwright and Felleisen s model is isomorphic to a cartesian closed category of sequential algorithms enriched with error elements. In this chapter, we construct a more abstract model for SPCF based on OS functions instead of concrete decision tree (sequential algorithm) ....

P.-L. Curien. Observable algorithms on concrete data structures. In Proc 7th Symposium on Logic in Computer Science, 1992.

....i ) Gamma error i Fig. 2. Rewriting Rules for KL f1 ; 1 Gamma ; left; right; cons; pair ; if0g. The combinators S and K are definable by recursive equations in KL, but the definitions are a bit tedious [4] Since the set of OS domains and OS functions form a cartesian closed category [8], we know that S and K are OS functions. For the proofs in this paper, it is more convenient to write programs as recursion equations. 6 Computability and Universality KL can be used to perform arbitrary sequential computations over an OS domain if it can express all computable OS functions. ....

P.-L. Curien. Observable algorithms on concrete data structures. In Proc 7th Symposium on Logic in Computer Science, 1992.

....constants for observing the behavior of higher order expressions as long as these languages include error values and appropriate control constructs. Instead of continuous functions, the CartwrightFelleisen model uses decision trees to assign meaning to procedures. In a subsequent paper, Curien [12] showed that the decision tree model is an extension of the sequential algorithm model [4, 11] and that the set of observably sequential domains and functions form a Cartesian closed category. In this paper, we continue the investigation of SPCF, Cartwright and Felleisen s extension of PCF. ....

....token, we write M[ M ] to denote the meaning of an SPCF program M , since it is closed term of type o. Figure 2 asserts that for any environment ae 2 E and a term M of type in SPCF, the meaning M[ M ] ae 2 T [ To prove this fact, we rely on standard techniques from Meyer [18] and Curien [11, 12]. In either case, the proof is laborious but uninteresting; the only interesting cases rely on the fact that OS functions are closed under composition and abstraction, which can be interpreted as a combinator [11, 7] Following Curien s path shows that the set of OS domains and OS functions form ....

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Curien, P.-L.. Observable algorithms on concrete data structures. In Proc 7th Symposium on Logic in Computer Science, 1992, to appear.

.... and Felleisen, subsequently built a fully abstract model for SPCF an extension of PCF that remains sequential but includes errors and a simple control operator out of question answer trees [CF92] Curien pointed out that they had defined on a subtly different version of sequential algorithms [Cur92, CCF94] While this result is not for PCF itself, the model is quite interesting in many respects. A fourth approach that is closely related to sequential algorithms is game semantics. Games based models are almost like sequential algorithms, except that the same question may be repeated in ....

P.-L. Curien. Observable algorithms on concrete data structures. In Logic in Computer Science, pages 432--443. IEEE Computer Society Press, 1992.

....forming the class of hereditarily sequential functionals, is based on a game in which each play describes the interaction between a functional and its arguments during a computation. This approach is influenced by the work of Kleene [Kle78] Gandy [Gan67] Kahn and Plotkin [KP78] Berry and Curien [BC82, Cur86, Cur92], and Cartwright and Felleisen [CF92] We characterize the computable elements in this model in two different ways: a) by recursiveness requirements for the game, and (b) as definability with the schemata (S1) S8) S11) which is related to definability in PCF. It turns out that both ....

P.-L. Curien. Observable algorithms on concrete data structures. In 7th IEEE Symposium on Logic in Computer Science, pp. 432--443, IEEE Computer Society Press, 1992.

....PCFP In [23] Plotkin proved that the standard model is not fully abstract for PCF, but it is for PCF extended by a parallel conditional construct. Here we shall prove full abstraction of the standard model for a different extension of PCF, PCF extended by parallel or. The proof is due to Curien [5]. 8.1 The standard model is not fully abstract for PCF Theorem 8.1 The standard model is not fully abstract for PCF. Proof Set C[X] j p o)o)o :cond o (p t Omega Gamma (cond o (p Omega t) cond o (p f f) Omega X) Omega Gamma Omega : Define T and F to be C[t] and C[f] respectively. ....

P.-L. Curien. Observable algorithms on concrete data structures. Information and Computation, 1993. To appear.

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