Gerard Boudol. Lambda-calculi for (strict) parallel functions. Information and Computation, 108(1):51--127, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... models (various forms of cpo constructions) because of criticisms [28, 6] related to the over defined top element, which somehow implies the presence of multivalues (for example what is the meaning of an upper bound of the truth values tt and ff ) Without entering the discussion see also [5] we only mention that here uncatchable errors behave as a black hole above any value and therefore give a natural justification for coming back to a lattice structure with a top element. In order to accept terms such as ( xy:y) we also take a different approach to typing. Section 6 defines a ....

G erard Boudol. Lambda-Calculi for (Strict) Parallel Functions. Information and Computation 108:51-127, 1994.

....cannot be considered fair in the following sense: out of two convergent branches, it always chooses the one requiring fewest reduction steps. Boudol examines full abstraction for two versions of the untyped lambda calculus (di ering in the kinds of abstractions allowed) plus a parallel combinator [Bou94]. The combinator isn t actually a choice operator, and the setting is a deterministic one, but parallel combinations may be viewed as collecting all possible behaviours of some choice operator. However, since only may converge properties are considered, that choice operator cannot be distinguished ....

G. Boudol. Lambda-calculi for (strict) parallel functions. Information and Computation, 108(1):51-127, January 1994.

....cannot be considered fair in the following sense: out of two convergent branches, it always chooses the one requiring fewest reduction steps. Boudol examines full abstraction for two versions of the untyped lambda calculus (di ering in the kinds of abstractions allowed) plus a parallel combinator [2]. The combinator isn t actually a choice operator, and the setting is a deterministic one, but parallel combinations may be viewed as collecting all possible behaviours of some choice operator. However, since only may converge properties are considered, that choice operator cannot be distinguished ....

G. Boudol. Lambda-calculi for (strict) parallel functions. Information and Computation, 1992.

....and type constructors allows us to search for a proof along the following lines. Suppose we were able to define for each type ff a test term T ff such that T ff M converges if and only if M has type ff. Then we would obtain an observational equivalence which coincides with the tree equality (see [8]) The paper is organized as follows. In Section 2 we shall recall the various definitions of tree. We will introduce the notion of approximant in Section 3. In Section 4 we will describe the type assignment systems which will be used for our main result and we will give a theorem of ....

G'erard Boudol. Lambda-calculi for (strict) parallel functions. Information and Computation, 108(1):51--127, 1994.

....and type constructors allows to search for a proof along the following lines. Suppose we were able to define, for each type ff, a test term T ff such that T ff M converges if and only if M has type ff. Then we would obtain an observational equivalence which coincides with the tree equality (see [8]) This paper is organized as follows. In Section 2 we shall recall the various definitions of tree. We will introduce the notion of approximant in Section 3. In Section 4 we will describe the type assignment systems which will be used for our main result and we will give a theorem of ....

Gerard Boudol. Lambda-calculi for (strict) parallel functions. Information and Computation, 108(1):51--127, 1994.

....and type constructors allows us to search for a proof along the following lines. Suppose we were able to define for each type ff a test term T ff such that T ff M converges if and only if M has type ff. Then we would obtain an observational equivalence which coincides with the tree equality (see [8]) The paper is organised as follows. In Section 2 we shall recall the various definitions of tree. We will introduce the notion of approximant in Section 3. In Section 4 we will describe the type assignment systems which will be used for our main result and we will give a theorem of ....

Gerard Boudol. Lambda-calculi for (strict) parallel functions. Information and Computation, 108(1):51--127, 1994.

....be considered fair in the following sense: out of two convergent branches, it always chooses the one requiring fewest reduction steps. Boudol examines full abstraction for two versions of the untyped lambda calculus (differing in the kinds of abstractions allowed) plus a parallel combinator [2]. The combinator isn t actually a choice operator, and the setting is a deterministic one, but parallel combinations may be viewed as collecting all possible behaviours of some choice operator. However, since only may converge properties are considered, that choice operator cannot be distinguished ....

G. Boudol. Lambda-calculi for (strict) parallel functions. Information and Computation, 1992.

....strategy is used e.g. Man95] Some other approaches [Ong93, DP95, San94] do not take care of sharing properties and thus suffer from the implicit choice problem mentioned in the introduction. Bottom avoiding choice operators like amb or por as investigated for example in [HO90, HO89, HM92, Bou94] corresponds to the operational semantics of (local) speculative evaluation; i.e. it is not like a committed choice. In [Bou94] confluence of a calculus using (por) is shown, which is related to the properties of nd , but the calculi are different, since nd builds upon a stronger contextual ....

....from the implicit choice problem mentioned in the introduction. Bottom avoiding choice operators like amb or por as investigated for example in [HO90, HO89, HM92, Bou94] corresponds to the operational semantics of (local) speculative evaluation; i.e. it is not like a committed choice. In [Bou94] confluence of a calculus using (por) is shown, which is related to the properties of nd , but the calculi are different, since nd builds upon a stronger contextual equivalence (Definition 5.1) and thus justifies a different set of valid program transformations. In [Pat91] similar ideas are ....

G. Boudol. Lambda-calculi for (strict) parallel functions. Information and Computation, 108:51--127, 1994.

....cannot be considered fair in the following sense: out of two convergent branches, it always chooses the one requiring fewest reduction steps. Boudol examines full abstraction for two versions of the untyped lambda calculus (dioeering in the kinds of abstractions allowed) plus a parallel combinator [Bou92]. The combinator isn t actually a choice operator, and the setting is a deterministic one, but parallel combinations may be viewed as collecting all possible behaviours of some choice operator. However, since only may converge properties are considered, that choice operator cannot be ....

G. Boudol. Lambda-calculi for (strict) parallel functions. Information and Computation, 1992.

....at higher types. Given this success, it is important to see how broad the scope of the approach initiated in [1, 4] actually is. In this paper, we consider the lazy calculus [2, 6] This calculus, and certain associated notions such as applicative simulation, have been quite widely influential [8, 11, 13, 14, 17, 19, 22]. In [6] a syntactic construction of a fully abstract model for the basic sequential language was given, and the canonical denotational model was shown to be fully abstract for a certain parallel extension of the language. However, the problem of giving a direct, syntax free, purely semantic ....

G. Boudol. A lambda calculus for (strict) parallel functions. Information and Computation, 108:51--127, 1994.

....parallel convergence testing combinator, this may be surprising because there is no parallel feature in our calculus with multiplicities. The same remark holds as well if, instead of using this combinator, we use a non deterministic choice, as in [4] or a parallel composition of functions, as in [6]. Therefore the sensitivity of the calculus to the lack of resources is much greater than one could think. Another consequence of our characterization result is that the observational equivalence M m N over terms, meaning that for any context C with multiplicities, C[M ] has a value if and ....

....consequence of Theorem 6.3. In [4] we have shown that the semantical preorder, that is, equivalently, the preorder p coincides with the observational preorder on the calculus extended with convergence testing (cM) and non deterministic choice (M Phi N) or parallel composition (M k N ) see [6]) For instance, the two terms G 1 and G 2 of Proposition 6.1 are distinguished using non deterministic choice: G 2 (U 2 1 Phi U 2 2 ) has a value, namely y(U 2 1 Phi U 2 2) Omega Gamma y, while G 1 (U 2 1 Phi U 2 2 ) always diverges (one does not really need the convergence testing ....

G. Boudol. Lambda-calculi for (strict) parallel functions. Information and Computation, 108(1):51--127, 1994.

....value is a closed normal form with respect to some reduction strategy. For instance, in the lazy calculus [1] a value is a (closed) abstraction. Here, we may have several agents computing in parallel, and therefore we regard as a value a term where at least one component is an abstraction, as in [7]. That is, a value is a (closed) term given by the following grammar: V . u)P j (V j P ) j (P j V ) j (u)V where P is any term. The set of values is denoted V. It is easy to see that this set is closed under structural manipulations and reduction. We say that P converges, in notation P , ....

G. Boudol, Lambda-calculi for (strict) parallel functions, Information and Computation 108 (1994) 51-127.

....which the full abstraction problem has been developed is the simply typed calculus. With the introduction of the operational equivalence resulting from applicative bisimulation, it can be neatly transferred to the untyped calculus and it has motivated elegant works by Abramsky, Ong and Boudol ([AO93, Bou94]) A denotational interpretation is said to be sound if it only equates operationally equivalent terms, complete if it equates all operationally equivalent terms, and fully abstract if it is sound and complete. We call the equality on terms induced by model D of the previous section ....

G. Boudol. A lambda calculus for (strict) parallel functions. Information and Computation, 108(1):51--127, 1994.

....higher types. 1 Given this success, it is important to see how broad the scope of the approach initiated in [1, 4] actually is. In this paper, we consider the lazy calculus [2, 6] This calculus, and certain associated notions such as applicative simulation, have been quite widely influential [8, 11, 13, 14, 17, 19, 22]. In [6] a syntactic construction of a fully abstract model for the basic sequential language was given, and the canonical denotational model was shown to be fully abstract for a certain parallel extension of the language. However, the problem of giving a direct, syntax free, purely semantic ....

G. Boudol. A lambda calculus for (strict) parallel functions. Information and Computation, 108:51--127, 1994.

....to the identity I, if and only if M converges or N converges, one could encode it as follows, using output guards: E [ P] u = def u(x; v)v(y; w) r) s) xr: ys: o) rxr: o j sys: o j o( E [ I] w) However, parallel convergence testing is not the most general way to deal with parallel functions. In [5, 7] we have shown that parallel composition of functions is simply the join in the denotational semantics, which may be represented in the syntax by internal choice. For instance, P = xy:Cx) Phi (xy:Cy) Obviously, to preserve some properties of the interpretation of internal choice as a join, ....

G. Boudol. Lambda-calculi for (strict) parallel functions. Information and Computation, 108(1):51--127, 1994.

....to the identity I, if and only if M converges or N converges, one could encode it as follows, using output guards: E [ P] u = def u(x; v)v(y; w) r) s) xr: ys: o) rxr: o j sys: o j o( E [ I] w) However, parallel convergence testing is not the most general way to deal with parallel functions. In [5, 7] we have shown that parallel composition of functions is simply the join in the denotational semantics, which may be represented in the syntax by internal choice. For instance, P = xy:Cx) Phi (xy:Cy) Obviously, to preserve some properties of the interpretation of internal choice as a join, ....

G. Boudol. Lambda-calculi for (strict) parallel functions. Information and Computation, 108(1):51--127, 1994.

....parallel convergence testing combinator, this may be surprising because there is no parallel feature in our calculus with multiplicities. The same remark holds as well if, instead of using this combinator, we use a non deterministic choice, as in [4] or a parallel composition of functions, as in [6]. Therefore the sensitivity of the calculus to the lack of resources is much greater than one could think. Another consequence of our characterization result is that the observational equivalence M m N over terms, meaning that for any context C with multiplicities, C[M ] has a value if and ....

....consequence of Theorem 6.3. In [4] we have shown that the semantical preorder, that is, equivalently, the preorder p coincides with the observational preorder on the calculus extended with convergence testing (cM ) and non deterministic choice (M Phi N ) or parallel composition (M k N ) see [6]) For instance, the two terms G 1 and G 2 of Proposition 6.1 are distinguished using non deterministic choice: G 2 (U 2 1 Phi U 2 2 ) has a value, namely y(U 2 1 Phi U 2 2) Omega Gamma y, while G 1 (U 2 1 Phi U 2 2 ) always diverges (one does not really need the convergence testing ....

G. Boudol. Lambda-calculi for (strict) parallel functions. Information and Computation, 108(1):51--127, 1994.

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Gerard Boudol. Lambda-calculi for (strict) parallel functions. Information and Computation, 108(1):51--127, 1994.

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G. Boudol. Lambda calculi for (strict) parallel functions. Information and Computation, 108:51--127, 1994.

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