A. Stoughton, Interdefinability of Parallel Operations in PCF, Th. Comp. Sci., Vol. 79, pp. 357--358, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....ground types. The constants of PCF are those of PCF together with por : o o o and 9 : o) o. Both of these languages appear essentially in [Plo77] The formulations here differ principally in the 4 use of a parallel or constant rather than a parallel conditional, for which see [Sto91]; it is also convenient to include constants Omega fl for undefined at ground types. The operational semantics of these languages may be given by rules specifying a binary relation M ) c of evaluation of programs to canonical form see, for example, Plo77, Mit96] Here a program is a closed ....

A. Stoughton, Interdefinability of Parallel Operations in PCF, Theoretical Comp. Sci., Vol. 79, pp. 357--358, 1991.

...., obtained by adding parallel operations. All three of these languages appear essentially in [14] the formulations here differ in two inessential respects: one is the absence of a Boolean type; the other is the use of a parallel or constant rather than a parallel conditional (for which see [18]) The types of PCF are built up from a single ground type (the natural numbers) using the right associative binary type constructor ; we write M : oe to mean M is a term of type oe . For each type oe we have a countably infinite set of variables of type oe, ranged over by x oe ; y oe ; ....

A. Stoughton, Interdefinability of Parallel Operations in PCF, Theoretical Comp. Sci., Vol. 79, pp. 357--358, 1991.

....def = 8 : d 1 if b = t, d 2 if b = f, d 1 if b = and d 1 = d 2 , otherwise. Parallel or may be simulated using parallel conditional by the term x o :y o :pif o xty: In fact, Stoughton showed that parallel or and parallel conditional are inter definable [ Stoughton, 1991b ] Since the standard model does not characterize pcf, is there any model that does More precisely, is there an order extensional, continuous, fully abstract model for pcf This question was answered by Milner [ Milner, 1977 ] who constructed one such model and showed that it is unique up to ....

A. Stoughton. Interdefinability of parallel operations in PCF. Theoretical Computer Science, 79:357--358, 1991.

....there is any extension of PCF that is fully abstract with respect, say, to the bc domains semantics. Plotkin [ Plotkin, 1976 ] showed that this is the case; indeed, the language PCF is fully abstract for the bc domains model if one adds a primitive for computing the parallel disjunction por [ Stoughton, 1991 ] Interestingly, no similar way of extending the language seems to work for the dI domains model [ Jim and Meyer, 1991 ] More details on full abstraction and related topics can be found in [ Gunter, 1992 ] Some current research directions are indicated in the concluding section of this ....

A. Stoughton. Interdefinability of parallel operations in PCF. Theoretical Computer Science, 79:357--358, 1991.

....0 isZero 0 tt isZero (n 1) ff ; for n 0 In addition to the standard constants listed above, we assume the existence of a constant time equality test for integers: o with the obvious operational semantics. Traditionally, the equality test is implemented using recursion (cf. [20]) but this would render some of the issues of interest to us moot, since we are not dealing with integers in unary representation. We find it useful to view PCF programs as circuits. There are several reasons for this. First, it enables us to reason based on the last gate used in the circuit. ....

....on booleans) and pif (parallel conditional on integers) The functions are defined as follows: por : o o o pif oe : o oe oe oe por tt = tt pif oe x x = x por tt = tt pif oe tt x = x por ff ff = ff pif oe ff x = x for oe = o. They are known to be extensionally equivalent [11, 20]. Interestingly, they are not intensionally equivalent. Obviously, por can be used to implement n ary disjunction as in CDSP, thus providing added intensional expressiveness over PCF. It turns out that pif o and pif can also be used for this purpose. However, por and pif o cannot implement pif ....

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A. Stoughton, Interdefinability of parallel operations in PCF, Theoretical Computer Science 79 (1991) 357-8.

....do coincide, the model is called fully abstract. Plotkin went on to establish that the model based on cpo s (Scott s model construction generated from base types without top) and continuous functions is fully abstract for PCF enhanced with a parallel if operation [Plo77] Stoughton [Sto91] later showed that adding a term for por was enough. But what about the original problem can one describe the sequential functional computation of PCF via some abstract denotational model The problem has fascinated semanticists for over 25 years, and is important for at least three reasons. ....

A. Stoughton. Interdefinability of parallel operations in PCF. Theoretical Computer Science, 79:357--358, 1991.

....the type inference rules, which are standard. In addition to the standard constants listed in Figure 1, we assume the existence of a constant time equality test for integers: o with the obvious operational semantics. Traditionally, the equality test is implemented using recursion (cf. [31]) but this would render some of the issues of interest to us moot. Let FV (M) stand for the set of free variables of term M . If FV (M ) then the term M is closed , else it is open . The closed terms of ground type are referred to as programs. tt ; ff : o isZero 0 tt n : integers, n ....

....abstract with respect to the standard denotational semantics. The functions are defined as follows: por : o o o pif oe : o oe oe oe por tt = tt pif oe x x = x por tt = tt pif oe tt x = x por ff ff = ff pif oe ff x = x for oe = o. They are known to be extensionally equivalent [11,31], i.e. one can be implemented in terms of another. The question we address is whether they are intensionally equivalent. Interestingly, the answer turns out to be negative. por j xy: query (x; y) is query : oe 1 Theta Delta Delta Delta Theta oe n ) tt , tt query (x oe 1 1 ; ....

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A. Stoughton, Interdefinability of parallel operations in PCF, Theoretical Computer Science 79 (1991) 357-8.

....shows that by adding a parallel conditional facility to PCF we get full abstraction. He also observed that using parallel conditional, parallel or is definable. Curien [Cu86] and Abramsky [Ab90] later showed that in order to get full abstraction for PCF, it suffices to add parallel or. Stoughton [St91] showed that parallel if and or are each definable in terms of the other. Here we show that we can get full abstraction for FP1 by adding parallel or; however, we cannot get full abstraction for FP in this way We then go on to show that using techniques similar to Plotkin s, we can further extend ....

....in D1 has a greatest lower bound, and that glb is a continuous function. We leave these details to the reader. Observe that or : hhx; yii = pcond : hhx; T; yii, so that that or is definable in terms of pcond. As we now show, the converse also holds. The proof is similar to the proof in [St91]. Lemma C.6: There exists an FPO1 (and, a fortiori, FP1 ) expression pcond such that 1 (pcond) pcond. Proof: Let nn be an abbreviation for y: isseq : y not : null : y) Thus, nn : y returns true exactly if 1 (y) is a nonempty sequence item. We now define pcond as follows. hx 1 ; x 2 ....

A. Stoughton, Interdefinability of parallel operations in PCF, Theoretical Computer Science 79 (1991), pp. 357--358.

....Let the equality test Eq of sort 2 be Y (zxy: If x (If y (z(Pred x) Pred y) 0) Not y) where Not of sort is x: If x 0 1. For n 2 , define operators And n of sort n by: And 0 = 1 and And n 1 = xy 1 Delta Delta Delta yn : If x (And n y 1 Delta Delta Delta yn ) 0: From [Sto3], we know that we can define the parallel if operator, PIf , of sort 3 by PIf = Y 4 H 0, where H of sort 4 4 is defined by H = fwxyz: If (POr (PAnd (Eq y w) Eq z w) PAnd x (Eq y w) PAnd (Not x) Eq z w) w (f (Succ w)x y z) Here, we have extended POr to three arguments ....

A. Stoughton. Interdefinability of parallel operations in PCF. Theoretical Computer Science, to appear, 1991.

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A. Stoughton, Interdefinability of Parallel Operations in PCF, Th. Comp. Sci., Vol. 79, pp. 357--358, 1991.

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