Colson, Loc. About primitive recursive algorithms. Theoretical Computer Science 83 (1991) 5769.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....related to computational complexity could be uncovered. In the long run such an approach should bring about novel criteria for expressibility, permitting one to certify that certain algorithms are not programmable in a given programming language interpretable in the model (in the spirit of [28]) or, possibly even more strongly, that certain functions are not representable and thus do not belong to the associated complexity class. Our model should also be employed to give another proof of the fact that (untyped) reduction in IMLAL2 can be implemented in polynomial time. This will become ....

Loic Colson. About primitive recursive algorithms. Theoretical Computer Science, 83:57-69, 1991.

....(on x) with substitution in parameter (y) which does not lead outside of primitive recursion. This reduction to primitive recursion is important extensionally for knowing that min is primitive recursive, but the function is to be computed directly from its 3 clauses used as rewrite rules. Colson [Col91] has proved that this kind of intensional behavior is impossible with any primitive recursive derivation of the minimum function where the identities are used intensionally as rewriting rules. The difference of the two definitions shows also in proofs of the property: x y min(x; y) y x y ....

L. Colson. About primitive recursive algorithms. Theoretical Computer Science, 83(1):57--69, 1991.

.... different level of abstraction that retains intensional information about programs (at a level appropriate to the task at hand) one should be able to use an intensional denotational semantics to reason about the intensional aspects of programs, such as laziness and complexity (see for instance [Col89] for a potential application) One of our initial goals in this study has been the definition of a richer intensional semantic model by generalizing Berry and Curien s sequential algorithms between concrete data structures to parallel algorithms. Our thesis is that, by analogy with the ....

L. Colson. About primitive recursive algorithms. In Proceedings of ICALP89, volume 372 of Lecture Notes in Computer Science, pages 194--206. Springer-Verlag, 1989.

....the cells are accessed in increasing order of index. We denote the states as follows: S n ( fb i = 1 j i ng and S n (0) fb i = 1 j i ng [ fb n = 0g; for n 0; and S ( fb i = 1 j i 0g. Thus (D(LNat) is isomorphic to the lazy natural numbers cpo, described for example in [Col89]. ffl 2.2 Product of DCDSs If c is a cell and i is a tag or label, we write c:i for the the labelled cell (c; i) This notation extends to sets of cells and sets of events: for C CM and y EM , C:i = fc:i j c 2 Cg and y:i = f(c:i;v) j (c; v) 2 yg. In defining products we use the labels 1 and ....

Loic Colson. About primitive recursive algorithms. In Proceedings of ICALP89, volume 372 of Lecture Notes in Computer Science, pages 194--206. Springer-Verlag, 1989.

.... fun suffixlist [ suffixlist(z as : l) z: suffixlist(l) e.g. suffixlist [1,2,3] 1,2,3] 2,3] 3] val suffixlist = fn : a list a list list fun flatten [ flatten( L) flatten L = flatten( x: l) L) x: flatten(l: L) e.g. flatten [ 1,2] [3,4,5]] 1,2,3,4,5] val flatten = fn : a list list a list fun merge ( z2) z2 = merge (z1, z1 = merge (x1: l1,x2: l2) x1: x2: merge(l1,l2) e.g. merge( 1,2,3,4] 5,6] 1,5,2,6,3,4] val merge = fn : a list a list a list In existing languages, the typing ....

.... [ suffixlist(z as : l) z: suffixlist(l) e.g. suffixlist [1,2,3] 1,2,3] 2,3] 3] val suffixlist = fn : a list a list list fun flatten [ flatten( L) flatten L = flatten( x: l) L) x: flatten(l: L) e.g. flatten [ 1,2] 3,4,5] [1,2,3,4,5] ) val flatten = fn : a list list a list fun merge ( z2) z2 = merge (z1, z1 = merge (x1: l1,x2: l2) x1: x2: merge(l1,l2) e.g. merge( 1,2,3,4] 5,6] 1,5,2,6,3,4] val merge = fn : a list a list a list In existing languages, the typing and operational ....

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L. Colson. About primitive recursive algorithms. In Proceedings of the 16th International Colloquium on Automata, Languages, and Programming, Stresa, July 1989, pages 194--206. Lecture Notes in Computer Science, Vol. 372, Springer-Verlag, 1989.

.... the need for unbounded computation for a host of well known canonical (read toy ) examples, and appears sufficient for situations where the pieces of code satisfy a sort of masterslave relationship: anomalous behavior evidenced by programs without this property have been investigated by Colson [Col91]. It would be interesting to see if the expressiveness of the primitive recursive iterators is sufficient to formalize more sophisticated arguments about subtyping, for instance [AC91] In reasoning about equational theories where the control is distributed, the robustness of the approach is not ....

L. Colson. About primitive recursive algorithms. Theoretical Computer Science 83 (1991), pp. 57--69.

....can compute the minimum of two numbers n and p in unary representation in time O(min(n; p) However, it cannot compute a natural version of this function. CDSP allows us to compute this function, as well as functions like parallel or. This work can be seen as an extension of the work of Colson [7, 8] with primitive recursive algorithms to the setting of sequential algorithms. In the second part, we show that deterministic parallelism adds intensional expressiveness, settling a folk conjecture from the literature in the negative. We show that CDSP is more expressive intensionally than CDS0. ....

....in reasoning about intensional aspects (e.g. complexity) so we need semantic models that contain more computational information. This can be achieved in many ways. We outline just a few possibilities: ffl We could take the meaning of a program to be a function on a richer domain (e.g. [4, 7]) whose structure permits us to deduce information about computation strategy. ffl We could take the meaning to be a pair consisting of a function and an object conveying intensional information; this object could represent the cost of evaluating the function, or could be a function from inputs ....

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L. Colson, About Primitive Recursive Algorithms, in: Proc. International Colloquium on Automata, Languages, and Programming, 1989, 194-206.

....2.3 An example: Primitive recursion and the lazy natural numbers As a simple example, we exhibit an intensional semantics for the primitive recursive (PR) algorithms [33] A PR algorithm can only recur on one input. The presentation of the algorithms is as a rewrite system following Colson [14]. Consider the following two algorithms for the addition of integers in unary representation (0, S(0) where S stands for successor) 15] add1(0;y) y add1(S(x) y) S(add1(x;y) add2(x;0) x add2(x;S(y) S(add2(x;y) The extensional denotational semantics for add1, add2 maps them ....

....= S(add1(x;y) add2(x;0) x add2(x;S(y) S(add2(x;y) The extensional denotational semantics for add1, add2 maps them into the same function, the addition function of type N 2 N , where N stands for the natural numbers. A simple intensional semantics is provided by the lazy natural numbers [14, 15, 16]. The domain LNAT is shown in Figure 1. LNAT captures the temporal aspect of finding out what an input is. At S k ( we don t know yet if we have the number S k (0) or something larger. S( S k ( S ( 0 S(0) S k (0) Gamma Gamma Gamma Gamma . ....

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L. Colson, About primitive recursive algorithms, in: G. Ausiello et al. eds., Proc. 16th International Colloquium on Automata, Languages and Programming (Springer-Verlag LNCS 372, 1989), 194-206.

.... classes (Clote s work on NC [7] the characterization of P in terms of bounded linear logic [15] and calculus [22] Jones s reconstruction of computability and complexity theory from a programming languages perspective [20,21] and Colson s work on the expressiveness of primitive recursion [8,9]. Our work can be seen in the same spirit, as an attempt to bridge the gap between these two areas of theoretical computer science. 1.2 Outline of the paper Section 2 describes PCF and the deterministic parallel extensions we are comparing. A first version of the circuit semantics is introduced ....

L. Colson, About Primitive Recursive Algorithms, in: Proc. International Colloquium on Automata, Languages, and Programming, 1989 , 194--206.

....for a set O with n elements. Hence it has at least the complexity of g(O) This inefficiency is in contrast to the higher order translation of Tannen and Subrahmanyam [7] which is both uniform and efficient (that is, preserves complexity classes) A similar situation was described by Colson [10]. He gave examples of functions that can be efficiently computed using primitive recursion when functional input and output are allowed, but cannot be efficiently computed when functional input and output are not allowed. This result, as well as that of Colson, indicates the important effects ....

L. Colson. About primitive recursive algorithms. Theoretical Computer Science, 83:57--69, 1991.

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Colson, Loc. About primitive recursive algorithms. Theoretical Computer Science 83 (1991) 5769.

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L. Colson. About primitive recursive algorithms. Theoretical Computer Science, 83:57--69, 1991.

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L. Colson. About primitive recursive algorithms. Theoretical Computer Science, 83:57--69, 1991.

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