R. S. Bird and L. G. L. T. Meertens. Two exercises found in a book on algorithmics. In L. G. L. T. Meertens, editor, TC2/WG 2.1 Working Conference on Program Speci cation and Transformation, Bad Tolz, Germany, pages 451-457. North-Holland, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....p list (X; Ym ; L 2 ; Yn ) 9L: append (L 1 ; L 2 ; L) p list(X; Y 0 ; L; Yn ) where append is the familiar predicate which speci es list concatenation. Notice that by Property 2 the predicate p list de nes a homomorphism w.r.t. the list structure of its third argument in the sense of [BiM87]. De nition 3 (Clause Extension) Let R be a goal list of length r based on goal M , and S be a goal list of length s based on the same goal M such that r s. We say that clause K Body [S] is an extension of clause H Body [R] De nition 4 (List Introduction Generalisation) Let A be the ....

R. S. Bird and L. G. L. T. Meertens. Two exercises found in a book on algorithmics. In L. G. L. T. Meertens, editor, TC2/WG 2.1 Working Conference on Program Speci cation and Transformation, Bad Tolz, Germany, pages 451-457. North-Holland, 1987.

....p list (X; Ym ; L 2 ; Yn ) 9L: append (L 1 ; L 2 ; L) p list(X; Y 0 ; L; Yn ) where append is the familiar predicate which speci es list concatenation. Notice that by Property 2, the predicate p list de nes a homomorphism w.r.t. the list structure of its third argument in the sense of [BiM87]. De nition 3 (Clause Extension) Let R be a goal list of length r based on goal M , and S be a goal list of length s based on the same goal M such that r s. We say that clause K Body [S] is an extension of clause H Body [R] De nition 4 (List Introduction Generalization) Let A be the ....

R. S. Bird and L. G. L. T. Meertens. Two exercises found in a book on algorithmics. In L. G. L. T. Meertens, editor, TC2/WG 2.1 Working Conference on Program Specication and Transformation, Bad Tolz, Germany, pages 451-457. North-Holland, 1987.

....rule is not violated. Thus, recursion in point wise rules is bounded to the number of entities in the simulation mesh. An example of a symmetric Gauss Seidel method implemented through point wise rule recursion will be illustrated in section 6.2. 3. 3 Reduction Rules The Bird Meertens formalism[7] provides a means of describing what are otherwise known as reduction operations 2 . In this abstraction, a reduction is described 2 These reduction operations may be computations of global simulation parameters such as the maximum stable time step, or of local accumulations such as summing ....

R. S. Bird and L. Meertens. Two exercises found in a book on algorithmics. In L. G. L. T. Meertens, editor, Program Specification and Transformation, pages pp 451--457. North-Holland, 1987.

....transformations can be found in [22, 23, 136] The schema transformations presented in these papers deal with recursion removal and the reduction of nondeterminism in generate and test programs. The schemata approach has been used also for deriving e cient functional programs (see, for instance, [10, 11]) In this case programs are denoted by applicative expressions and it is possible to improve their e ciency by replacing some subexpressions by more e cient, semantically equivalent new subexpressions. These equivalences are stated as theorems in suitable theories. The schemata approach has been ....

R. S. Bird and L. G. L. T. Meertens. Two exercises found in a book on algorithmics. In L. G. L. T. Meertens, editor, TC2/WG 2.1 Working Conference on Program Specication and Transformation, Bad Tlz, Germany, pages 451457. North-Holland, 1987.

....irrelevant fact that my implementation language was imperative whereas Bird s was recursively defined functions but I had no time to explore the connections in detail. Only recently did I make the time for such an exploration, spurred on by the observation that Bird together with Meertens [10, 6, 11, 20] seemed to have been making remarkable progress. The goal that I set myself, the outcome of which is reported here, was self evident. In my book [1] I had identified a number of properties of quantifications over finite domains that experience showed were central to the calculation of a number of ....

R.S. Bird and L. Meertens. Two exercises found in a book on algorithmics. In L.G.L.T. Meertens, editor, Program Specification and Transformations, pages 451--457, Elsevier Science Publishers B.V., North Holland, 1987.

....our own conception of it) is a calculus of total functions based on a small number of primitives and a hierarchy of types including trees and lists. The theory was set out in an inspiring paper by Meertens [22] and has been further refined and applied in a number of papers by Bird and Meertens [8, 9, 11, 12, 13]. Its beauty derives from the small scale of the theory itself compared with the large scale of applications. Essentially there are just three primitive operators in the theory reduce , map and filter . Actually, the names used by Meertens for the first two of these operators were ....

R.S. Bird and L. Meertens. Two exercises found in a book on algorithmics. In L.G.L.T. Meertens, editor, Program Specification and Transformations, pages 451457. Elsevier Science Publishers B.V., North Holland, 1987.

....our own conception of it) is a calculus of total functions based on a small number of primitives and a hierarchy of types including trees and lists. The theory was set out in an inspiring paper by Meertens [23] and has been further refined and applied in a number of papers by Bird and Meertens [9, 10, 13, 11, 14]. Essentially there are just three primitive operators in the theory reduce , map and filter . These operators are defined at each level of a hierarchy of types called the Boom hierarchy 1 after H.J. Boom to whom Meertens attributes the concept. The Boom hierarchy begins at the level ....

....our own conception of it) is a calculus of total functions based on a small number of primitives and a hierarchy of types including trees and lists. The theory was set out in an inspiring paper by Meertens [23] and has been further refined and applied in a number of papers by Bird and Meertens [9, 10, 13, 11, 14]. Essentially there are just three primitive operators in the theory reduce , map and filter . Actually, the names used by Meertens for the first two of these operators were inserted in and applied to all . Moreover, just the first two are primitive since filter is defined in terms of ....

R.S. Bird and L. Meertens. Two exercises found in a book on algorithmics. In L.G.L.T. Meertens, editor, Program Specification and Transformations, pages 451--457. Elsevier Science Publishers B.V., North Holland, 1987.

....variables are sent to a single processor to evaluate this function, a performance bottle neck would develop. Instead, these functions are evaluated by way of a reduction operation. A reduction operation can be defined by any function that can be decomposed according the the Bird Meertens formalism[36]. This formalism identifies a class of functions that can be decomposed into two distinct components: a function that is applied to each element of a container, and an associative binary operator that has an associated identity element. The advantage of identifying this structure is in the ....

R. S. Bird and L. Meertens, "Two exercises found in a book on algorithmics," in Program Specification and Transformation (L. G. L. T. Meertens, ed.), pp. pp 451--457, NorthHolland, 1987.

....computation that provides value for the parameters specified in its head. Like point wise rules, these rules can not violate the uniqueness of parameter values. Any conflict of value assignment are reported as ambiguity errors during scheduling. 3. 4 Reduction Rules The Bird Meertens formalism[7] provides a means of describing what are otherwise known as reduction operations 2 . In this abstraction, a reduction is described by a function composed of three components: a function that is applied to a set of values, an associative and commutative operator Phi that is defined on the type ....

R. S. Bird and L. Meertens. Two Exercises Found in a Book on Algorithmics. In L. G. L. T. Meertens, editor, Program Specification and Transformation, pages pp 451--457. North-Holland, 1987.

....of expressions in the imperative programming paradigm due to this lack of referential transparency. A comprehensive study of these and other topics associated with the functional programming paradigm can be found in [4] 5.1. 1 The Bird Meertens Formalism The formalism of Bird and Meertens [5] is often referenced in the literature on functional programming paradigms. The Bird Meertens formalism describes the properties of several operations performed on trees, sequences, and bags (unordered sets) The basis of these operations is that they construct a homomorphism described on these ....

R. S. Bird and L. Meertens. Two Exercises Found in a Book on Algorithmics. In L. G. L. T. Meertens, editor, Program Specification and Transformation, pages pp 451--457. North-Holland, 1987.

....replace in a clause C a goal of the form p list (T 0 ; L 1 ; T ; Y ) p list (T ; L 2 ; T n ; Y ) by the goal append (L 1 ; L 2 ; L) p list (T 0 ; L; T n ; Y ) 17. By Property P2, the predicate p list de nes a homomorphism w.r.t. the list structure in its second argument, in the sense of [3]. Let us now describe the list introduction strategy. This strategy combined with the tupling strategy can be used in the case when a variable number of calls should be tupled together to allow for a folding step which collects all calls of a given predicate in a clause. The application of the ....

R. S. Bird and L. G. L. T. Meertens, \Two exercises found in a book on algorithmics," in TC2/WG 2.1 Working Conference on Program Specication and Transformation, Bad Tolz, Germany (L. G. L. T. Meertens, ed.), pp. 451-457, North-Holland, 1987.

....to program with. In functional programming the role of such universals combined with the calculational style entailed by category theory has been fundamental to a whole discipline of algorithm derivation and transformation. This can be traced back to the so called Bird Meertens formalism [5,6] and the foundational work of T. Hagino [8] Since then, the area has known a remarkable progress, as witnessed by the vast bibliography published both on theory and applications see [12,13,4] among many others references. This paper reports on an attempt to apply the same reasoning principles ....

R. S. Bird and L. Meertens. Two exercises found in a book on algorithmics. In L. Meertens, editor, Program Specication and Transformation, pages 451-458. North-Holland, 1987.

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Bird, R. S., Meertens, L. G. L. T.: Two Exercises Found in a Book on Algorithmics. TC2 WG 2.1 Working Conference on Program Specification and Transformation, Bad Tolz (Germany) North Holland (1987) 451--457

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Richard S. Bird et al. Two exercises found in a book on algorithmics. In Lambert Meertens, editor, Program Specification and Transformation. Elsevier Science Publishers B.V. (North-Holland), 1987.

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