D. R. Smith. Structure and design of problem reduction generators. In B. Moller, editor, Constructing Programs from Speci cations, pages 91-124. North-Holland, Amsterdam, 1991.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....tabular solution method, and has been studied extensively since [58] Bird [5] de Moor [17] and others have studied it in the context of program transformation. While some work addresses the derivation of recursive equations, including the original work by Bellman [4] and the later work by Smith [57], our work addresses the derivation of ecient programs that use tabulation. Previous methods for this problem either apply to speci c subclasses of problems [11, 13, 14, 24, 46, 48] or give general frameworks or strategies rather than precise derivation algorithms [3, 5, 6, 8, 9, 16, 17, 45, 47, ....

D. R. Smith. Structure and design of problem reduction generators. In B. Moller, editor, Constructing Programs from Speci cations, pages 91-124. North-Holland, Amsterdam, 1991.

....tabular solution method, and has been studied extensively since [58] Bird [5] de Moor [17] and others have studied it in the context of program transformation. While some work addresses the derivation of recursive equations, including the original work by Bellman [4] and the later work by Smith [57], our work addresses the derivation of ecient programs that use tabulation. Previous methods for this problem either apply to speci c subclasses of problems [11, 13, 14, 24, 46, 48] or give general frameworks or strategies rather than precise derivation algorithms [3, 5, 6, 8, 9, 16, 17, 45, 47, ....

D. R. Smith. Structure and design of problem reduction generators. In B. Moller, editor, Constructing Programs from Specications, pages 91-124. North-Holland, Amsterdam, 1991.

....that it cannot be used very usefully for a whole class of particular cases. 4 Pioneering work in program synthesis among classes of problems has been done mainly by D. Smith. He already studied the following classes: Divideand Conquer [Smi85a, Smi85b, Smi87a] Problem Reduction Generators [Smi91] abstracting general Branch and Bound and Dynamic Programming, Global Search [Smi87b] abstracting bactrack and Branch and Bound and Local Search (hillclimbing) Low91] with M. Lowry (see Appendix) D. Smith inserted those different classes into a taxonomy [Smi93] He also showed in [Smi93] how ....

....to help a user in the design of a greedy algorithm solving a given formal specification. Our approach will be similar to Smith s one in the way to: ffl characterize the class of Greedy Algorithms as Smith already did for Divide and Conquer [Smi85a, Smi85b, Smi87a] Problem Reduction Generators [Smi91] Global Search [Smi87b] and with Lowry for Local Search (see Appendix) ffl formalize the synthesis process [SL90, Smi92, Smi93] Two interesting consequences of this study would be: 1. to create a framework which could help in synthesizing greedy algorithms; 2. to improve Smith s hierarchy ....

D.R. Smith. Structure and design of problem reduction generators. In B. Moller, editor, Constructing Programs from Specifications, pages 91--124. North-Holland, 1991.

....concerned with list partitions. Most recent work in this area has been concerned with improving the time complexity of naive dynamic programming solutions, by exploiting special properties of the cost function [12] In programming methodology, our work is very much akin to that of Smith and Lowry [23, 24]. Smith s notion of problem reduction generators is quite similar to the generic algorithm presented here, but his results are more concerned with similarities in the derivation process, and not with a single generic program. There has recently been a surge of interest in polytypic programming. ....

D.R. Smith. Structure and design of problem reduction generators. In B. Moller, editor, Proc. of the IFIP TC2 Working Conference on Constructing Programs from Specifications. North--Holland, 1991.

....and potentially so ubiquitously applicable, that it is impossible a priori to operationalize it for all cases to which it is applicable. On the other hand, that method has indeed been successfully operationalized for specific domains, such as for composition of a certain class of algorithms (Smith, 1991). Then there are methods, such as Propose and Revise, and Heuristic Classification, that are also quite general and may be applied to almost any problem in principle. These methods have been operationalized: if we can find domain knowledge to suit their requirements, they can be applied. Almost ....

Smith, 1991 Smith, Douglas, R. (1991). Structure and design of problem reduction generators. In Möller , B., (Ed.) , Constructing Programs from Specifications. North Holland.

....specification to the problem domain. The view construction can proceed incrementally by working downwards in the hierarchy, from more general algorithm classes to more specific ones. These algorithm class specifications (also known as algorithm theories ) have been developed over several years [14, 12, 15]. See Section 4.4 and 4.5 for an example of a divide and conquer algorithm theory. 4 Synthesis as Diagram Growing We present a concise discussion of an application problem in the Air Traffic Control domain and the derivation of a suitable algorithm. The emphasis is not on the algorithm per se, ....

.... e j track Theta plot axioms ds, dw are user specified ds(e) feg dw(d; e) d with e cs, cw are system deduced cs(e) mk tpa(e) cw(r; e) min tpa (r; mk tpa(e) end Figure 5: Domain dependent development axioms in a complete divide and conquer theory; for full details, see [14, 15]. Each choice of operations leads to a different divide and conquer theory, since the number of operations and their signature can vary. Therefore the synthesis environment contains, in its knowledge base, a meta level representation of a family of divide and conquer theories from which it ....

Smith, D. R. Structure and design of problem reduction generators. In Constructing Programs from Specifications, B. Moller, Ed. NorthHolland, Amsterdam, 1991, pp. 91--124.

....conclusion Dynamic programming was first formulated by Bellman [4] and has been studied extensively since [51] Bird [5] de Moor [16] and others have studied it in the context of program transformation. While some works address the derivation of recursive equations, notably the work by Smith [50], our work addresses the derivation of efficient programs that use tabulation. Previous methods for this problem either apply to specific subclasses of problems [13, 40, 10, 12, 42, 21] or give general frameworks and strategies rather than precise algorithms [52, 9, 5, 48, 6, 3, 39, 49, 8, 16, 41, ....

D. R. Smith. Structure and design of problem reduction generators. In B. Moller, editor, Constructing Programs from Specifications, pages 91--124. North-Holland, Amsterdam, 1991.

...., then z 0 is a feasible solution to input x 0 . Similar comments hold for the other Soundness axioms. The operator is a well founded order on D to assure termination (axioms are omitted for simplicity) A general scheme for problem reduction theories (including divide and conquer) is given in [6]. The idea is to have a different first order divide and conquer theory for each possible abstract constructor signature. The codomain spec of the divide and conquer refinement (Figure 3) contains a schematic definition for the top level divide and conquer functions and schematic requirement ....

Smith, D. R. Structure and design of problem reduction generators. In Constructing Programs from Specifications, B. Moller, Ed. North-Holland, Amsterdam, 1991, pp. 91--124.

.... has tactics for simple problem reduction (reducing a specification to a library routine) 33] divide and conquer [33] global search (binary search, backtrack, branch and bound) 34] problem reduction generators (dynamic programming, general branch and bound, and game tree search algorithms) [36], and local search (hillclimbing algorithms) 23] 4. Apply optimizations The KIDS system allows the application of optimization techniques such as expression simplification, partial evaluation, finite differencing, case analysis, and other transformations [42] The user selects an ....

....6.2.1) extends problem theory with the basic concepts of backtracking: subspace descriptors, initial space, the splitting and extraction operations, filters, and so on. A divide and conquer theory would extend problem theory with concepts such as decomposition operators and composition operators [33, 36]. 5.2 Synthesizing a Scheduler There are two basic approaches to computing a schedule: local and global. Local methods focus on individual schedules and similarity relationships between them. Once an initial schedule is obtained, it is iteratively improved by moving to neighboring structurally ....

Smith, D. R. Structure and design of problem reduction generators. In Constructing Programs from Specifications, B. Moller, Ed. North-Holland, Amsterdam, 1991, pp. 91-- 124.

....pattern. To express the essence of divide and conquer, we define a divide and conquer theory comprised of various sorts, function, predicates, and axioms that assure that the above scheme correctly solves a given problem. A simplified divideand conquer theory is as follows (for more details see [8, 9]) Theory Divide Gammaand GammaConquer Sorts D;R domain and range of a problem Operations I : D Boolean input condition O : D Theta R Boolean output condition primitive : D Boolean control predicate ODecompose : D Theta D Theta D Boolean output condition for Decompose OCompose : ....

Smith, D. R. Structure and design of problem reduction generators. In Constructing Programs from Specifications, B. Moller, Ed. North-Holland, Amsterdam, 1991, pp. 91--124.

....0 , then z 0 is a feasible solution to input x 0 . Similar comments hold for the other Soundness axioms. The operator is a well founded order on D to assure termination (axioms are omitted for simplicity) A general scheme for problem reduction theories (including divide and conquer) is given in [6]. The idea is to have a different first order divide and conquer theory for each possible abstract constructor signature. The codomain spec of the divide and conquer refinement (Figure 3) contains a schematic definition for the top level divide and conquer functions and schematic requirement ....

Smith, D. R. Structure and design of problem reduction generators. In Constructing Programs from Specifications, B. Moller, Ed. North-Holland, Amsterdam, 1991, pp. 91--124.

....the algorithm theories for divide and conquer and dynamic programming involve a soundness axiom that relates decomposition and composition operators. Given a simple decomposition operator, the soundness axiom is unskolemized and used to derive a corresponding composition operator (Smith (1985) Smith (1991)) The pruning mechanisms of global search algorithms are also derived via unskolemization (Smith (1987) The principle of divide and conquer algorithms is to solve small problem instances directly, and to solve larger problem instances by decomposing them, solving the pieces, and composing the ....

....the resulting solutions. Part of a specification for a simple divide and conquer theory is given next. It provides the structure for a binary decomposition operator and corresponding composition operator. A general scheme for problem reduction theories (including divide and conquer) is given in (Smith (1991)) Spec Divide and Conquer Theory Sorts D input domain R output domain Operations I : D boolean input condition O : D Theta R boolean output condition Decompose : D Theta D Theta D boolean decomposition relation Compose : R Theta R Theta R boolean composition relation ....

Smith, D. R. (1991). Structure and design of problem reduction generators. In Moller, B., editor, Constructing Programs from Specifications, pages 91--124. North-Holland, Amsterdam.

....resulting solutions. Part of a specification for a simple divide and conquer design theory is given next. It provides the structure for a binary decomposition operator and corresponding composition operator. A general scheme for problem reduction theories (including divide and conquer) is given in [19]. generate and test Problems (CSP) steepest ascent simulated annealing repair methods binary search backtrack Structure Linear Programming Programming simplex primal dual specialized simplex NW Algorithm Reduction sieves Generators dynamic programming game tree search divide and conquer ....

....unskolemizing another axiom to obtain a translation for the prim predicate, and translating and proving other axioms. The resulting algorithm is a variant of Quicksort. Once the classification arrow is complete, divide and conquer programtheory can be instantiated to obtain concrete code; see [16, 19] for details. 4.2 Distribution of Goods Problem Theory = I 0 ) GDP 0 Constraint Satisfaction = I 1 ) GDP 1 Integer Programming = I 2 ) GDP 2 Integer Linear Programming = I 3 ) GDP 3 Network Flow = I 4 ) GDP 4 GDP Ladder Construction Suppose that a ....

Smith, D. R. Structure and design of problem reduction generators. In Constructing Programs from Specifications, B. Moller, Ed. North-Holland, Amsterdam, 1991, pp. 91--124.

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