Pascal Van Hentenryck
Abstract
Existing CLP languages support backtracking by generalizing traditional Prolog implementations: modifications to the constraint system are trailed and restored on backtracking. Although simple and efficient, trailing may be very demanding in memory space, since the constraint system may potentially be saved at each choice point.
This article proposes a new implementation scheme for backtracking in CLP languages over linear (rational or real) arithmetic. The new scheme, called semantic backtracking, does not use trailing but rather exploits the semantics of the constraints to undo the effect of newly added constraints. Semantic backtracking reduces the space complexity compared to implementations based on trailing by making it essentially independent of the number of choice points. In addition, semantic backtracking introduces negligible space and time overhead on deterministic programs. The price for this improvement is an increase in backtracking time, although constraint-solving time may actually decrease. The scheme has been implemented as part of a complete CLP system CLP (ℜLin) and compared analytically and experimentally with optimized trailing implementations. Experimental results on small and real-life problems indicate that semantic backtracking produces significant reduction in memory space, while keeping the time overhead reasonably small.
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Index Terms
- Backtracking without trailing in CLP (ℜLin)
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Reviews
Jeffrey B. Putnam
Constraint logic programming (CLP) is a powerful mechanism for solving combinatorial search problems, combining ease of use with powerful problem-solving techniques. CLP, as usually implemented, operates in the same way as Prolog, using backtracking to explore the search space, and saving states as they are encountered in order to enable backtracking (“trailing backtracking”). This paper offers a different method for backtracking, called “semantic backtracking,<__?__Pub Caret>” in CLP R Lin (a version of CLP that operates over the real numbers with constraints in the form of linear equations, inequalities, and disequations). This backtracking method uses the semantics of the operations applied at choice points, and effectively computes the inverse operator for the decisions made. The paper gives the algorithms in excellent detail as well as in overview. The authors provide information on time and space usage that shows just how the semantic backtracking compares to the usual backtracking methods. It uses less space, at the expense of time. The space requirements in the semantic scheme range from slightly over half the space requirements in the usual scheme to about one tenth, with the time requirements ranging from slightly faster, in one or two cases, to about 1.6 times the original time. This seems (especially with the numbers presented) to be a favorable exchange, since machines seem to be speeding up faster than memory is growing. The tables confused me for a moment, because they show the cost for semantic divided by the cost for trailing in the space section and the cost for trailing divided by the cost for semantic in the time section. I was left with one question. The backtracking techniques presented both depend on the actual operations performed, with backtracking just reversing the order of those operations. Is it possible (rather in the spirit of a truth maintenance system) to run them backwards in a different order__?__
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