# Tech Report CS-93-51

## Backtracking Without Trailing in CLP(${\Re}_{\rm Lin}$)

### Abstract:

Constraint logic programming (CLP) is a generalization of logic programming where unification is replaced by constraint solving as the basic operation of the language. The combination of constraint solving and nondeterminism (approximated by backtracking) makes these languages appealing for a variety of combinatorial search problems. 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 paper proposes a fundamentally 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 by an order of magnitude compared to implementations based on trailing and makes space complexity 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 (about 20,000 lines of C) and compared analytically and experimentally with an optimized trailing implementation. Experimental results indicate that semantic backtracking produces significant reduction in memory space, while keeping the time overhead reasonably small.

(complete text in pdf or gzipped postscript)