The limits of fixed-order computation

Abstract

Fixed-order computation rules, used by Prolog and most deductive database systems, do not suffice to compute the well-founded semantics (Van Gelder et al., J. ACM 38(3) (1991) 620– 650) because they cannot properly resolve loops through negation. This inadequacy is reflected both in formulations of SLS-resolution (Przymusinski, in: Proc. 8th ACM SIGACT-SIGMOD-SIGART Symp. on Principles of Database Systems, ACM Press, Philadelphia, Pennsylvania, March 1989, pp. 11–21; Ross, J. Logic Programming 13(1) (1992) 1–22) which is an ideal search strategy, and in more practical strategies like SLG (Chen and Warren, J. ACM 43(1) (1996) 20–74), or Well-Founded Ordered Search (Stucky and Sudarshan, J. Logic Programming 32(3) (1997) 171– 206). Typically, these practical strategies combine an inexpensive fixed-order search with a relatively expensive dynamic search, such as an alternating fixed point (Van Gelder, J. Comput. System Sci. 47(1) (1993) 185–221). Restricting the search space of evaluation strategies by maximizing the use of fixed-order computation is of prime importance for efficient goal-directed evaluation of the well-founded semantics. Towards this end, the theory of modular stratification (Ross, J. ACM 41(6) (1994) 1216–1266), formulates a subset of normal logic programs whose literals can be statically reordered so that the program can be evaluated using a fixed-order computation rule. The class of modularly stratified programs, however, is not closed under simple program transformations such as the HiLog transformation. We address the limits of fixed-order computation by adapting results of Przymusinski (1992) to formulate the class of left-to-right dynamically stratified programs, and show that this class properly includes other classes of fixed-order stratified programs. We then introduce SLG strat , a variant of SLG resolution that uses a fixed-order computation rule, and prove that it correctly evaluates ground left-to-right dynamically stratified programs. Finally, we indicate how SLG strat can be used as a basis for computing the well-founded semantics through a search strategy called SLG RD , for SLG with Reduced use of Delaying.