Justifying control for logic programs

Abstract

A pure prolog program (with goal) consists of a definite clause part P and an expression G where P essentially assigns recursive definitions to predicate names and G is evaluated with respect to that assignment. This can be described using a term calculus for defining predicators p:πn and goal expressions g:o. Predicate names can be bound using recursion operators, and programs correspond to expressions without free predicate names. Evaluation of a goal expression g(ξ) with free program variables ξ enumerates a stream of answer substitutions {τ/ξ}.The denotational semantics of this language is employed to show limits of the expressiveness: While pure prolog is computationally complete in the sense that for every partial recursive function there is a closed predicator p so that [p](n,y) = [{f(n)/y}] (here n and f(n) stand for representations of numbers and [α] is a singleton stream), the set of definable functionals λP [[πn]. [t(p)]ϕ[p<-P] is extremely small. It is shown that even simple functionals as λP.λt.[P(t)(1)] (if defined otherwise ⊥), λP.λt.[P(t)(2)],… (i.e. ‘pick the first, second,… answer") are not definable. While ‘pick the first’ is usually obtained by adding the control construct once or cut or if-then-else to pure prolog, this does not in general account for ‘pick the n-th’.In full prolog, where imperative features are available, the situation changes. The classical prolog control, however, like cut etc. prove valuable as well as insufficient.