Smallest Horn Clause Programs

The simplest non{trivial program pattern in logic programming is the following one : 8 < : p(fact) : p(left) p(right) : p(goal) : where fact, goal, left and right are arbitrary terms. Because the well known append program matches this pattern, we will denote such programs \append{like". In spite of their simple appearance, we prove in this paper that termination and satissability (i.e the existence of answer{substitutions, called the emptiness problem) for append{like programs are undecidable. We also study some subcases depending on the number of occurrences of variables in fact, goal, left or right. Moreover, we prove that the computational power of append{like programs is equivalent to the one of Turing machines ; we show that there exists an append{like universal program. Thus, we propose an equivalent of the BB ohm{Jacopini theorem for logic programming. This result connrms the expressiveness of logic programming. The proofs are based on program transformations and encoding of problems , unpredictable iterations within number theory deened by J.H. Con-way or the Post correspondence problem. / This paper is a survey which covers the results of 19], 20], 21] and 28].