Flattening and Implication

Flattening is a method to make a de nite clause function-free. For a de nite clause C, attening replaces every occurrence of a term f(t 1 ; ; t n ) in C with a new variable v and adds an atom p f (t 1 ; ; t n ; v) with the associated predicate symbol p f with f to the body of C. Here, we denote the resulting function-free de nite clause from C by at(C). In this paper, we discuss the relationship between attening and implication. For a de nite program and a de nite clause D, it is known that if at( ) j= at(D) then j= D, where at( ) is the set of at(C) for each C 2 . First, we show that the converse of this statement does not hold even if = fCg, that is, there exist de nite clauses C and D such that C j= D but at(C) 6j= at(D). Furthermore, we investigate the conditions of C and D satisfying that C j= D if and only if at(C) j= at(D). Then, we show that, if (1) C is not self-resolving and D is not tautological, (2) D is not ambivalent, or (3) C is singly recursive, then the statement holds.