| OCR Text |
Show 59 4.2.1 Some Notation We often need to refer to the subterms of t . in a unifier e = I {x1 <-t1 , ... ,xm <-tm}. By an address, say ll, however, we cannot identify which D(t.} 1l belongs to. Furthermore, for each subterm in t, we need to identify the I I corresponding subterms in 9H{P,O} that are introduced by the substitution e on H(P,O}. Since this notation is very important in the discussion that follows, it needs to be explained in detail. We first illustrate is by an example. Let H(P,O) = H(f{g(x1 ),x2), h{a,x2}) e = { x 1 < - b, x2 < - g (a}}. The addresses that represent a variable occurrence in are H(P,Q) = H(f(g(x1 },x2}, h(a,x2)} 1.1.1 (representing the first occurrence of x1} 1.2 (representing the first occurrence of x2} 2.2 (representing the second occurrence of x2). After applying the substitution e to H(P,O), we get 9H(P,O) = H(f(g(b),g(a)), h(a,g(a))), where the addresses 1.1.1, 1.2 and 2.2 now represent the subterms b, g(a), and g(a), respectively. We say that these addresses and the addresses that are in-ner to them are introduced by the substitution e on H(P,O). The set of the in-traduced addresses for this example is thus { 1.1.1, 1.2, 1.2.1, 2.2, 2.2.1}. In the following discussion, the reader should pay particular attention to the set of introduced addresses and their images (along reduction sequences). The following definition formally establishes the connection between the terms in the unifier and those in 9(H(P,Q)). |
