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Semantics of infinite tree logic programming. (English) Zbl 0621.68054
The logic programs considered here consist of clauses $$A\leftarrow S[ ]B$$, where A is an atom, B is a finite sequence of atoms and S is a finite sequence of equations and inequations. Unification is made as in PROLOG, i.e. without ”occurs check”, so f(x) and x are unifiable and the result is the infinite term f(f(...)). The main difference from the treatment in the literature [cf. J. W. Lloyd, Foundations of logic programming (1984; Zbl 0547.68005)] concerns inequalities. The equation (resp. inequation) of terms u, v is true for a substitution s if us, vs coincide (resp. do not coincide). A satisfier of a system S of equations and inequations is a substitution s such that any element of S is true under any ground instance of s. Derivation sequences, success set SS(P), finite failure set FF(P), immediate consequence function $$T_ p$$ for a program P are defined in a familiar way. In the absence of inequations $$T_ p$$ is continuous in Baire topology on infinite terms and atoms containing such terms, but for the program $$Q=\{p(x)\leftarrow (x=a[ ]q(y));q(x)\leftarrow (x=f(y),x\neq f(x)[ ]q(y))\}$$, the $$T_ Q$$ is discontinuous at q(f(f(...))). So, instead of simply considering closed Herbrand interpretations to prove standard results on fixed points, the authors have to check that the results of T-operations have rational cover: any ground atom in a set is an instance of a rational atom (having only a finite number of subterms). This turns out to be possible since unification (without the occurs check) preserves rationality and any finite system S having ground satisfier s, has also a rational satisfier s’$$\geq s$$. Defining a (Clark-)completed program $$P^*$$ for a program P in the usual way, the authors prove that $$P^*\vDash A$$ iff A belongs to SS(P). Here $$\vDash$$ means consequence in all Herbrand models. Similar equivalence for $$\sim A$$ and FF(P) holds exactly when P is derivation compact: the sequence S’ of equations and inequations arising in any (possibly infinite) P-derivation is solvable iff each of its finite stumps is solvable. This is true for programs containing no inequations, but is false for the program Q above.
Reviewer: G.Mints

##### MSC:
 68T15 Theorem proving (deduction, resolution, etc.) (MSC2010) 68N01 General topics in the theory of software
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