The finite stages of inductive definitions

In general, the least fixed point of a positive elementary inductive definition over the Herbrand universe is Π 1 and has no computational meaning. The finite stages, however, are computable, since validity of equality formulas in the Herbrand universe is decidable. We set up a formal system BID for the finite stages of positive elementary inductive definitions over the Herbrand universe and show that the provably total functions of the system are exactly that of Peano arithmetic. The formal system BID contains the so-called inductive extension of a logic program as a special case. This first-order theory can be used to prove termination and correctness properties of pure Prolog programs, since notions like negation-as-failure and left-termination can be turned into positive inductive definitions. 1 Why inductive definitions over the Herbrand universe? In traditional logic programming, the semantics of a program is always given by the least fixed point of a monotonic operator over the Herbrand universe. The first example is the well-known van Emden-Kowalski operator for definite Horn clause programs in [25]. This operator is defined by a purely existential formula and is therefore continuous. The least fixed point of the operator is recursively enumerable. Moreover, the finite stages of the inductive definition are exactly what is computed by SLD-resolution. In [11], Fitting has generalized the van Emden-Kowalski operator using threevalued logic to programs which may also contain negation in the bodies of the clauses. Although Fitting’s operator is still monotonic it is no longer continuous. It follows from Blair [2] and Kunen [14] that the least fixed point of this operator can be Π 1 -complete and that the closure ordinal can be ω CK 1 even for definite Horn clause programs. The finite stages of Fitting’s operator, however, are decidable and correspond to what is computed by SLDNF-resolution. This has been shown by Kunen in [15] for allowed logic programs and by the author in [21] for mode-correct programs. The class of allowed programs is considered as too restrictive in general. The ? Research supported by the Swiss National Science Foundation. This article has been written at the Department of Mathematics, Stanford University. Appeared in P. Hájek, editor, GÖDEL’96, Logical Foundations of Mathematics, Computer Science and Physics — Kurt Gödel’s Legacy, Brno, Czech Republic. Springer-Verlag, Lecture Notes in Logic 6, pages 267–290, 1996. class of mode-correct programs, however, contains most programs of practical interest, since a programmer has always modes in mind when he writes a program. Moreover, every allowed program is also mode-correct. Even though the use of three-valued logic can be eliminated in Fitting’s operator (cf. eg. [13]), the completeness results of [15] and [21] cannot be applied to existing implementations of logic programming like, for example, Prolog. The reason is that these systems use special search-strategies which depend on the order of clauses in the program and on the order of literals in the bodies of clauses. Apt and Pedreschi, however, have observed in [1] that only the order of literals in the bodies is important. They say that most programs used in practice terminate independently of the order of clauses in the program — at least for the intended inputs. Based on this observation we assign in [22] to a predicate R of a logic program three positive elementary inductive definitions for new relations R, R and R. The relation R corresponds to Fitting’s truth-value true; R corresponds to the truth-value false; R expresses left-termination in the sense of Apt and Pedreschi. The corresponding formal system is called the inductive extension of a logic programs and can be used for proving termination and correctness properties of Prolog programs (see also [23]). A natural question is then: What is the proof-theoretic strength of the inductive extension? — We answer this problem below and show that the provably total functions of the inductive extension are exactly those of Peano Arithmetic. As a byproduct we obtain a proof-theoretic proof of a key lemma used in the completeness proofs in [22] and [24]. Our proof-theoretical analysis of the inductive extension in terms of provably total functions is similar to Pohlers’ treatment of ID1 in [7] and [18]. The plan of this article is as follows. After recalling some well-known facts on Clark’s equality theory CET in Sect. 2, we present a framework for inductive definitions over the Herbrand universe in Sect. 3 and set up a formal system for such in Sect. 4. Using the number-theoretic ordinal functions of Sect. 5 we embed the formal system into an infinitary sequent calculus in Sect. 6. Partial cutelimination allows then to consider the positive/negative fragment of the calculus only and to perform an asymmetric interpretation that yields the main results of the article. Sect. 7 finally provides some hints how the general framework for inductive definitions relates to the so-called inductive extension of a logic program and to notions like negation as failure and left-termination.