Stan Raatz

Extending SLD resolution to equational horn clauses using E -unification

We study the role of unification modulo a set of equations, or E-unification, in the context of refutation methods for sets of Horn clauses with equality. Two extensions of SLD resolution based on E-unification are presented, and rigorous completeness results are shown, including an analysis of the ground case for insight into the computational implications. The concept of a congruence closure generalized to sets of ground Horn clauses is central to these completeness results. The first method is general, in that it applies to arbitrary sets of equational Horn clauses, but is not practical, as it assumes a procedure which gives an explicit sequence of substitutions for each E-unifier. A second method uses a procedure enumerating a complete set of E-unifiers, and appears to be well suited to a class of “well-behaved” equational logic programs which allows a clean and natural integration of the functional and logic-programming paradigms. Using this second method, we have formalized the refutation method used in Eqlog for this class of programs, and a theorem establishes rigorously the completeness of this method. We compare these methods in detail with related work, and show that other methods either explicitly include E-unification or simulate it in some manner.

An algorithm for finding canonical sets of ground rewrite rules in polynomial time

In this paper, it is shown that there is an algorithm that, given by finite set <italic>E</italic> of ground equations, produces a reduced canonical rewriting system <italic>R</italic> equivalent to <italic>E</italic> in polynomial time. This algorithm based on congruence closure performs simplification steps guided by a total simplification ordering on ground terms, and it runs in time <italic>O(n<supscrpt>3</supscrpt>)</italic>.

Finding Canonical Rewriting Systems Equivalent to a Finite Set of Ground Equations in Polynomial Time

In this paper, it is shown that there is an algorithm which, given any finite set E of ground equations, produces a reduced canonical rewriting system R equivalent to E in polynomial time. This algorithm based on congruence closure performs simplification steps guided by a total simplification ordering on ground terms, and it runs in time O(n3).

Hornlog: a graph-based interpreter for general horn clauses

Abstract This paper presents hornlog , a general Horn-clause proof procedure that can be used to interpret logic programs. The system is based on a form of graph rewriting, and on the linear-time algorithm for testing the unsatisfiability of propositional Horn formulae given by Dowling and Gallier [8]. hornlog applies to a class of logic programs which is a proper superset of the class of logic programs handled by PROLOG systems. In particular, negative Horn clauses used as assertions and queries consisting of disjunctions of negations of Horn clauses are allowed. This class of logic programs admits answers which are indefinite, in the sense that an answer can consist of a disjunction of substitutions. The method does not use the negation-by- failure semantics [6] in handling these extensions and appears to have an immediate parallel interpretation.

Rigid E-Unification and, Its Applications to Equational Matings

Publisher Summary The method of matings can be extended to the first-order languages with equality, and this extension is both sound and complete. The method of matings exploits the fundamental property given by the Skolem–Herbrand–Godel theorem. The extension to equational matings is nontrivial and requires proving of the Skolem–Herbrand–Godel theorem. It also requires extending the concept of a mating so that an equational mating is a set of sets of literals mated sets, where a mated set consists of several positive equations and a single negated equation and a form of unification modulo equational theories, or E-unification. The method of matings is an incremental proof procedure that interleaves two steps: quantifier-duplication steps and search for matings.

Error correction in constructive induction

ABSTRACT This paper reports on MIRO, a constructive induction method that achieves substantive noise resistance with a small training set. It is possible to regard MIRO as a learning method that constructs and then performs induction in an abstraction space. After outlining the concept formation algorithm, we describe a postprocessor which identifies positive instances that are likely to contain noise, corrects them, and repairs noise-induced errors in the concept description.

Economy In Expert System Construction: The AEGIS Combat System Maintenance Advisor

We present the design philosophy used in constructing the AEGIS Combat System Maintenance Advisor, a rule-based expert system with over 2000 rules that was completed in less than two years. We attribute part of the success of this project to the decision to have the domain expert actively involved in the process of writing rules, and to the development of a knowledge acquisition tool that checked his work. The project was also noteworthy because it includes a compiler which retargets the knowledge base to alternate run-time environments. The techniques used in this project are general and apply to any propositional rule language for expert systems.

A Semantics for the Hornlog System

We start by presenting some examples of logic programs in the Hornlog system. The body P of a logic program in this system consists of any arbitrary set of Horn clauses over one-sorted Horn clause logic without equality. In chapters 6 through 8 we will examine equational extensions to the Hornlog method which apply to many-sorted Horn clause logic with equality. The procedural interpretation of logic programs [54] is extended so that negative clauses in the body of the logic program are interpreted as negative constraints. That is, ← B 1,..., B m , is interpreted as false ← B 1,..., B m , or “not the case that B 1,...,B m , all hold simultaneously.” The crucial difference between this interpretation and the negation by failure semantics used in Prolog is that in this interpretation any substitutions computed using substitution instances of negative clauses may participate in the construction of the answer substitution. Thus the Hornlog system incorporates a form of classical negation restricted to Horn clauses.

An Equational Extension

It has been recognized for some time that Prolog will need to be extended in order for it to realize the vision of declarative programming. One example is the so-called problem of equality and functions. The underlying deduction mechanism for Prolog, SLD-resolution, is defined only for languages without equality. As mentioned in chapter 1, current Prolog systems include a collection of interpretations of the equality symbol which are basically inherited from LISP, and that it is even possible for the “equality” predicate to cause the side-effect of binding an uninstantiated variable to an instantiated variable. The absence of equality also means that value-returning functions cannot be defined equationally. Instead, an n-ary function f(x 1,..., x, n ) is defined by an n+1-ary predicate p(x 1,..., x n , x n+l) which “collects” the value in the n + 1 st argument. In order for Prolog to reason equationally, rules for rewriting one term to another must be expressed via a distinguished predicate (say rewrite).

Soundness and Completeness Results II

In this chapter we will show that the Hornlog proof method is sound and complete, and that it enjoys the adequacy property defined in chapter 3. We will follow the following guideline in showing these results. First, the method in the ground case is presented and proved complete. Then first-order extension outlined in the previous chapter is shown to be complete by appealing to the completeness of the ground case and the Skolem-Herbrand-Godel theorem. Finally, the soundness and completeness of the method as a computational procedure is established by showing that the set of substitutions returned exactly coincides with the model-theoretic semantics defined for the language. This methodology is particularly appropriate for logic programming because it yields a lifting lemma which constructively illustrates the relationship between the ground and first-order cases.