A Quasi-Metric for Machine Learning

The subsumption relation is crucial in the Machine Learning systems based on a clausal representation. In this paper we present a class of operators for Machine Learning based on clauses which is a characterization of the subsumption relation in the following sense: The clause C 1 subsumes the clause C 2 iff C 1 can be reached from C 2 by applying these operators. In the second part of the paper we give a formalization of the closeness among clauses based on these operators and an algorithm to compute it as well as a bound for a quick estimation.Ministerio de Ciencia y Tecnología TIC 2000-1368-C03-0Junta de Andalucía TIC-13

Generalizing Programs via Subsumption

In this paper we present a class of operators for Machine Learning based on Logic Programming which represents a characterization of the subsumption relation in the following sense: The clause C 1 subsumes the clause C 2 iff C 1 can be reached from C 2 by applying these operators. We give a formalization of the closeness among clauses based on these operators and an algorithm to compute it as well as a bound for a quick estimation. We extend the operator to programs and we also get a characterization of the subsumption between programs. Finally, a weak metric is presented to compute the closeness among programs based on subsumption.Ministerio de Ciencia y Tecnología TIC 2000-1368-C03-0Junta de Andalucía TIC-13

Unification of terms with exponents

In an ICALP (1991) paper, H. Chen and J. Hsiang introduced a notion that allows for a finite representation of certain infinite sets of terms. These so called w-terms find an application in logic programming, where they can serve to represent finitely an infinite number of answers or to avoid nontermination in certain cases. Another application is in the field of equational logic. Using w-terms, it is possible to avoid a certain type of divergence of ordered completion. In all cases, unification is the basic computational aspect of this notation. Chen and Hsiang give a complete and terminating unification algorithm for w-terms. Recently, H. Comon introduced terms with exponents, thus significantly extending Chen and Hsiang's notion of w-terms. He provides a fairly complicated unification algorithm. This paper introduces a further syntactic generalization of Comon's notion together with a comparatively simple inference system for unification

Logical Reduction of Metarules

International audienceMany forms of inductive logic programming (ILP) use metarules, second-order Horn clauses, to define the structure of learnable programs and thus the hypothesis space. Deciding which metarules to use for a given learning task is a major open problem and is a trade-off between efficiency and expressivity: the hypothesis space grows given more metarules, so we wish to use fewer metarules, but if we use too few metarules then we lose expressivity. In this paper, we study whether fragments of metarules can be logically reduced to minimal finite subsets. We consider two traditional forms of logical reduction: subsumption and entailment. We also consider a new reduction technique called derivation reduction, which is based on SLD-resolution. We compute reduced sets of metarules for fragments relevant to ILP and theoretically show whether these reduced sets are reductions for more general infinite fragments. We experimentally compare learning with reduced sets of metarules on three domains: Michalski trains, string transformations, and game rules. In general, derivation reduced sets of metarules outperform subsumption and entailment reduced sets, both in terms of predictive accuracies and learning times

Non-Termination Inference of Logic Programs

We present a static analysis technique for non-termination inference of logic programs. Our framework relies on an extension of the subsumption test, where some specific argument positions can be instantiated while others are generalized. We give syntactic criteria to statically identify such argument positions from the text of a program. Atomic left looping queries are generated bottom-up from selected subsets of the binary unfoldings of the program of interest. We propose a set of correct algorithms for automating the approach. Then, non-termination inference is tailored to attempt proofs of optimality of left termination conditions computed by a termination inference tool. An experimental evaluation is reported. When termination and non-termination analysis produce complementary results for a logic procedure, then with respect to the leftmost selection rule and the language used to describe sets of atomic queries, each analysis is optimal and together, they induce a characterization of the operational behavior of the logic procedure.Comment: Long version (algorithms and proofs included) of a paper submitted to TOPLA

Generalization of Clauses under Implication

In the area of inductive learning, generalization is a main operation, and the usual definition of induction is based on logical implication. Recently there has been a rising interest in clausal representation of knowledge in machine learning. Almost all inductive learning systems that perform generalization of clauses use the relation theta-subsumption instead of implication. The main reason is that there is a well-known and simple technique to compute least general generalizations under theta-subsumption, but not under implication. However generalization under theta-subsumption is inappropriate for learning recursive clauses, which is a crucial problem since recursion is the basic program structure of logic programs. We note that implication between clauses is undecidable, and we therefore introduce a stronger form of implication, called T-implication, which is decidable between clauses. We show that for every finite set of clauses there exists a least general generalization under T-implication. We describe a technique to reduce generalizations under implication of a clause to generalizations under theta-subsumption of what we call an expansion of the original clause. Moreover we show that for every non-tautological clause there exists a T-complete expansion, which means that every generalization under T-implication of the clause is reduced to a generalization under theta-subsumption of the expansion.Comment: See http://www.jair.org/ for any accompanying file

Unifying cycles

Two-literal clauses of the form L leftarrow R occur quite frequently in logic programs, deductive databases, and - disguised as an equation - in term rewriting systems. These clauses define a cycle if the atoms L and R are weakly unifiable, i.e., if L unifies with a new variant of R. The obvious problem with cycles is to control the number of iterations through the cycle. In this paper we consider the cycle unification problem of unifying two literals G and F modulo a cycle. We review the state of the art of cycle unification and give new results for a special type of cycles called unifying cycles, i.e., cycles L leftarrow R for which there exists a substitution sigma such that sigmaL = sigmaR. Altogether, these results show how the deductive process can be efficiently controlled for special classes of cycles without losing completeness

Proceedings of Sixth International Workshop on Unification

Swiss National Science Foundation; Austrian Federal Ministry of Science and Research; Deutsche Forschungsgemeinschaft (SFB 314); Christ Church, Oxford; Oxford University Computing Laborator