The subsumption lattice and query learning

Abstract

The paper identifies several new properties of the lattice induced by the subsumption relation over first-order clauses and derives implications of these for learnability. In particular, it is shown that the length of subsumption chains of function free clauses with bounded size can be exponential in the size. This suggests that simple algorithmic approaches that rely on repeating minimal subsumption-based refinements may require a long time to converge. It is also shown that with bounded size clauses the subsumption lattice has a large branching factor. This is used to show that the class of first-order length-bounded monotone clauses is not properly learnable from membership queries alone. Finally, the paper studies pairing, a generalization operation that takes two clauses and returns a number of possible generalizations. It is shown that there are clauses with an exponential number of pairing results which are not related to each other by subsumption. This is used to show that recent pairing-based algorithms can make exponentially many queries on some learning problems.

Anti-unification in Constraint Logic Programming

“…The approach presented in this paper could easily be extended to take this property into account, but for the sake of clarity we will keep the approach purely syntactic on that point of view, only considering non-commutative symbols in textual representations of constraints. Despite their differences, our generalization relation shares the following property with the usual θ -subsumption order from (Khardon and Arias 2006).…”

Automated code generation for programmable logic controllers based on knowledge acquisition from engineering artifacts: Concept and case study

“…According to Khardon and Arias [21] and Ligeza [23], the rule r j generally subsumes r s , denoted by r j r s , if there exists a substitution ς such that Ant(r j )ς ⊆ Ant(r s ) and Con(r s ) ⊆ Con(r j )ς. In this case the set of inputs x j is a subset of x s , and y s ⊆ y j .…”