Inductive logic programming as satisfiability modulo theories

Thesis

At the intersection of machine learning, program synthesis and automated reasoning lies the field of Inductive Logic Programming (ILP). The aim of ILP is to automatically learn relational programs from input/output examples, typically through logic-based techniques. Inspired by Karl Popper’s falsification perspective on science, this dissertation sets out a new approach to ILP: Learning From Failures (LFF).

In science, starting from a huge set of a priori viable hypotheses, we select a hypothesis to test. This hypothesis typically gets falsified due to failing in some specific way. By examining the failure we learn that an entire space of related hypotheses is now ruled out. Having thus reduced our set of viable hypotheses, we subsequently select from just those that remain.

LFF applies this methodology to program induction, codifying it as a three-stage loop. The generate stage maintains a formula whose satisfying assignments correspond to the set of viable hypotheses. The test stage takes a satisfying assignment, interprets it as a logic program and tests it against training examples – imperfect fit is considered a failure. The constrain stage turns a failure into constraints to add to the generate stage’s formula, thereby eliminating a class of hypotheses which will fail for the same reason.

The thesis of this dissertation is three-fold. The first claim is that LFF frames the ILP problem as one of Satisfiability Modulo Theories (SMT). With the search for viable hypotheses handed-off to a SAT-solver, the test stage can be regarded as a theory solver collaborating with the SAT-solver, effectively making ILP’s notion of background knowledge into a (Horn) background theory.

The second claim is that LFF’s three-stage loop is an effective basis for falsification-based program induction. Chapter 4 develops the above correspondence into a feature-rich and flexible three-stage ILP system. Experimental evidence is provided for this system going beyond the state-of-the-art in ILP, e.g., by supporting large hypothesis spaces and large domains.

The third claim is that the LFF-as-SMT-perspective helps apply satisfiability solving techniques to ILP, in particular to reduce hypothesis space exploration. In Chapter 5, we show that SMT-based techniques for explaining conflicts have a natural analog in terms of explaining which parts of a hypothesis are responsible for its failure. In Chapter 6, we incorporate other theory solvers alongside the test stage to reason about the (satisfiability of) over-approximating properties of hypotheses. We show that both of these techniques can significantly reduce the number of iterations of the three-stage loop.

Authors: Morel, R

• DOI: 10.5287/ora-vmnpzyw1x
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: constraint solving; inductive logic programming; satisfiability modulo theories; automated reasoning; program synthesis; machine learning
• Subjects: Programming by example (Computer science)