Language Learning from Positive Data and Negative Counterexamples

In this paper we introduce a paradigm for learning in the limit of potentially infinite languages from all positive data and negative counterexamples provided in response to the conjectures made by the learner. Several variants of this paradigm are considered that reflect different conditions/constraints on the type and size of negative counterexamples and on the time for obtaining them. In particular, we consider the models where 1) a learner gets the least negative counterexample; 2) the size of a negative counterexample must be bounded by the size of the positive data seen so far; 3) a counterexample may be delayed. Learning power, limitations of these models, relationships between them, as well as their relationships with classical paradigms for learning languages in the limit (without negative counterexamples) are explored. Several surprising results are obtained. In particular, for Gold’s model of learning requiring a learner to syntactically stabilize on correct conjectures, learners getting negative counterexamples immediately turn out to be as powerful as the ones that do not get them for indefinitely (but finitely) long time (or are only told that their latest conjecture is not a subset of the target language, without any specific negative counterexample). Another result shows that for behaviourally correct learning (where semantic convergence is required from a learner) with negative counterexamples, a learner making just one error in almost all its conjectures has the “ultimate power”: it can learn the class of all recursively enumerable languages. Yet another result demonstrates that sometimes positive data and negative counterexamples provided by a teacher are not enough to compensate for full positive and negative data

A Theory of Formal Synthesis via Inductive Learning

Formal synthesis is the process of generating a program satisfying a high-level formal specification. In recent times, effective formal synthesis methods have been proposed based on the use of inductive learning. We refer to this class of methods that learn programs from examples as formal inductive synthesis. In this paper, we present a theoretical framework for formal inductive synthesis. We discuss how formal inductive synthesis differs from traditional machine learning. We then describe oracle-guided inductive synthesis (OGIS), a framework that captures a family of synthesizers that operate by iteratively querying an oracle. An instance of OGIS that has had much practical impact is counterexample-guided inductive synthesis (CEGIS). We present a theoretical characterization of CEGIS for learning any program that computes a recursive language. In particular, we analyze the relative power of CEGIS variants where the types of counterexamples generated by the oracle varies. We also consider the impact of bounded versus unbounded memory available to the learning algorithm. In the special case where the universe of candidate programs is finite, we relate the speed of convergence to the notion of teaching dimension studied in machine learning theory. Altogether, the results of the paper take a first step towards a theoretical foundation for the emerging field of formal inductive synthesis

Are There Good Mistakes? A Theoretical Analysis of CEGIS

Counterexample-guided inductive synthesis CEGIS is used to synthesize programs from a candidate space of programs. The technique is guaranteed to terminate and synthesize the correct program if the space of candidate programs is finite. But the technique may or may not terminate with the correct program if the candidate space of programs is infinite. In this paper, we perform a theoretical analysis of counterexample-guided inductive synthesis technique. We investigate whether the set of candidate spaces for which the correct program can be synthesized using CEGIS depends on the counterexamples used in inductive synthesis, that is, whether there are good mistakes which would increase the synthesis power. We investigate whether the use of minimal counterexamples instead of arbitrary counterexamples expands the set of candidate spaces of programs for which inductive synthesis can successfully synthesize a correct program. We consider two kinds of counterexamples: minimal counterexamples and history bounded counterexamples. The history bounded counterexample used in any iteration of CEGIS is bounded by the examples used in previous iterations of inductive synthesis. We examine the relative change in power of inductive synthesis in both cases. We show that the synthesis technique using minimal counterexamples MinCEGIS has the same synthesis power as CEGIS but the synthesis technique using history bounded counterexamples HCEGIS has different power than that of CEGIS, but none dominates the other.Comment: In Proceedings SYNT 2014, arXiv:1407.493

Learning-Based Synthesis of Safety Controllers

We propose a machine learning framework to synthesize reactive controllers for systems whose interactions with their adversarial environment are modeled by infinite-duration, two-player games over (potentially) infinite graphs. Our framework targets safety games with infinitely many vertices, but it is also applicable to safety games over finite graphs whose size is too prohibitive for conventional synthesis techniques. The learning takes place in a feedback loop between a teacher component, which can reason symbolically about the safety game, and a learning algorithm, which successively learns an overapproximation of the winning region from various kinds of examples provided by the teacher. We develop a novel decision tree learning algorithm for this setting and show that our algorithm is guaranteed to converge to a reactive safety controller if a suitable overapproximation of the winning region can be expressed as a decision tree. Finally, we empirically compare the performance of a prototype implementation to existing approaches, which are based on constraint solving and automata learning, respectively

Learning Residual Finite-State Automata Using Observation Tables

We define a two-step learner for RFSAs based on an observation table by using an algorithm for minimal DFAs to build a table for the reversal of the language in question and showing that we can derive the minimal RFSA from it after some simple modifications. We compare the algorithm to two other table-based ones of which one (by Bollig et al. 2009) infers a RFSA directly, and the other is another two-step learner proposed by the author. We focus on the criterion of query complexity.Comment: In Proceedings DCFS 2010, arXiv:1008.127

Towards unsupervised ontology learning from data

Data-driven elicitation of ontologies from structured data is a well-recognized knowledge acquisition bottleneck. The development of efficient techniques for (semi-)automating this task is therefore practically vital - yet, hindered by the lack of robust theoretical foundations. In this paper, we study the problem of learning Description Logic TBoxes from interpretations, which naturally translates to the task of ontology learning from data.In the presented framework, the learner is provided with a set of positive interpretations (i.e., logical models) of the TBox adopted by the teacher. The goal is to correctly identify the TBox given this input. We characterize the key constraints on the models that warrant finite learnability of TBoxes expressed in selected fragments of the Description Logic ε λ and define corresponding learning algorithms.This work was funded in part by the National Research Foundation under Grant no. 85482