Learning formulas in finite variable logics

Abstract

We consider grammar-restricted exact learning of formulas and terms in finite variable logics. We propose a novel and versatile automata-theoretic technique for solving such problems. We first show results for learning formulas that classify a set of positively- and negatively-labeled structures. We give algorithms for realizability and synthesis of such formulas along with upper and lower bounds. We also establish positive results using our technique for other logics and variants of the learning problem, including first-order logic with least fixed point definitions, higher-order logics, and synthesis of queries and terms with recursively-defined functions.

Languages with Decidable Learning: A Meta-theorem

“…We advocate an automata-theoretic technique for learning problems of this kind, inspired by recent work on learning formulas in finite-variable logics [Krogmeier and Madhusudan 2022], which is summarized as follows:…”

“…Expression Learning and Program Synthesis. Our approach is inspired by recent results for learning in finite-variable logics [Krogmeier and Madhusudan 2022]. Proofs in that work involve direct automata constructions, and the results can be obtained with our meta-theorem by writing suitable evaluators.…”

“…Our work generalizes the tree automata approach to general symbolic languages by separating decidable learning theorems into two parts: (1) identifying the underlying semantic aspects of the language in question and (2) programming with this new datatype in order to evaluate arbitrary expressions. The finite-variable restriction for the logics considered in [Krogmeier and Madhusudan 2022] leads to finitely-many aspects-a finite set of assignments to some 𝑘 variables. But, as our work shows, this restriction is not necessary for decidable learning; several languages we consider do not use variables and it is unclear what a corresponding variable restriction would mean.…”

“…Consequently, the result on decidable learning for finite-variable FO is an immediate corollary of our meta-theorem. In fact, all such results for finite-variable logics [Krogmeier and Madhusudan 2022] are obtained as corollaries 1 .…”

“…Learning in Finite-Variable Logics. Our work draws inspiration from a recent result that showed classical logics, e.g., first-order logic (FO), have decidable learning when restricted to use finitelymany variables [Krogmeier and Madhusudan 2022]. The technique underlying this result uses tree automata.…”