On Finding a First-Order Sentence Consistent with a Sample of Strings

TL;DR

This paper presents an algorithm to find the simplest first-order logical sentence consistent with a sample of classified strings, using Ehrenfeucht-Fra"issé game theory to distinguish strings.

Contribution

It introduces a novel method combining Ehrenfeucht-Fra"issé games with algorithmic techniques to identify minimal quantifier rank sentences consistent with string samples.

Findings

Algorithm effectively finds minimal quantifier rank sentences.

Uses Ehrenfeucht-Fra"issé game theory for string distinction.

Provides a boolean combination of distinguishing formulas.

Abstract

We investigate the following problem: given a sample of classified strings, find a first-order sentence of minimal quantifier rank that is consistent with the sample. We represent strings as successor string structures, that is, finite structures with unary predicates to denote symbols in an alphabet, and a successor relation. We use results of the Ehrenfeucht-Fra\"iss\'e game over successor string structures in order to design an algorithm to find such sentence. We use conditions characterizing the winning strategies for the Spoiler on successor strings structures in order to define formulas which distinguish two strings. Our algorithm returns a boolean combination of such formulas.