Membership Testing for Semantic Regular Expressions

TL;DR

This paper introduces an efficient NFA-based algorithm for membership testing of semantic regular expressions involving external oracles, with theoretical analysis, experimental validation, and insights into computational complexity and oracle query costs.

Contribution

It presents the first practical algorithm for SemRE membership testing, analyzes its complexity, and explores the connection to graph theory and oracle query minimization.

Findings

The algorithm runs in quadratic time relative to expression and string size.

Experimental results show the algorithm outperforms baseline methods.

A lower bound on oracle queries necessary for membership testing is established.

Abstract

SMORE (Chen et al., 2023) recently proposed the concept of semantic regular expressions that extend the classical formalism with a primitive to query external oracles such as databases and large language models (LLMs). Such patterns can be used to identify lines of text containing references to semantic concepts such as cities, celebrities, political entities, etc. The focus in their paper was on automatically synthesizing semantic regular expressions from positive and negative examples. In this paper, we study the membership testing problem: First, We present a two-pass NFA-based algorithm to determine whether a string $w$ matches a semantic regular expression (SemRE) $r$ in $O(|r|^2 |w|^2 + |r| |w|^3)$ time, assuming the oracle responds to each query in unit time. In common situations, where oracle queries are not nested, we show that this procedure runs in $O(|r|^2 |w|^2)$ time. Experiments with a prototype implementation of this algorithm validate our theoretical analysis, and show that the procedure massively outperforms a dynamic programming-based baseline, and incurs a $\approx 2 \times$ overhead over the time needed for interaction with the oracle. Next, We establish connections between SemRE membership testing and the triangle finding problem from graph theory, which suggest that developing algorithms which are simultaneously practical and asymptotically faster might be challenging. Furthermore, algorithms for classical regular expressions primarily aim to optimize their time and memory consumption. In contrast, an important consideration in our setting is to minimize the cost of invoking the oracle. We demonstrate an $\Omega(|w|^2)$ lower bound on the number of oracle queries necessary to make this determination.