Automated Pattern Detection--An Algorithm for Constructing Optimally Synchronizing Multi-Regular Language Filters

TL;DR

This paper introduces two algorithms for multi-regular language filtering, one ideal but computationally intensive, and a second efficient finite-state approximation suitable for real-time applications.

Contribution

It presents a novel finite-state transducer construction that approximates an ideal stack-based filtering algorithm for multi-regular languages.

Findings

The transducer runs in linear time and requires finite memory.

It provides immediate output for each input symbol.

It is the optimal finite-state approximation of the ideal algorithm.

Abstract

In the computational-mechanics structural analysis of one-dimensional cellular automata the following automata-theoretic analogue of the \emph{change-point problem} from time series analysis arises: \emph{Given a string $\sigma$ and a collection $\{\mc{D}_i\}$ of finite automata, identify the regions of $\sigma$ that belong to each $\mc{D}_i$ and, in particular, the boundaries separating them.} We present two methods for solving this \emph{multi-regular language filtering problem}. The first, although providing the ideal solution, requires a stack, has a worst-case compute time that grows quadratically in $\sigma$'s length and conditions its output at any point on arbitrarily long windows of future input. The second method is to algorithmically construct a transducer that approximates the first algorithm. In contrast to the stack-based algorithm, however, the transducer requires only a finite amount of memory, runs in linear time, and gives immediate output for each letter read; it is, moreover, the best possible finite-state approximation with these three features.