Deterministic Suffix-reading Automata

TL;DR

This paper introduces deterministic suffix-reading automata (DSA), a new automaton model that recognizes regular languages more concisely by jumping along blocks of letters, and explores the complexity of minimizing such automata.

Contribution

The paper formally defines DSAs, compares their expressiveness with DFAs, and proves that finding minimal DSAs is NP-complete, highlighting fundamental challenges in automata minimization.

Findings

DSAs can recognize regular languages more concisely than DFAs.

The smallest DSA derived from a DFA may not be minimal.

Deciding the existence of a DSA with bounded total-size is NP-complete.

Abstract

We introduce deterministic suffix-reading automata (DSA), a new automaton model over finite words. Transitions in a DSA are labeled with words. From a state, a DSA triggers an outgoing transition on seeing a word ending with the transition's label. Therefore, rather than moving along an input word letter by letter, a DSA can jump along blocks of letters, with each block ending in a suitable suffix. This feature allows DSAs to recognize regular languages more concisely, compared to DFAs. In this work, we focus on questions around finding a "minimal" DSA for a regular language. The number of states is not a faithful measure of the size of a DSA, since the transition-labels contain strings of arbitrary length. Hence, we consider total-size (number of states + number of edges + total length of transition-labels) as the size measure of DSAs. We start by formally defining the model and providing a DSA-to-DFA conversion that allows to compare the expressiveness and succinctness of DSA with related automata models. Our main technical contribution is a method to derive DSAs from a given DFA: a DFA-to-DSA conversion. We make a surprising observation that the smallest DSA derived from the canonical DFA of a regular language L need not be a minimal DSA for L. This observation leads to a fundamental bottleneck in deriving a minimal DSA for a regular language. In fact, we prove that given a DFA and a number k, the problem of deciding if there exists an equivalent DSA of total-size at most k is NP-complete.