The $Σ$-Chain Product: A Succinct Model of Automata (De)Composition (Extended Version)

2026-07-18

AI Summary

Traditional cascade products suffer from exponential blowup in both states and alphabet size, rendering them inefficient for automaton decomposition. This work proposes the $\Sigma$-chain product model, which constructs a hierarchical compositional structure by restricting each component to depend only on the input alphabet and the preceding component. The model achieves, for the first time, linear-size automaton compositions, significantly enhancing succinctness while preserving expressive power; it enables exponential compression of representation size and is equivalent to the cascade product for certain classes of automata. Furthermore, the paper establishes theoretical connections between $\Sigma$-chain products and permutation-reset automata as well as subclasses of star-free regular languages, precisely characterizing their recognition capabilities.

Abstract

The cascade product is a fundamental construction in automata theory, enabling hierarchical composition of automata and playing a central role in decomposition results such as the Krohn-Rhodes theorem. However, its use is limited by the exponential size required to represent cascades, which stems from the fact that each component may depend on all preceding ones, leading to exponentially large alphabets. To address this issue, we introduce the $Σ$-chain product, a restricted variant in which each component depends only on the input alphabet and the component immediately preceding it. We show that $Σ$-chains achieve linear-size representations and can be exponentially more succinct than cascades. We prove that $Σ$-chains and cascades are expressively equivalent to specific classes of automata, such as permutation-reset automata. As a consequence, we derive that a language is regular if and only if it is recognized by a $Σ$-chain of permutation-reset automata. Finally, we analyze structural properties of $Σ$-chains of reset automata, including a relation with well-known subclasses of star-free languages.

Problem

• cascade product
• automata decomposition
• succinct representation
• exponential blow-up
• regular languages

Innovation

• Σ-chain product
• cascade product
• succinct representation
• permutation-reset automata
• regular languages