Soft state reduction of fuzzy automata over residuated lattices

TL;DR

This paper proposes a soft state reduction method for fuzzy automata over residuated lattices, enabling approximate minimization that balances accuracy and computational efficiency, especially for complex non-locally finite structures.

Contribution

It introduces an approximate reduction technique using fuzzy invariances and thresholds, addressing challenges in minimizing nondeterministic fuzzy automata over complex residuated lattices.

Findings

Effective reduction of fuzzy automata where existing methods fail

Balances reduction accuracy with computational feasibility

Applicable to automata over non-locally finite residuated lattices

Abstract

State reduction of finite automata plays a significant role in improving efficiency in formal verification, pattern recognition, and machine learning, where automata-based models are widely used. While deterministic automata have well-defined minimization procedures, reducing states in nondeterministic fuzzy finite automata (FfAs) remains challenging, especially for FfAs over non-locally finite residuated lattices like the product and Hamacher structures. This work introduces soft state reduction, an approximate method that leverages a small threshold $\varepsilon$ possibly combined with a word length bound $k$ to balance reduction accuracy and computational feasibility. By omitting fuzzy values smaller than $\varepsilon$, the underlying residuated lattice usually becomes locally finite, making computations more tractable. We introduce and study approximate invariances, which are fuzzy relations that allow merging of almost equivalent states of an FfA up to a tolerance level $\varepsilon$ and, optionally, to words of bounded length $k$. We further present an algorithm which iteratively applies such invariances to achieve reduction while preserving approximate language equivalence. Our method effectively reduces FfAs where existing techniques fail.