Soft state reduction of fuzzy automata over residuated lattices

TL;DR

The paper tackles state reduction for nondeterministic fuzzy finite automata over general linear and complete residuated lattices, including non-locally finite ones like product and Hamacher. It introduces soft state reduction, based on an epsilon-threshold approximation and optional word-length bounds, and formalizes approximate invariances to merge near-equivalent states, resulting in reduced automata that preserve approximate language behavior. A key contribution is the A_{Z,ε} construction and the associated theorems showing ε-(k)-equivalence under right invariance, along with a terminating algorithm SoftStateReduction that repeatedly applies right and left invariances. The approach enables practical reductions where existing methods fail, is supported by theoretical guarantees and complexity analysis, and is complemented by a publicly available Python implementation (SRFA-prog) with extensive experimental results across multiple 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.

Figures (9)

• Figure 1: An illustration of the fuzzy automaton $\mathpzc{A}$ discussed in the introduction section and used in Example \ref{['example: HEJSA']}.
• Figure 2: ReductionByRightInvariance($\mathpzc{A}, \varepsilon, k$)
• Figure 3: SoftStateReduction$_0$($\mathpzc{A}, \varepsilon, k$)
• Figure 4: An illustration of the fuzzy automaton $\mathpzc{A}$ used in Example \ref{['example: HDIAJ']}.
• Figure 5: An illustration of the fuzzy automaton $\mathpzc{A}_G$ mentioned in Example \ref{['example: JHJKS']}.

Theorems & Definitions (58)

• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Lemma 3.3
• proof
• Proposition 3.4
• proof
• Definition 4.1
• Remark 4.2