[Paper Review] Decidability of minimization of fuzzy automata

Lvzhou Li, Daowen Qiu|arXiv (Cornell University)|May 18, 2013

Formal Methods in Verification15 references3 citations

TL;DR

This paper proves the decidability of the minimization problem for fuzzy automata over totally ordered lattices by introducing a novel framework based on solving systems of fuzzy polynomial equations. The key contribution is a decision procedure that determines whether a given fuzzy automaton has an equivalent k-state automaton, establishing the problem as decidable and at least PSPACE-complete.

ABSTRACT

State minimization is a fundamental problem in automata theory. The problem is also of great importance in the study of fuzzy automata. However, most work in the literature considered only state reduction of fuzzy automata, whereas the state minimization problem is almost untouched for fuzzy automata. Thus in this paper we focus on the latter problem. Formally, the decision version of the minimization problem of fuzzy automata is as follows: \begin{itemize} \item Given a fuzzy automaton $\mathcal{A}$ and a natural number $k$, that is, a pair $\langle \mathcal{A}, k angle$, is there a $k$-state fuzzy automaton equivalent to $\mathcal{A}$? \end{itemize} We prove for the first time that the above problem is decidable for fuzzy automata over totally ordered lattices. To this end, we first give the concept of systems of fuzzy polynomial equations and then present a procedure to solve these systems. Afterwards, we apply the solvability of a system of fuzzy polynomial equations to the minimization problem mentioned above, obtaining the decidability. Finally, we point out that the above problem is at least as hard as PSAPCE-complete.

Motivation & Objective

• To address the long-standing open problem of whether state minimization is decidable for fuzzy automata.
• To formalize the decision version of the minimization problem: given a fuzzy automaton 𝒜 and integer k, does an equivalent k-state fuzzy automaton exist?
• To establish a procedure for deciding the existence of a k-state equivalent fuzzy automaton using solvable systems of fuzzy polynomial equations.
• To analyze the computational complexity of the minimization problem, showing it is at least as hard as PSPACE-complete.

Proposed method

• Define the decision problem as determining the non-emptiness of the set 𝕊(𝒜,k), which contains all possible k-state automata equivalent to 𝒜.
• Construct a system of fuzzy polynomial equations (25) encoding the equivalence constraints between the original automaton and candidate k-state automata.
• Use Algorithm I to check the solvability of the system of fuzzy polynomial equations, which determines whether a valid k-state automaton exists.
• Compute the right-hand side values of the equations via iterative composition of transition functions δ and output functions η up to length c = |V|^{n+k}−1.
• Evaluate the left-hand side for all possible state assignments X ∈ V^{2k+|Σ|k²} to test solvability of the system.
• Apply Theorem 4 to verify whether the system of equations has a solution, thereby deciding the existence of a k-state equivalent automaton.

Experimental results

Research questions

• RQ1 Is the minimization problem for fuzzy automata decidable over totally ordered lattices?
• RQ2 Can the existence of an equivalent k-state fuzzy automaton be algorithmically determined?
• RQ3 What is the computational complexity of the fuzzy automata minimization problem?
• RQ4 Can systems of fuzzy polynomial equations be effectively solved to enable minimization?

Key findings

• The minimization problem for fuzzy automata over totally ordered lattices is decidable, providing the first positive result on this long-open problem.
• A decision procedure is constructed based on the solvability of a system of fuzzy polynomial equations, which can determine the existence of a k-state equivalent automaton.
• The algorithm runs in time O(|V|^{2k+|Σ|k²} N k² + N n²), where N = 1 + |Σ| + ⋯ + |Σ|^{c} and c = |V|^{n+k}−1.
• The problem is at least as hard as PSPACE-complete, as shown by reduction from the NFA minimization problem, which is known to be PSPACE-complete.
• The proposed method generalizes the concept of state merging via indistinguishability to fuzzy automata using algebraic constraints encoded in fuzzy polynomial systems.