Soft State Reduction in Fuzzy & Nonlinear Systems

• Soft state reduction is an approximation method that merges nearly equivalent states based on user-defined thresholds, reducing complex automata and dynamic systems.
• It employs truncated lattice operations and approximate invariances to balance accuracy and computational efficiency in fuzzy finite automata.
• This approach is applicable to fuzzy, nondeterministic, and nonlinear systems, providing practical state-space reductions with quantifiable trade-offs.

Soft state reduction refers to a class of approximate methodologies for minimizing the number of states in automata, dynamic systems, and related models while retaining an acceptable approximation of the original behavior. This concept has been developed primarily for fuzzy finite automata (FfAs) over residuated lattices, where exact minimization is computationally intractable or conceptually too stringent. Soft state reduction generalizes classical notions by introducing tunable thresholds and word length bounds, enabling practical reductions and direct applicability to nondeterministic, fuzzy, or nonlinear systems.

1. Formal Definition and Mathematical Foundations

At its core, soft state reduction is an approximation process that merges states which are "close enough" in their behavior according to user-specified tolerances. In the context of FfAs over residuated lattices $\mathcal L = (L,\leq,\land,\lor,\otimes,\to,0,1)$, a soft reduction relies on a threshold $\varepsilon>0$ and—optionally—a maximum word length $k$. State pairs whose behavioral difference is bounded by $\varepsilon$ (and matches for all words of length up to $k$) are merged. This approach utilizes truncated lattice operations $$(\leq_\varepsilon,\,\otimes_\varepsilon,\,\to_\varepsilon)$$, where all values below $\varepsilon$ are considered negligible and rounded up.

For fuzzy finite automata $A=(Q,\Sigma,I,\delta,F)$, the $\varepsilon$-approximate language recognized is defined as

$L_{A,\varepsilon}(w) = I \circ \delta_w \circ F$

where composition is performed with truncated operations and the comparison is up to length $k$ if specified. The goal is to produce a reduced automaton $A_{Z,\varepsilon}$ such that

$L_{A_{Z,\varepsilon}}(w) \!=_\varepsilon\! L_A(w)$

for all $w$ (or for all $|w|\leq k$), where $=_\varepsilon$ denotes equivalence modulo the threshold $\varepsilon$ (Nguyen et al., 7 Dec 2025).

2. Methodologies for Soft State Reduction in Fuzzy Automata

Several algorithmic schemes have been developed for soft state reduction:

2.1 Approximate Invariances

Key to soft reduction is the construction of approximate invariances—fuzzy relations $Z$ that allow merging "almost equivalent" states. For right $\varepsilon$-invariance, the principal requirement is

$$Z \circ F_w =_\varepsilon F_w, \quad \forall w \in \Sigma^*$$

and for right $(\varepsilon,k)$-invariance, the condition is only for $|w|\leq k$. The greatest right invariance is constructed as

$$Z^* = \bigwedge_{w \in \Sigma^*} (F_w =_\varepsilon F_w)$$

There is a dual construction for left invariance by reversing the automaton.

2.2 Reduction Algorithm

Given $(\varepsilon, k)$, soft state reduction proceeds via:

1. Removal of unreachable/unproductive states.
2. Iterative computation of truncated language profiles $(F_w)_\varepsilon$ for $|w|\leq k$ (or possibly until no new profiles appear).
3. Formation of the largest merging relation $Z$ consistent with the invariance conditions.
4. Construction of the quotient automaton $A_{Z,\varepsilon}$.

A two-pass algorithm may apply the reduction from both directions (right, then left) and return the smaller automaton (Nguyen et al., 7 Dec 2025). Termination is ensured in practice if either $k<\infty$ or the truncated lattice becomes locally finite.

2.3 Trade-offs

The $(\varepsilon, k)$ parameters directly control reduction aggressiveness:

• Lower $\varepsilon$ yields finer behavioral distinctions, less reduction.
• Shorter $k$ provides coarser equivalence, allowing more merging. Empirical case studies demonstrate substantial state-space shrinkage (e.g., 28-state FfA reduced to 19 for $\varepsilon = 0.01$) where exact methods fail (Nguyen et al., 7 Dec 2025).

3. ℓ-Approximate State Reduction for Fuzzy Automata

A related approach, termed "ℓ-approximate" state reduction (Ćirić et al., 2023), emphasizes matching automata behaviors on all input words up to a fixed length $k$ rather than over the entire language. Four constructions using fuzzy quasi-orders enable $k$-equivalent reductions:

• Method I: Iterative right-invariant sequence.
• Method II: Iterative left-invariant sequence.
• Method III & IV: Weak invariance via all behaviors up to length $k$.

Each method employs residual operations over residuated lattices, using the row-automaton construction to define the reduced automaton. The trade-off is explicit: smaller $k$ means greater reduction but only matching for short words, while increasing $k$ approaches full equivalence (Ćirić et al., 2023).

4. Soft Reduction of Nondeterministic and Structured Automata

Soft state reduction is not limited to fuzzy automata. Heizmann et al. (Heizmann et al., 2017) demonstrate "soft" minimization for nondeterministic weakly-hierarchical visibly pushdown automata (VPA) using reachability-aware quotient (RAQ) equivalence. Their methodology merges states that are provably equivalent in their stack-based behaviors over reachable configurations. The process encodes reduction as a partial Max-SAT problem, balancing language preservation (hard constraints) with maximal merge (soft objectives).

This produces locally optimal reductions scaling to automata with tens of thousands of states, evidenced by improved verification tool performance (Ultimate Automizer) and 10–30% typical state count reductions (Heizmann et al., 2017).

5. Soft State Reduction in Model Order Reduction for Nonlinear Systems

Within soft robotics, dimensionality reduction—sometimes referred to as soft state reduction—has been achieved using Proper Orthogonal Decomposition (POD) and Koopman operator techniques. For hybrid soft-rigid robots (Alkayas et al., 21 May 2024), POD extracts low-dimensional strain modes ("soft synergies") from high-order physical models. The resulting reduced-order model (ROM) preserves dynamic and static behaviors with significant computational speed-ups, validated experimentally at sub-cm accuracy.

A data-driven Koopman-based approach uses Hankel DMD to embed nonlinear, infinite-dimensional soft robotic dynamics in lifted linear coordinates. Top Koopman modes are selected by an energy criterion, and projection onto this reduced basis permits LQR control and real-time execution with up to $10\times$ computational savings (Haggerty et al., 2020).

6. Applications and Practical Considerations

Soft state reduction methods have seen deployment in:

• Formal verification with fuzzy or nondeterministic transitions (Nguyen et al., 7 Dec 2025, Heizmann et al., 2017)
• Pattern recognition and grammatical inference under uncertainty (Nguyen et al., 7 Dec 2025)
• Large-scale automata-based software verification (Heizmann et al., 2017)
• Real-time simulation and control of soft/hybrid robotic systems (Alkayas et al., 21 May 2024, Haggerty et al., 2020)
• Approximate verification in hybrid or cyber-physical systems (Nguyen et al., 7 Dec 2025)

Advantages include handling of non-locally finite structures, flexible accuracy-performance trade-offs, and guaranteed termination under mild conditions.

7. Limitations and Tunable Parameters

Critical limitations and tuning aspects include:

• Excessively large $\varepsilon$ values can induce over-collapse, yielding unsatisfactory equivalence.
• Small $\varepsilon$ (or large $k$) approaches exactness but with little reduction.
• Computational complexity can grow exponentially with alphabet size or word length bounds ($s^k$ or $l^n$ for local finiteness).
• The success of soft reduction depends strongly on the underlying lattice structure—Gödel, Łukasiewicz, and nilpotent are tractable; product and Hamacher require nonzero $\varepsilon$ for local finiteness (Nguyen et al., 7 Dec 2025).

Future work involves adaptive selection of reduction parameters, generalization to non-visibly pushdown and weighted models, and integration with model checking or grammatical inference pipelines (Nguyen et al., 7 Dec 2025, Heizmann et al., 2017).