GFM State-Space Reduction Technique

• GFM state-space reduction techniques are methods that compress high-dimensional dynamic models into smaller, functionally equivalent representations, enhancing computational efficiency.
• They employ structured steps such as alphabet embedding, acceptance swapping, and POD/DEIM to achieve significant reductions, with empirical improvements of 2–4× in automata and 10²–10³× in FEM.
• These methods ensure preservation of essential system properties or controlled approximations, making them valuable for automata-based planning, nonlinear finite element discretizations, and symmetry-based reductions in mean field games.

The GFM state-space reduction technique encompasses a range of methods for reducing the computational state-space in dynamic models, particularly in the contexts of automata-based planning for Markov Decision Processes (MDPs), nonlinear finite element discretizations, and mean field games. At its core, GFM (Good-for-MDP or Group Finite Element Method, depending on the context) reduction targets the elimination of redundancy or non-essential structure in order to yield lower-dimensional, more tractable models while retaining essential system properties within specified tolerances.

1. Formal Definition and Motivation

The principal objective of GFM state-space reduction is to transform an automaton or dynamical system specification into a functionally equivalent but smaller representation, thereby improving scalability in synthesis, verification, and control. In stochastic LTL planning, a good-for-MDP (GFM) automaton is a nondeterministic B-automaton (NBA) whose nondeterminism is “angelic” relative to the MDP: every optimal strategy in the product MDP-automaton can resolve the automaton’s nondeterminism without overestimating satisfaction probabilities. The analogous reduction in nonlinear finite elements (GFM/EGFEM) involves the use of interpolation and projection-based model order reduction to compress the algebraic system produced by Galerkin discretization, and in mean field games it involves symmetry-based dimensionality reduction when the system data are invariant under a given transformation (Weinhuber et al., 12 Nov 2025, Tolle et al., 2021, Lasry et al., 2021).

2. GFM State-Space Reduction in Automata-Based LTL Planning

For LTL objectives in MDPs, the GFM state-space reduction pipeline, as formalized by Weinhuber et al. (Weinhuber et al., 12 Nov 2025), proceeds as follows:

1. Alphabet Embedding (GFM→DBA): The nondeterminism in a GFM NBA is resolved by embedding a choice index in the input alphabet, yielding a deterministic B-automaton.
2. Acceptance Swapping (DBA→DCA): The B-acceptance condition is reinterpreted into a deterministic co-B rejecting automaton.
3. GFG Minimisation: Polynomial-time Good-for-Games (GFG) minimisation is applied, typically yielding a much smaller state set.
4. Probabilistic Resolution (GFG→0/1-PA): Remaining nondeterminism is made uniform-random, yielding a 0/1-probabilistic B-automaton.

Each step preserves the acceptance language in the product with any MDP, and the final automaton maintains semantic and syntactic satisfaction probability equivalence for all finite MDPs, as formalized by the following statements:

$$P_{\rm sem}(M,A) = P_{\rm syn}(M,A) = P_{\rm syn}(M',A_{\rm DBA}) = P_{\rm sem}(M',A_{\rm DBA}) = P_{\rm syn}(M',P) = P_{\rm sem}(M',P)$$

This ensures that optimal policies synthesised on the reduced product yield optimal satisfaction probabilities in the original specification.

3. Complexity and Specialised Constructions

The worst-case complexity for the overall GFM reduction pipeline matches the classical doubly-exponential blow-up of LTL-to-automaton translation. However, each reduction stage is polynomial in the automaton size and often results in a dramatic reduction in practice. Of special significance is the single-exponential construction for formulae of the form $GF\,\varphi$ (co-safety), which proceeds by constructing a reset automaton based on the NFA for $\varphi$, producing a GFM NBA whose size is $2^{O(n)}$ for $n=|\varphi|$ (Weinhuber et al., 12 Nov 2025).

Scenario/Formula Class	Classical Size	GFM-Reduced Size
General LTL	$2^{2^{O(n)}}$	$2^{2^{O(n)}}$ (but smaller leading constants)
$GF\,\varphi$, co-safety	$2^{O(n)}$	$2^{O(n)}$ (single-exponential)

4. Empirical Results and Practical Impact

Empirical benchmarks on standard LTL pattern suites demonstrate typical GFM state-space reductions of 2–4× compared to prior GFM automaton constructions (e.g., “Slim” and “Owl” case studies):

• Example: TDR$G\,F\,(a\wedge X^8b)$" title="" rel="nofollow" data-turbo="false" class="assistant-link">8: Slim 257 $\rightarrow$ Slim-Red 130, Owl 256 $\rightarrow$ Owl-Red 256.
• LIB6: Slim 129 $\rightarrow$ 66, Owl 25 $\rightarrow$ 14.
• Specialised $GF\varphi$ construction can yield an order of magnitude smaller automata.

Product MDP–automata compositions and subsequent policy synthesis benefit proportionally, with synthesis runtimes decreasing with automaton size reduction (Weinhuber et al., 12 Nov 2025).

5. Model Order Reduction in Nonlinear Finite Elements

In nonlinear FEM, the group finite element method (GFM/EGFEM) uses interpolation onto auxiliary FE grids and a two-stage state-space reduction (Tolle et al., 2021):

1. Snapshot-based POD: Proper Orthogonal Decomposition (POD) is applied to solution trajectories, yielding a truncated reduced basis $V_u$ that captures the dominant modes.
2. Complexity Reduction via DEIM: The Discrete Empirical Interpolation Method (DEIM) further compresses the nonlinear term evaluation by projecting nonlinear vectors onto a small number of selected sample points.

The fully reduced system thus has dimension $n_u\ll N_u$ (where $N_u$ is the original number of FE degrees of freedom) and is solved at a cost proportional to $O(n_u^3+n_f n_u)$ per step, where $n_f\ll N_f$ is the number of DEIM interpolation points. Numerical experiments show end-to-end speed-ups of $10^2$–$10^3\times$ with prescribed accuracy (Tolle et al., 2021).

6. Symmetry-Based State Reductions in Mean Field Games

Dimension reduction in deterministic mean field games (MFG) is achieved whenever system Hamiltonian, cost, and coupling are invariant under a linear surjective projection (for instance, $S:\mathbb{R}^d\to\mathbb{R}^k$) (Lasry et al., 2021). Provided $H(x, p) = \tilde H(Sx, Sp)$ and all costs depend only on $Sx$ and the measure-pushforward under $S$, the forward–backward MFG PDEs reduce to their analogues in $\mathbb{R}^k$ with the ansatz $u(t, x) = \tilde u(t, Sx)$, $m(t, x) = \tilde m(t, Sx)$. This phenomenon is robust to small noise, by way of small-noise expansions that justify approximate reductions when the structural invariance is perturbed.

Reduction Type	Criterion	Target System
Exact	Data invariant under projection $S$	PDEs/Odes in $\mathbb{R}^k$
Approximate	Small perturbation of invariance	Perturbed reduced PDEs

In practical modelling, even approximate invariance can serve to inform construction of tractable surrogate MFGs.

7. Theoretical Guarantees and General Guidelines

Across the domains in which GFM state-space reduction is applied, a unifying property is the preservation of the essential quantitative semantics (e.g., satisfaction probability in automata–MDP products, leading-order error bounds in PDE reduction). The pipeline in automata-based planning preserves satisfaction probabilities by construction; the model reduction with POD/DEIM controls projection and interpolation errors via singular value truncation; in symmetry-based MFG reductions, invariant or nearly-invariant structure guarantees equivalence or controlled approximation.

Practical application of these schemes is contingent upon checking that the target problem instance possesses either the requisite structural invariance (MFG), non-adversarial nondeterminism (automata), or dominant low-dimensional modes (PDEs), and that the computational savings justify the reduction overhead.

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References:

• "Good-for-MDP State Reduction for Stochastic LTL Planning" (Weinhuber et al., 12 Nov 2025)
• "Nonlinear Reduction using the Extended Group Finite Element Method" (Tolle et al., 2021)
• "Dimension reduction techniques in deterministic mean field games" (Lasry et al., 2021)