We prove that Moore machines can be exactly realized as state-space models (SSMs), establishing a formal correspondence between symbolic automata and these continuous machine learning architectures. These Moore-SSMs preserve both the complete symbolic structure and input-output behavior of the original Moore machine, but operate in Euclidean space. With this correspondence, we compare the training of SSMs with both passive and active automata learning. In recovering automata from the SYNTCOMP benchmark, we show that SSMs require orders of magnitude more data than symbolic methods and fail to learn state structure. This suggests that symbolic structure provides a strong inductive bias for learning these systems. We leverage this insight to combine the strengths of both automata learning and SSMs in order to learn complex systems efficiently. We learn an adaptive arbitration policy on a suite of arbiters from SYNTCOMP and show that initializing SSMs with symbolically-learned approximations learn both faster and better. We see 2-5 times faster convergence compared to randomly initialized models and better overall model accuracies on test data. Our work lifts automata learning out of purely discrete spaces, enabling principled exploitation of symbolic structure in continuous domains for efficiently learning in complex settings.
Figures & tables
Table 1
Figure 1: Warm-starting SSM using symbolic structure VS. learning using traditional random initialization on an dynamic round robin arbitration policy with 4 channels.
Figure 2: Highlevel training pipeline visualization for learning regular languages from SYNTCOMP
Figure 3: Methodology for generating assumption following traces by randomly picking transitions from states, generating assumption abiding traces
Figure 5: SSM architecture used to learn SYNTCOMP regular languages
Figure 6: Accuracy and sample complexity (Log Scale) for learning a random subset of 20 SYNTCOMP examples
Learning Method
Perfect Emulation
Gradient-based SSMs
33.3%
L ∗
77.3%
RPNI
56.0%
Table 2: Percentage of SYNTCOMP benchmarks solved with 100% accuracy by each learning method.
Figure 7: Training and Testing Accuracy on Predicting Next Step and Trace Acceptance for collector/parametric/collector_v2, N=3
Figure 8: PCA projection of SSM hidden states after processing 1,000 random length-5 traces, colored by ground-truth automaton state. Counts indicate traces terminating in each state.
A\leftarrow I+\mathcal{N}(0,\varepsilon),\text{ where A_{ij}=0}
Algorithm 1 Symbolic Initialization of an SSM from a Moore Machine
Figure 9: Learning and performance of SSMs learning dynamic arbitration policies over various arbiter settings. Warm-start VS. random initialization. N denotes parameterization
Appendix figures & tables4 assets
Supplementary material from the paper’s appendix.
Appendix
Benchmark
SSM Acc (%)
L* Acc (%)
RPNI Acc (%)
SSM Samples
L* Samples
RPNI Samples
01
26.9
100.0
100.0
8,100,000
6,346
5,000
02
7.6
100.0
100.0
8,100,000
6,440
5,000
03
2.3
99.0
0.0
8,100,000
79,764
5,000
EscalatorSmart
0.0
100.0
100.0
8,100,000
6,959
5,000
Gamemodule
0.2
100.0
100.0
8,100,000
4,721
5,000
KitchenTimerV1
100.0
100.0
100.0
4,500,000
1,878
5,000
Appendix
Table 3: Comparison of learning methods on SYNTCOMP benchmarks.
Figure 14: Adaptive arbiter results 1. Shown in main paper
Transformers and state-space models (SSMs) are two prominent sequential learning architectures, yet their comparison remains largely empirical and existing theoretical analyses are typically task-specific or architecturally restricted. In this paper, we develop belief geometry, a unified analytical framework for comparing the representational capabilities of broad classes of attention and SSMs. Starting from a generalized formulation of in-context linear regression and using cumulative Bayes regret as our measure, we abstract three capabilities required by many sequential learning problems in our belief geometry: evidence assembly, belief maintenance, and addressing. We then study three cases of our formulation that isolate these capabilities and yield sharp architectural lessons: For belief maintenance, SSMs attain the optimal regret over stationary aggregation kernels; for positional assembly, SSMs have a memory advantage; and for content addressing, softmax attention has an exponential width advantage over sigmoid-selective SSMs. Experiments with LLaMA-type Transformers and Mamba-2 show that these architectural insights extend beyond our analytically tractable classes and linear-regression testbed.
Enes Arda, Semih Cayci, Atilla Eryilmaz
The Ohio State University · RWTH Aachen University
Active automata learning (AAL) algorithms can learn a behavioral model of a system from interacting with it. The primary challenge remains scaling to larger models, in particular in the presence of many possible inputs to the system. Modern AAL algorithms fail to scale even if, in every state, most inputs lead to errors. In various challenging problems from the literature, these errors are observable, i.e., they emit a known error output. Motivated by these problems, we study learning these systems more efficiently. Further, we consider various degrees of knowledge about which inputs are non-error producing at which state. For each level of knowledge, we provide a matching adaptation of the state-of-the-art AAL algorithm L# to make the most of this domain knowledge. Our empirical evaluation demonstrates that the methods accelerate learning by orders of magnitude with strong but realistic domain knowledge to a single order of magnitude with limited domain knowledge.
Loes Kruger, Sebastian Junges, Jurriaan Rot
Institute for Computing and Information Sciences, Radboud University, Nijmegen, the Netherlands
State Space Models (SSMs) have emerged as a compelling alternative to Transformers, enabling sequence modeling with constant memory and linear compute. Although SSMs exhibit reasonable performance and favorable computational characteristics, they continue to lag behind Transformers on tasks that require in-context learning and precise retrieval, slowing their adoption for large-scale language modeling. In this work, we demonstrate that both the success and failure of SSMs in these domains can be explained by studying the role of the gating mechanism, a prevalent component in modern recurrent networks. Specifically, we show through theory and experiments that this gating mechanism causes SSMs to first learn an in-weights "memorization" solution, while delaying, or even preventing, convergence to a correct in-context learning solution. Importantly, this happens even in cases where there are no fundamental limitations due to the architecture or its memory capacity. On the other hand, we find that gating is often beneficial for improving generalization to long sequence lengths. Our results illuminate the crucial role of the gating mechanism in shaping both the training dynamics and generalization of SSMs, and provide a basis for understanding and improving linear-time models.