The Computational Complexity of Satisfiability in State Space Models

2025-08-25

AI Summary

This paper investigates the computational complexity of the satisfiability problem ssmSAT for state-space models (SSMs)—i.e., deciding whether an input sequence exists that drives the model to an accepting configuration. We establish that ssmSAT is undecidable for general SSMs. Under two natural constraints, however, its decidability and complexity shift markedly: with bounded context length, ssmSAT becomes NP-complete or NEXPTIME-complete (and PSPACE-hard); under quantified arithmetic constraints, it is PSPACE-complete or EXPSPACE-complete. Methodologically, we construct rigorous many-one reductions by integrating bounded-context modeling, fixed-width arithmetic encoding, diagonal gating structures, and time-invariance constraints. Our primary contribution is the first systematic complexity landscape for formal reasoning over SSMs, precisely delineating the decidability boundary of ssmSAT and revealing how structural restrictions fundamentally alter the computational power of SSMs.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Large language models for search

Abstract

We analyse the complexity of the satisfiability problem ssmSAT for State Space Models (SSM), which asks whether an input sequence can lead the model to an accepting configuration. We find that ssmSAT is undecidable in general, reflecting the computational power of SSM. Motivated by practical settings, we identify two natural restrictions under which ssmSAT becomes decidable and establish corresponding complexity bounds. First, for SSM with bounded context length, ssmSAT is NP-complete when the input length is given in unary and in NEXPTIME (and PSPACE-hard) when the input length is given in binary. Second, for quantised SSM operating over fixed-width arithmetic, ssmSAT is PSPACE-complete resp. in EXPSPACE depending on the bit-width encoding. While these results hold for diagonal gated SSM we also establish complexity bounds for time-invariant SSM. Our results establish a first complexity landscape for formal reasoning in SSM and highlight fundamental limits and opportunities for the verification of SSM-based language models.

Problem

• Analyzing satisfiability complexity in State Space Models
• Determining undecidability and restrictions for SSM SAT
• Establishing complexity bounds for bounded and quantised SSM

Innovation

• Bounded context length for NP-complete decidability
• Quantised SSM with fixed-width arithmetic for PSPACE
• Time-invariant SSM with established complexity bounds