Configuration Identity as a Formal Parameter in Deterministic State Counting

Standard models of deterministic computation count configurations as distinct states whenever their concrete encodings differ. Although pervasive, this practice is rarely isolated as an explicit formal choice. In particular, arguments about reachable state-space size, traversal cost, and related complexity bounds are typically formulated relative to a fixed but tacit notion of configuration identity.

This paper isolates configuration identity as a formal parameter in deterministic state counting. We study deterministic transition systems equipped with an equivalence relation on configurations and a canonical representative map, and use this structure to define computation over canonical states. In this setting, quotient computation is used as a precise formal device for making the operative identity criterion explicit and operational.

Three facts are established. First, under acceptance-preserving and successor-compatible equivalence, quotient dynamics are well-defined and deterministic. Second, finite path projection and lifting preserve acceptance behavior between concrete and canonical computation. Third, if successor generation and canonicalization are effectively computable and the reachable canonical state space is finite, respectively polynomially bounded on an input family, then the resulting canonical traversal is finite, respectively polynomial-time.

The result provides a general formal framework for model-relative state counting. It applies to deterministic systems equipped with an admissible equivalence and an effective canonical representative map. Problem-specific upper bounds require separate proofs that these structures exist, are computationally accessible, and yield a suitably bounded reachable canonical state space.