Recursive Operator Machines: A New Algebraic Framework for Complexity Classes

Recursive Operator Machines (ROM) introduce a simple, modular way to describe computation using small “operators” that act on a bounded, layered state. A dependency graph (called Causality Matrix Alignment, CMA) enforces race-free ordering when multiple operators might interact. We also budget how many operators may fire at once, so each computational step is predictable and efficiently checkable. Why this matters. The famous P vs NP question asks whether problems whose solutions are easy to check are also easy to find. ROM gives a clean algebraic setting to restate and study that question. It sits alongside Turing machines and Boolean circuits but emphasizes composable actions and finite algebra, which lets us bring powerful tools from semigroup theory and descriptive logic to complexity classes. Main contributions Model: A fully specified, finite-state, operator model (ROM) with race-free step semantics, activation budgets, and cross-layer updates governed by CMA. Complexity landmarks: Deterministic ROMs capture polynomial-time simulation under standard resource bounds. With existential operator choices, a natural Reach-Closure problem is NP-complete; with bounded alternation, Σ_k^P/Π_k^P completeness follows (QBF encodings). In logical terms, deterministic ROM properties are FO(LFP) (≈ P) while existential schedule properties are ESO (≈ NP), giving an internal P/NP mirror. Algebraic lower bound: Using the transformation monoid generated by operators, we prove an aperiodic-monoid barrier: ROM fragments whose action monoids have no nontrivial groups cannot compute modular counting (e.g., MOD_p/parity). This recovers classic small-depth barriers within ROM. Resource hierarchies: We formalize and separate power by depth (CMA layers per step), operator count, state width, and modulus, providing sample strictness results and uniform, size-accounted encodings. Takeaway. ROM offers a compact, algebraic lens on computation where P/NP-style phenomena are visible as differences in fixed-point definability, operator alternation, and the group structure of the action monoid. The paper provides full proofs, explicit reductions, and uniform encodings to make the framework immediately testable. Keywords: computational complexity, P vs NP, descriptive complexity (FO(LFP), ESO), finite semigroups/monoids, circuit-style lower bounds, modular arithmetic, race-free semantics.