Approximate Computation via Le Cam Simulability

We propose a decision-theoretic framework for computational complexity, complementary to classical theory: moving from syntactic exactness (Turing / Shannon) to semantic simulability (Le Cam). While classical theory classifies problems by the cost of exact solution, modern computation often seeks only decision-valid approximations. We introduce a framework where "computation" is viewed as the efficient simulation of a target statistical experiment within a bounded risk distortion (Le Cam deficiency). We formally define computational deficiency ($δ_{\text{poly}}$) and use it to construct the complexity class LeCam-P (Decision-Robust Polynomial Time), characterizing problems that may be syntactically hard but semantically easy to approximate. We show that classical Karp reductions can be viewed as zero-deficiency simulations, and that approximate reductions correspond to bounded deficiency. Furthermore, we establish the No-Free-Transfer Inequality, showing that strictly invariant representations inevitably destroy decision-relevant information. This framework offers a statistical perspective on approximation theory, bridging the gap between algorithmic complexity and decision theory.

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