Reasoning about “reasoning about reasoning”: semantics and contextual equivalence for probabilistic programs with nested queries and recursion

Metareasoning can be achieved in probabilistic programming languages (PPLs) using agent models that recursively nest inference queries inside queries. However, the semantics of this powerful, reflection-like language feature has defied an operational treatment, much less reasoning principles for contextual equivalence. We give formal to a core PPL with continuous distributions, scoring, general recursion, and nested Unlike prior work, presence recursion makes it impossible stratify definition sampling-based measure-theoretic semantics—the two must defined mutually recursively. A key yet challenging property we establish is programs have well-defined meanings: limits exist step-indexed measures they induce. Beyond semantics, offer relational making construct step-indexed, biorthogonal logical-relations model. soundness theorem establishes logical relatedness implies demonstrate usefulness by proving novel equivalences practical relevance—in particular, game-playing decisionmaking agents. mechanize our technical developments leading proof Coq assistant. Nested are important theoretically underdeveloped linguistic PPLs; first them provide sound