We formulate an alternation-indexed family of robust monitor-synthesis problems for hierarchical double-bind computation. The base case is the robust escape synthesis problem with quantifier prefix p\, w\, c, which is complete at level ⁰₃ when acceptance is expressed by witnessed certificates. We define, for each integer k 1, a family RESₖ in which strategic parameter choices and adversarial input choices alternate for exactly k alternations, with acceptance witnessed by an explicit certificate. We prove a parametric completeness theorem: if k is odd, then RESₖ is ⁰₊+₂-complete; if k is even, then RESₖ is ⁰₊+₂-complete, all under computable many-one reductions. The proof is encoding-complete: every matrix predicate is decidable because certificates contain explicit halting-time bounds for every monitor evaluation. The theorem yields a calculus: each additional strategic-environment alternation raises the arithmetical-hierarchy level by exactly one, and the polarity is determined by which player moves first. We extend an alternation-based arithmetical-hierarchy classification for robust monitor synthesis by removing the special assumption that base acceptance is a ⁰₁ predicate witnessed by a single existential certificate. First, we prove a parameterized compositional law with an exact block-merger correction: if the base acceptance predicate lies in ⁰ₐ then k alternations between existential strategic parameters and universal environment inputs yield a synthesis problem in ⁰₊+₀+₁ (odd k) or ⁰₊+₀+₁ (even k) ;if the base acceptance predicate lies in ⁰ₐ the corresponding levels are ⁰₊+₀ (odd k) or ⁰₊+₀ (even k). Second, we prove matching completeness under a uniform base universality hypothesis that captures the required uniformity for tight lower bounds. Third, we stress-test the certificate paradigm under divergence-permissive monitor semantics. We give a new two-sided certificate system (run certificates versus refutation certificates) with decidable verification and prove that the base acceptance predicate becomes ⁰₂-complete, forcing an irreducible one-level upward shift that propagates additively to the full alternation-indexed hierarchy.
Kevin Fathi (Fri,) studied this question.