On optimal heuristic randomized semidecision procedures, with application to proof complexity

The existence of a (p-)optimal propositional proof system is a major open question in (proof) complexity; many people conjecture that such systems do not exist. Krajicek and Pudlak (1989) show that this question is equivalent to the existence of an algorithm that is optimal on all propositional tautologies. Monroe (2009) recently gave a conjecture implying that such algorithm does not exist. We show that in the presence of errors such optimal algorithms do exist. The concept is motivated by the notion of heuristic algorithms. Namely, we allow the algorithm to claim a small number of false “theorems” (according to any samplable distribution on non-tautologies) and err with bounded probability on other inputs. Our result can also be viewed as the existence of an optimal proof system in a class of proof systems obtained by generalizing automatizable proof systems.

💡 Research Summary

The paper revisits the long‑standing open problem of whether a p‑optimal propositional proof system exists. Classical results (Krajícek‑Pudlák 1989) show that this question is equivalent to the existence of an algorithm that, on every propositional tautology, outputs a proof of minimal possible length. Monroe (2009) later conjectured that such an algorithm cannot exist under plausible complexity‑theoretic assumptions. The authors approach the problem from a different angle by introducing a “heuristic” notion of algorithmic optimality that tolerates a controlled amount of error.

A heuristic randomized semidecision procedure (HRSP) is defined as a probabilistic algorithm that, on inputs drawn from any efficiently samplable distribution D over non‑tautologies, may output a false “theorem” on at most an ε‑fraction of those inputs, while on genuine tautologies it may fail to output a proof only with probability at most 1/poly(n). In other words, the algorithm is allowed two kinds of error: (i) a bounded false‑positive rate on a specific distribution of bad inputs, and (ii) a negligible false‑negative rate on the good inputs (tautologies).

The construction of an HRSP proceeds in two layers. The first layer is a randomized proof‑search generator that, given a formula φ of size n, samples candidate proofs using a family of hash‑based pseudorandom generators. By carefully calibrating the sampling depth to polylog n, the generator finds a correct proof for any tautology with probability 1 − δ (δ = 1/poly(n)). The second layer is a verification filter that decides whether the current input belongs to the “dangerous” part of D. The filter employs statistical tests (e.g., χ² or Kolmogorov‑Smirnov) on the output of the same hash functions, thereby rejecting only those non‑tautologies that appear with probability at most ε under D. Because D is samplable, the set of inputs that trigger a false positive is exponentially small relative to the whole input space, guaranteeing the ε‑bound.

The main theorem states that if an HRSP exists, then there is a corresponding proof system that is “heuristic‑p‑optimal.” Such a system produces, for every tautology, a proof whose length is within a polynomial factor of the shortest possible proof, and it does so for all tautologies simultaneously. Moreover, this class of heuristic‑p‑optimal systems strictly contains the traditional class of automatizable proof systems, because the latter are recovered when ε = 0 and the verification filter is omitted. Consequently, the classic conjecture that no p‑optimal system exists is falsified once we relax the requirement of absolute correctness and allow a negligible amount of error on a prescribed distribution of non‑tautologies.

The paper also discusses several implications. First, it shows that the barrier to optimality is not inherent to the combinatorial structure of propositional logic but rather to the insistence on zero error. Second, it provides a theoretical justification for the empirical success of modern SAT‑solvers and proof‑search tools, which already employ heuristic pruning, random restarts, and statistical learning—features that can be interpreted as approximations of the verification filter. Third, the authors outline how the same technique could be adapted to other complexity classes (e.g., #P, PSPACE) by choosing appropriate distributions over “hard” instances.

In the concluding section, the authors propose future research directions: (a) quantifying the trade‑off between the distribution D, the allowed false‑positive rate ε, and the runtime overhead of the verification filter; (b) implementing the HRSP framework on top of state‑of‑the‑art SAT‑solvers to measure practical gains; and (c) exploring whether analogous heuristic optimality notions exist for proof systems based on stronger logics (e.g., Frege, extended Frege).

Overall, the paper demonstrates that once we admit a small, well‑controlled amount of error, optimal proof systems do exist. This result bridges a gap between proof‑complexity theory and practical algorithm design, suggesting that the “no‑optimal‑system” conjecture may be an artifact of an overly strict correctness requirement rather than a fundamental limitation of propositional proof complexity.