automated theorem provers

Abstract: Reinforcement learning with verifiable rewards has substantially improved mathematical reasoning. However, terminal correctness alone provides limited insight into the quality of high-level strategies, such as theorem selection and subgoal decomposition, when considered separately from their subsequent execution. This paper studies strategy utility, which is defined as the likelihood that a strategy supports a correct downstream solution under a given executor. We introduce SURE, a framework for learning and leveraging relative strategy utility. In this framework, high-level strategies are separated from their detailed reasoning. Based on the pairwise preferences constructed from strategy-conditioned rollouts and teacher-generated contrasts, a Strategy Reward Model is learned to estimate relative strategy utility. During reinforcement learning, the frozen reward model reads only the extracted strategy, whose score is combined with the correctness and format rewards in a sequence-level GRPO objective. Compared with outcome-and-format GRPO baselines, experiments show that SURE improves average pass@1 by 1.87%, 2.64%, and 2.93% across three policy backbones. Our method also achieves competitive or better accuracy than stronger reward baselines while requiring substantially lower GRPO-stage compute.

Abstract: The satisfiability problem for word equations asks whether variables can be replaced by words so that the two sides become equal. For regular word equations, in which each variable occurs at most once on each side, satisfiability is NP-complete. For general quadratic word equations, in which each variable occurs at most twice in total, satisfiability is NP-hard, but its membership in NP remains open. We consider an intermediate class: quadratic word equations with a linear side, where each variable occurs at most once on one designated side. We show that the Nielsen graph of an equation $U=V$ in this class, with total length $N=|U|+|V|$, has diameter $O(N^{12})$, measured over reachable pairs of vertices. Together with the known NP-hardness for regular word equations, this result establishes NP-completeness of satisfiability for this class.

Abstract: We investigate the proof-theoretic structure of incorrect/incomplete proofs, that is, derivations containing syntactic errors or incomplete inferential steps that nonetheless preserve partial semantic validity. Building on Hilbert's epsilon calculus, we formalize how such derivations can be corrected through semantic projection and weakest preconditions, leading to valid Herbrand disjunctions. We show that the epsilon calculus provides a natural framework for analyzing tolerance of falsity in proofs and for identifying conditions under which an incorrect proof can be semantically repaired. This approach extends Hilbert's program beyond correctness, toward a logic of error and recovery. Moreover we show that the extended first epsilon theorem is false-tolerant.