Asuccessful autoformalization system could advance the fields of formal verification, program synthesis, and artificial intelligence. While the long-term goal of autoformalization seemed elusive for a long time, we show large language models provide new prospects towards this goal.

We develop a theoretical analysis of LLM-guided formal theorem proving in interactive proof assistants (e.g., Lean) by modeling tactic proposal as a stochastic policy in a finite-horizon deterministic MDP. To capture modern representation learning, we treat the state and action spaces as general compact metric spaces and assume Lipschitz policies. To explain the gap between worst-case hardness and empirical success, we introduce problem distributions generated by a reference policy $q$, including a latent-variable model in which proofs exhibit reusable cut/lemma/sketch structure represented by a proof DAG. Under a top-$k$ search protocol and Tsybakov-type margin conditions, we derive lower bounds on finite-horizon success probability that decompose into search and learning terms, with learning controlled by sequential Rademacher/covering complexity. Our main separation result shows that when cut elimination expands a DAG of depth $D$ into a cut-free tree of size $Ω(Λ^D)$ while the cut-aware hierarchical process has size $O(λ^D)$ with $λ\llΛ$, a flat (cut-free) learner provably requires exponentially more data than a cut-aware hierarchical learner. This provides a principled justification for subgoal decomposition in recent agentic theorem provers.

Let q(Y;Θ) and cK(Y,X) be two smooth, decomposable circuits that are compatible overY then computing their product as a circuit rΘ,K(X,Y) = q(Y;Θ) cK(Y,X) that is decomposable overY can be done inO(|q||c|). Letr(X,Y)beacircuitthat is smooth and decomposable and deterministic overY then for a configurationx its MAP state argmaxyr(x,y)canbecomputedintimeO(|r|). For our experiments we use standard compilation tools toobtain aconstraint circuit starting from a propositional logical formula in conjunctive normal form. We now illustrate step-by-step one example of such a compilation for a simple logical formula. Deterministic sum units representdisjoint solutions to the logical formula, meaning there exists distinct assignments, characterized by the children, that satisfy the logical constraint e.g.

Neural Information Processing Systems http://nips.cc/

For example, a prediction market on whether GPT -4 will be able to consistently solve "easy" Sudoku puzzles from the LA Times has remained open for several months at the time of

The nineteenth century saw the emergence of proofs so involved that they could only be verified by a handful of specialists.