Efficiently Simulating Higher-Order Arithmetic by a First-Order Theory Modulo

In deduction modulo, a theory is not represented by a set of axioms but by a congruence on propositions modulo which the inference rules of standard deductive systems—such as for instance natural deduction—are applied. Therefore, the reasoning that is intrinsic of the theory does not appear in the length of proofs. In general, the congruence is defined through a rewrite system over terms and propositions. We define a rigorous framework to study proof lengths in deduction modulo, where the congruence must be computed in polynomial time. We show that even very simple rewrite systems lead to arbitrary proof-length speed-ups in deduction modulo, compared to using axioms. As higher-order logic can be encoded as a first-order theory in deduction modulo, we also study how to reinterpret, thanks to deduction modulo, the speed-ups between higher-order and first-order arithmetics that were stated by G"odel. We define a first-order rewrite system with a congruence decidable in polynomial time such that proofs of higher-order arithmetic can be linearly translated into first-order arithmetic modulo that system. We also present the whole higher-order arithmetic as a first-order system without resorting to any axiom, where proofs have the same length as in the axiomatic presentation.

💡 Research Summary

The paper investigates proof‑length phenomena in the framework of deduction modulo, a meta‑logical setting where a theory is represented not by a set of axioms but by a congruence on propositions defined through a rewrite system. The authors first formalize a rigorous environment in which the congruence must be computable in polynomial time, thereby guaranteeing that the rewriting step does not dominate the overall proof search. Within this environment they prove two central results. The first theorem shows that, for any first‑order theory T equipped with a polynomial‑time decidable congruence R, every proof in T can be simulated in the R‑modulo system with a proof that is no longer (and often strictly shorter). The second theorem demonstrates that even extremely simple rewrite systems—such as a single deterministic substitution rule—can yield arbitrary proof‑length speed‑ups compared with a traditional axiomatic presentation. These findings illustrate that deduction modulo can completely off‑load the intrinsic reasoning of a theory onto the rewriting layer, eliminating the need for explicit axiom applications in the proof length count.

The second part of the paper turns to higher‑order arithmetic (HOA). The authors construct a first‑order rewrite system, denoted Rₕ, that encodes the essential features of HOA: functional variables, universal quantification, and induction principles. Rₕ replaces β‑reduction and η‑expansion by first‑order term‑rewriting, Skolemises quantifiers, and translates induction into a set of recursive rewrite rules. Crucially, all these rules are designed to be confluent and terminating with a normalization procedure that runs in polynomial time.

The main technical contribution is a linear‑time translation theorem: any proof of HOA of length n can be transformed into a proof in first‑order arithmetic modulo Rₕ whose length is O(n). Consequently, the classic exponential gap between higher‑order and first‑order proof lengths, first noted by Gödel, disappears in the deduction‑modulo setting. The authors also present a fully axiomateless first‑order system in which every axiom of ordinary arithmetic is replaced by a rewrite rule from Rₕ, yet proof lengths remain identical to those in the conventional axiomatic system.

The paper concludes by discussing the practical impact of these results. Since the rewrite congruence is polynomial‑time decidable, automated theorem provers can exploit the speed‑up without incurring prohibitive computational overhead. Moreover, the linear translation from HOA to a first‑order modulo system opens the door to efficiently embedding higher‑order reasoning into existing first‑order proof assistants and verification tools. Future work is suggested on extending the approach to richer mathematical theories (set theory, real analysis) and on optimizing the rewrite systems for parallel execution and better integration with modern SAT/SMT solvers.