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

TL;DR

This paper introduces a framework for proof length analysis in deduction modulo, demonstrating how simple rewrite systems can significantly speed up proofs and enabling efficient encoding of higher-order arithmetic within first-order theories.

Contribution

It provides a polynomial-time decidable rewrite system that allows linear translation of higher-order arithmetic proofs into first-order arithmetic modulo, eliminating the need for axioms.

Findings

Simple rewrite systems can cause arbitrary proof-length speed-ups.

Higher-order arithmetic can be encoded as a first-order theory with equivalent proof lengths.

Proofs in higher-order arithmetic can be efficiently translated into first-order proofs modulo the rewrite system.

Abstract

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.