On Learning to Prove

Abstract

In this paper, we consider the problem of learning a first-order theorem prover that uses a representation of beliefs in mathematical claims to construct proofs. The inspiration for doing so comes from the practices of human mathematicians where "plausible reasoning" is applied in addition to deductive reasoning to find proofs.Towards this end, we introduce a representation of beliefs that assigns probabilities to the exhaustive and mutually exclusive first-order possibilities found in Hintikka's theory of distributive normal forms. The representation supports Bayesian update, induces a distribution on statements that does not enforce that logically equivalent statements are assigned the same probability, and suggests an embedding of statements into an associated Hilbert space.We then examine conjecturing as model selection and an alternating-turn game of determining consistency. The game is amenable (in principle) to self-play training to learn beliefs and derive a prover that is complete when logical omniscience is attained and sound when beliefs are reasonable. The representation has super-exponential space requirements as a function of quantifier depth so the ideas in this paper should be taken as theoretical. We will comment on how abstractions can be used to control the space requirements at the cost of completeness.

Elementary Logic in Linear Space

“…The construction of the Hilbert space is inspired by the one given in [15] (Section 3.1 and Section 3.2). The main difference is that we take the space view as primary whereas [15] defines a probability distribution on first-order sentences and derives the corresponding space.…”

“…The primary building blocks for this paper come from Hintikka's work on distributive normal forms [12][13][14] and Huang's work [15] on using them to assign probabilities to first-order 12 We follow the convention that −1 → 1 and 1 → 0 when converting between {−1, 1} and 2 (e.g., see [29]). 13 As a reminder, Ω(•) is an asymptotic lower-bound.…”

“…Logical uncertainty Logical uncertainty and the problem of logical omniscience is a problem of interest in philosophy (e.g., see [3,11,30,35]) and there have been many solutions proposed for addressing it. One approach proposes assigning probabilities to sentences such that logically equivalent statements are not necessarily assigned the same probability (e.g., see [8] and [15]). 14 Another approach syntactically models an agent's knowledge (e.g., see [6] and [7]).…”

“…We suggest using pca with respect to an agent's knowledge, which is constructed with model-checking, as a heuristic. Huang [15] proposes conjecturing as hypothesis selection with respect to a probability distribution on first-order sentences.…”

“…Thus our first task is to identify a space that holds the information content of firstorder sentences. Towards this end, we build upon ideas found in [15] which introduces a probability distribution on firstorder sentences and identifies a corresponding space. We organize our exploration as a series of questions.…”