Theorem Proving as Constraint Solving with Coherent Logic

Abstract

In contrast to common automated theorem proving approaches, in which the search space is a set of some formulae and what is sought is again a (goal) formula, we propose an approach based on searching for a proof (of a given length) as a whole. Namely, a proof of a formula in a fixed logical setting can be encoded as a sequence of natural numbers meeting some conditions and a suitable constraint solver can find such sequence. The sequence can then be decoded giving a proof in the original theory language. This approach leads to several unique features, for instance, it can provide shortest proofs. In this paper, we focus on proofs in coherent logic, an expressive fragment of first-order logic, and on SAT and SMT solvers for solving sets of constraints, but the approach could be tried in other contexts as well. We implemented the proposed method and we present its features, perspectives and performances. One of the features of the implemented prover is that it can generate human understandable proofs in natural language and also machine verifiable proofs for the interactive prover Coq.

Automated Completion of Statements and Proofs in Synthetic Geometry: an Approach based on Constraint Solving

“…. , y l−1 ; B j (for 0 ≤ j ≤ m − 1) denotes a conjunction of atomic formulae (involving zero or more of the variables from⃗ x and⃗ y) [14]. If there are no formulae A i , then the left-hand side of the implication is assumed to be ⊤.…”

“…"Theorem proving as constraint solving" is a paradigm for automated theorem proving recently proposed [14]. In contrast to common automated theorem proving approaches, in which the search space is a set of some formulae and what is sought is again a (goal) formula, this new approach is based on searching for a proof (of a given length) as a whole.…”

“…Completing Proofs using Larus. Larus can be instructed to look for a proof of a given conjecture (also possibly only partially specified) meeting some conditions (that we call "hints") [14]. Therefore, Larus can try, for instance, to reconstruct a proof given only in outline (like proofs in textbooks).…”

“…Because we aim at producing intelligible and readable proofs, with a similar level of granularity as paper-and-pencil proofs, our approach is logic-based, uses a fragment of first-order logic called coherent logic, and is focused on synthetic geometry (in contrast to algebraic methods). Our approach for dealing with partial conjectures and partial proofs is implemented as an extension of the automated theorem prover Larus developed previously [14]. The approach is uniform for all three kinds of completion tasks, flexible and, to our knowledge, unique such approach.…”

“…https://github.com/janicicpredrag/Larus3 The corresponding C++ implementation is an improved version of the implementation presented earlier[14].…”