nanoCoP: Natural Non-clausal Theorem Proving

Abstract

Most efficient fully automated theorem provers implement proof search calculi that require the input formula to be in a clausal form, i.e. disjunctive or conjunctive normal form. The translation into clausal form introduces a significant overhead to the proof search and modifies the structure of the original formula. Translating a proof in clausal form back into a more readable non-clausal proof of the original formula is not straightforward. This paper presents a non-clausal automated theorem prover for classical first-order logic. It is based on a non-clausal connection calculus and implemented with a few lines of Prolog code. Working entirely on the original structure of the input formula yields not only a speed up of the proof search, but the resulting non-clausal proofs are also shorter.

Theorem Proving as Constraint Solving with Coherent Logic

“…-Vampire 15 4.4 using CASC mode [44], a superposition-based theorem prover; -E 16 2.2 in auto mode, a superposition-based theorem prover; -Iprover 17 3.1e [43], an instantiation-based theorem prover for first-order logic; -leanCoP 18 2.1 [58], a prover based on the connection (tableau) calculus; -nanoCoP 19 1.1 [57], a prover based on the non-clausal connection calculus for classical logic; -Geo 20 2007f, a prover based on geometric resolution; -ChewTPTP 21 1.0.0 [15], which proves rigid first-order theorems by encoding the existence of a first-order connection tableau in SAT; -Zenon 22 0.8.4 [16], a prover based on the tableau method and capable of producing Coq proofs; -Standard first-order tactic of Coq, a reflexive implementation of a first-order prover [21]; -Coq Hammer Tactics 23 v1.3 [22].…”