Twee: An Equational Theorem Prover

Twee is an automated theorem prover for equational logic. It implements unfailing Knuth-Bendix completion with ground joinability testing and a connectedness-based redundancy criterion. It came second in the UEQ division of CASC-J10, solving some problems that no other system solved. This paper describes Twee’s design and implementation.

CSE_E 1.0: An Integrated Automated Theorem Prover for First-Order Logic

First-order logic is an important part of mathematical logic, and automated theorem proving is an interdisciplinary field of mathematics and computer science. The paper presents an automated theorem prover for first-order logic, called C S E _ E 1.0, which is a combination of two provers contradiction separation extension (CSE) and E, where CSE is based on the recently-introduced multi-clause standard contradiction separation (S-CS) calculus for first-order logic and E is the well-known equational theorem prover for first-order logic based on superposition and rewriting. The motivation of the combined prover C S E _ E 1.0 is to (1) evaluate the capability, applicability and generality of C S E _ E , and (2) take advantage of novel multi-clause S-CS dynamic deduction of CSE and mature equality handling of E to solve more and harder problems. In contrast to other improvements of E, C S E _ E 1.0 optimizes E mainly from the inference mechanism aspect. The focus of the present work is given to the description of C S E _ E including its S-CS rule, heuristic strategies, and the S-CS dynamic deduction algorithm for implementation. In terms of combination, in order not to lose the capability of E and use C S E _ E to solve some hard problems which are unsolved by E, C S E _ E 1.0 schedules the running of the two provers in time. It runs plain E first, and if E does not find a proof, it runs plain CSE, then if it does not find a proof, some clauses inferred in the CSE run as lemmas are added to the original clause set and the combined clause set handed back to E for further proof search. C S E _ E 1.0 is evaluated through benchmarks, e.g., CASC-26 (2017) and CASC-J9 (2018) competition problems (FOFdivision). Experimental results show that C S E _ E 1.0 indeed enhances the performance of E to a certain extent.

Implementing Polymorphism in Zenon

Extending first-order logic with ML-style polymorphism allows to definegeneric axioms dealing with several sorts. Until recently, mostautomated theorem provers relied on preprocess encodings intomono/many-sorted logic to reason within such theories. In this paper, wediscuss the implementation of polymorphism into thefirst-order tableau-based automated theorem prover Zenon. Thisimplementation leads to slightly modify some basic parts of the code,from the representation of expressions to the proof-search algorithm.

Dialogues for proof search

Dialogue games are a two-player semantics for a variety of logics, including intuitionistic and classical logic. Dialogues can be viewed as a kind of analytic calculus not unlike tableaux. Can dialogue games be an effective foundation for proof search in intuitionistic logic (both first-order and propositional)? We announce Kuno, an automated theorem prover for intuitionistic first-order logic based on dialogue games.

Automated Deduction in the B Set Theory using Typed Proof Search and Deduction Modulo

We introduce an encoding of the set theory of the B method using polymorphic types and deduction modulo, which is used for the automated verification of proof obligations in the framework of the BWare project. Deduction modulo is an extension of predicate calculus with rewriting both on terms and propositions. It is well suited for proof search in theories because it turns many axioms into rewrite rules. We also present the associated automated theorem prover Zenon Modulo, an extension of Zenon to polymorphic types and deduction modulo, along with its backend to the Dedukti universal proof checker, which also relies on types and deduction modulo, and which allows us to verify the proofs produced by Zenon Modulo. Finally, we assess our approach over the proof obligation benchmark provided by the BWare project.

Embedding of Quantified Higher-Order Nominal Modal Logic into Classical Higher-Order Logic

In this paper, we present an embedding of higher-order nominal modal logicinto classical higher-order logic, and study its automation. There exists no automated theorem prover for first-order or higher-order nominal logic at the moment, hence, this is the first automation for this kind of logic.In our work, we focus on nominal tense logic and have successfully proven some first theorems.

EPR-based k-induction with Counterexample Guided Abstraction Refinement

In recent years it was proposed to encode bounded model checking (BMC) into the effectively propositional fragment of first-order logic (EPR). The EPR fragment can provide for a succinct representation of the problem and facilitate reasoning at a higher level.In this paper we present an extension of the EPR-based bounded model checkingwith k-induction which can be used to prove safety properties of systems overunbounded runs. We present a novel abstraction-refinement approach based onunsatisfiable cores and models (UCM) for BMC and k-induction in the EPR setting.We have implemented UCM refinements for EPR-based BMC and k-induction in a first-order automated theorem prover iProver. We also extended iProver with the AIGER format and evaluated it over the HWMCC'14 competition benchmarks. The experimental results are encouraging. We show that a number of AIG problems can be verified until deeper bounds with the EPR-based model checking.

Metis-based Paramodulation Tactic for HOL Light

Metis is an automated theorem prover based on ordered paramodulation.It is widely employed in the interactive theorem provers Isabelle/HOL and HOL4to automate proofs as well as reconstruct proofs found by automated provers.For both these purposes, the tableaux-based MESON tactic is frequently usedin HOL Light. However, paramodulation-based provers such as Metisperform better on many problems involving equality.We created a Metis-based tactic in HOL Light which translates HOL problemsto Metis, runs an OCaml version of Metis, and reconstructs proofsin Metis' paramodulation calculus as HOL proofs.We evaluate the performance of Metis as proof reconstruction methodin HOL Light.

SAT solving experiments in Vampire

In order to better understand how well a state of the art SAT solver would behave in the framework of a first-order automated theorem prover we have decided to integrate Lingeling, best performing SAT solver, inside Vampire’s AVATAR framework. In this paper we propose two ways of integrating a SAT solver inside of Vampire and evaluate overall performance of this combination. Our experiments show that by using a state of the art SAT solver in Vampire we manage to solve more problems. Surprisingly though, there are cases where combination of the two solvers does not always prove to generate best results.