Automated Theorem Proving Systems Research Articles

The CADE-29 Automated Theorem Proving System Competition – CASC-29

The CADE ATP System Competition (CASC) is the annual evaluation of fully automatic, classical logic, Automated Theorem Proving (ATP) systems – the world championship for such systems. CASC-29 was the twenty-eighth competition in the CASC series. Twenty-four ATP systems competed in the various divisions. This paper presents an outline of the competition design and a commentated summary of the results.

The 11th IJCAR automated theorem proving system competition – CASC-J11

The CADE ATP System Competition (CASC) is the annual evaluation of fully automatic, classical logic, Automated Theorem Proving (ATP) systems. CASC-J11 was the twenty-seventh competition in the CASC series. Twenty-four ATP systems competed in the various competition divisions. This paper presents an outline of the competition design and a commentated summary of the results.

Reinforcement Learning for Guiding the E Theorem Prover

Automated Theorem Proving (ATP) systems search for aproof in a rapidly growing space of possibilities. Heuristicshave a profound impact on search, and ATP systems makeheavy use of heuristics. This work uses reinforcement learn-ing to learn a metaheuristic that decides which heuristic to useat each step of a proof search in the E ATP system. Proximalpolicy optimization is used to dynamically select a heuristicfrom a fixed set, based on the current state of E. The approachis evaluated on its ability to reduce the number of inferencesteps used in successful proof searches, as an indicator of in-telligent search.

Fully reusing clause deduction algorithm based on standard contradiction separation rule

An automated theorem proving (ATP) system's capacity for reasoning is significantly influenced by the inference rules it uses. The recently introduced standard contradiction separation (S-CS) inference rule extends binary resolution to a multi-clause, dynamic, contradiction separation inference mechanism. The S-CS rule is used in the present work to provide a framework for fully clause reusing deductions. Accordingly, a fully reusing clause deduction algorithm (called the FRC algorithm) is built. The FRC algorithm is then incorporated as an algorithm module into the architecture of a top ATP, Vampire, creating a single integrated ATP system dubbed V_FRC. The objective of this integration is to enhance Vampire's performance while assessing the FRC algorithm's capacity for reasoning. According to experimental findings, V_FRC not only outperforms Vampire in a variety of aspects, but also solves 46 problems in the TPTP benchmark database that have a rating of 1, meaning that none of the existing ATP systems are able to resolve them.

The CADE-28 Automated Theorem Proving System Competition – CASC-28

The CADE ATP System Competition (CASC) is the annual evaluation of fully automatic, classical logic Automated Theorem Proving (ATP) systems. CASC-28 was the twenty-sixth competition in the CASC series. Twenty-two ATP systems competed in the various competition divisions. This paper presents an outline of the competition design and a commentated summary of the results.

The 10th IJCAR automated theorem proving system competition – CASC-J10

The CADE ATP System Competition (CASC) is the annual evaluation of fully automatic, classical logic Automated Theorem Proving (ATP) systems. CASC-J10 was the twenty-fifth competition in the CASC series. Twenty-four ATP systems and system variants competed in the various competition divisions. This paper presents an outline of the competition design, and a commentated summary of the results.

A multi-clause dynamic deduction algorithm based on standard contradiction separation rule

In the past decades, automated theorem proving (ATP) for first-order logic has made good progress, in which binary resolution inference rule plays a crucial role. However, as shown in the latest benchmark library of the ATP system, there are still many practical problems that have not been resolved or cannot be effectively resolved. Recently, in order to overcome the limitations of ATP based on binary resolution inference rules, a novel multi-clause dynamic standard contradiction separation (S-CS) inference rule and its automated deduction theory have been proposed. Based on this theory, this paper first clarifies the generality of this S-CS rule by comparing it with some well-known variants of the binary resolution rule, and then focuses on how to design a specific and effective algorithm along with search strategies to realize the S-CS based deductive theory with its implementation. Specifically, the present work proposes a novel S-CS dynamic deduction algorithm (in short SDDA) based on different strategies and summarizes its implementation procedures. In addition, we focus on evaluating whether SDDA, as a novel perspective multi-clause dynamic automatic deduction algorithm, can be applied on top of the current leading ATP system architectures to further improve their performances. Therefore, SDDA is applied to the current leading first-order ATP systems, i.e., Vampire and E, respectively forming two integrated APT systems, denoted as SDDA_V and SDDA_E. Then the capabilities of SDDA_V and SDDA_E are evaluated on the latest benchmark database TPTP, such as the CASC-J9 problems (FOF division) as well as the hard problems with a rating of 1 in the TPTP benchmark database. The experimental results show the effectiveness of SDDA: SDDA_V outperforms Vampire itself, and SDDA_E, outperforms E itself, and the two improved ATP systems have solved a number of hard problems with the rating of 1 in TPTP, that is, some problems in the latest benchmark database TPTP which have not yet been solved by other current first-order ATP systems.

The CADE-25 Automated Theorem Proving system competition – CASC-25

The CADE ATP System Competition (CASC) is an annual evaluation of fully automatic, classical logic Automated Theorem Proving (ATP) systems. CASC-25 was the twentieth competition in the CASC series. Twenty-seven ATP systems and system variants competed in the various competition divisions. An outline of the competition design, and a commentated summary of the results, are presented.

The 7th IJCAR automated theorem proving system competition – CASC-J7

The CADE ATP System Competition (CASC) is an annual evaluation of fully automatic, first order Automated Theorem Proving (ATP) systems. CASC-J4 was the thirteenth competition in the CASC series. Twenty-six ATP systems and system variants competed in the various competition and demonstration divisions. An outline of the competition design, and a commentated summary of the results, are presented.

The CADE-24 automated theorem proving system competition – CASC-24

The CADE ATP System Competition (CASC) is an annual evaluation of fully automatic, classical logic Automated Theorem Proving (ATP) systems. CASC-24 was the eighteenth competition in the CASC series. Thirty-two ATP systems and sys- tem variants competed in the various competition divisions. An outline of the competition design, and a commentated summary of the results, are presented.

A Tableau Based Automated Theorem Prover Using High Performance Computing

Automated Theorem Proving systems are enormously powerful computer programs capable of solving immensely difficult problems. The extreme capabilities of these systems lie on some well-established proof systems, such as Semantic tableau. It is used to prove the validity of a formula by contradiction and it can produce a counterexample if it fails. It can also be used to prove whether a formula is a logical consequence of a set of formulas. Tableau can be used in propositional logic, predicate logic, modal logic, temporal logic, and in other non-classical logics. In this article, we describe a detailed implementation of a sequential tableau algorithm for propositional and first order logic using a procedural language rather then logic programming language. We also illustrate a tableau based proof system in a distributed environment using the Message Passing Interface. This paper also investigates two distinct approaches for parallel and distributed implementation and describes the experimental formula generation procedure. The proposed high performance approach will un-wrap an efficient paradigm for automated theorem proving.

The CADE-23 Automated Theorem Proving System Competition – CASC-23

The CADE ATP System Competition (CASC) is an annual evaluation of fully automatic, classical logic Automated Theorem Proving (ATP) systems. CASC-23 was the sixteenth competition in the CASC series. Thirty-six ATP systems and system variants competed in the various competition and demonstration divisions. An outline of the competition design, and a commentated summary of the results, are presented.

The 5th IJCAR automated theorem proving system competition - CASC-J5

The CADE ATP System Competition (CASC) is an annual evaluation of fully automatic, classical logic Automated Theorem Proving (ATP) systems. CASC-J5 was the fifteenth competition in the CASC series....

The 5th IJCAR automated theorem proving system competition – CASC-J5

The CADE ATP System Competition (CASC) is an annual evaluation of fully automatic, classical logic Automated Theorem Proving (ATP) systems. CASC-J5 was the fifteenth competition in the CASC series. Thirty-nine ATP systems and system variants competed in the various competition and demonstration divisions. An outline of the competition design, and a commentated summary of the results, are presented.

Automated Reasoning in Higher-Order Logic using the TPTP THF Infrastructure

The Thousands of Problems for Theorem Provers (TPTP) problem library is the basis of a well known and well established infrastructure that supports research, development, and deployment of Automated Theorem Proving (ATP) systems. The extension of the TPTP from first-order form (FOF) logic to typed higher-order form (THF) logic has provided a basis for new development and application of ATP systems for higher-order logic. Key developments have been the specification of the THF language, the addition of higher-order problems to the TPTP, the development of the TPTP THF infrastructure, several ATP systems for higher-order logic, and the use of higher-order ATP in a range of domains. This paper describes these developments.

The CADE-22 automated theorem proving system competition – CASC-22

The CADE ATP System Competition (CASC) is an annual evaluation of fully automatic, classical logic Automated Theorem Proving (ATP) systems. CASC-22 was the fourteenth competition in the CASC series. Twenty-nine ATP systems and system variants competed in the various competition and demonstration divisions. An outline of the competition design, and a commentated summary of the results, are presented.

Solving quantified verification conditions using satisfiability modulo theories

First-order logic provides a convenient formalism for describing a wide variety of verification conditions. Two main approaches to checking such conditions are pure first-order automated theorem proving (ATP) and automated theorem proving based on satisfiability modulo theories (SMT). Traditional ATP systems are designed to handle quantifiers easily, but often have difficulty reasoning with respect to theories. SMT systems, on the other hand, have built-in support for many useful theories, but have a much more difficult time with quantifiers. One clue on how to get the best of both worlds can be found in the legacy system Simplify which combines built-in theory reasoning with quantifier instantiation heuristics. Inspired by Simplify and motivated by a desire to provide a competitive alternative to ATP systems, this paper describes a methodology for reasoning about quantifiers in SMT systems. We present the methodology in the context of the Abstract DPLL Modulo Theories framework. Besides adapting many of Simplify's techniques, we also introduce a number of new heuristics. Most important is the notion of instantiation level which provides an effective mechanism for prioritizing and managing the large search space inherent in quantifier instantiation techniques. These techniques have been implemented in the SMT system CVC3. Experimental results show that our methodology enables CVC3 to solve a significant number of quantified benchmarks that were not solvable with previous approaches.

Automated verification of refinement laws

Demonic refinement algebras are variants of Kleene algebras. Introduced by von Wright as a light-weight variant of the refinement calculus, their intended semantics are positively disjunctive predicate transformers, and their calculus is entirely within first-order equational logic. So, for the first time, off-the-shelf automated theorem proving (ATP) becomes available for refinement proofs. We used ATP to verify a toolkit of basic refinement laws. Based on this toolkit, we then verified two classical complex refinement laws for action systems by ATP: a data refinement law and Back’s atomicity refinement law. We also present a refinement law for infinite loops that has been discovered through automated analysis. Our proof experiments not only demonstrate that refinement can effectively be automated, they also compare eleven different ATP systems and suggest that program verification with variants of Kleene algebras yields interesting theorem proving benchmarks. Finally, we apply hypothesis learning techniques that seem indispensable for automating more complex proofs.

The ILTP Problem Library for Intuitionistic Logic

The Intuitionistic Logic Theorem Proving (ILTP) library provides a platform for testing and benchmarking automated theorem proving (ATP) systems for intuitionistic propositional and first-order logic. It includes about 2,800 problems in a standardized syntax from 24 problem domains. For each problem an intuitionistic status and difficulty rating were obtained by running comprehensive tests of currently available intuitionistic ATP systems on all problems in the library. Thus, for the first time, the testing and evaluation of ATP systems for intuitionistic logic have been put on a firm basis.

SEMANTIC DERIVATION VERIFICATION: TECHNIQUES AND IMPLEMENTATION

Automated Theorem Proving (ATP) systems are complex pieces of software, and thus may have bugs that make them unsound. In order to guard against unsoundness, the derivations output by an ATP system may be semantically verified by trusted ATP systems that check the required semantic properties of each inference step. Such verification needs to be augmented by structural verification that checks that inferences have been used correctly in the context of the overall derivation. This paper describes techniques for semantic verification of derivations, and reports on their implementation and testing in the GDV verifier.