scholarly journals An Inference Rule for the Acyclicity Property of Term Algebras

10.29007/tlw4 ◽  
2018 ◽  
Author(s):  
Simon Robillard
Keyword(s):  
Computer Science ◽  
Inference Rule ◽  
Theorem Prover ◽  
Science Reasoning ◽  
First Order ◽  
Theorem Provers ◽  
Smt Solvers ◽  
Indexing Technique ◽  
Superposition Calculus

Term algebras are important structures in many areas of mathematics and computer science. Reasoning about their theories in superposition-based first-order theorem provers is made difficult by the acyclicity property of terms, which is not finitely axiomatizable. We present an inference rule that extends the superposition calculus and allows reasoning about term algebras without axioms to describe the acyclicity property. We detail an indexing technique to efficiently apply this rule in problems containing a large number of clauses. Finally we experimentally evaluate an implementation of this extended calculus in the first-order theorem prover Vampire. The results show that this technique is able to find proofs for difficult problems that existing SMT solvers and first-order theorem provers are unable to solve.

scholarly journals A Clausal Normal Form Translation for FOOL

10.29007/ltkk ◽  
2018 ◽  
Author(s):  
Evgenii Kotelnikov ◽  
Laura Kovács ◽  
Martin Suda ◽  
Andrei Voronkov
Keyword(s):  
Normal Form ◽  
Programming Languages ◽  
Program Analysis ◽  
State Of The Art ◽  
Order Logic ◽  
Theorem Prover ◽  
First Order Logic ◽  
First Order ◽  
Theorem Provers ◽  
Superposition Calculus

Automated theorem provers for first-order logic usually operate on sets of first-order clauses. It is well-known that the translation of a formula in full first-order logic to a clausal normal form (CNF) can crucially affect performance of a theorem prover. In our recent work we introduced a modification of first-order logic extended by the first class boolean sort and syntactical constructs that mirror features of programming languages. We called this logic FOOL. Formulas in FOOL can be translated to ordinary first-order formulas and checked by first-order theorem provers. While this translation is straightforward, it does not result in a CNF that can be efficiently handled by state-of-the-art theorem provers which use superposition calculus. In this paper we present a new CNF translation algorithm for FOOL that is friendly and efficient for superposition-based first-order provers. We implemented the algorithm in the Vampire theorem prover and evaluated it on a large number of problems coming from formalisation of mathematics and program analysis. Our experimental results show an increase of performance of the prover with our CNF translation compared to the naive translation.

scholarly journals Making Automatic Theorem Provers more Versatile

10.29007/n6j7 ◽  
2018 ◽  
Author(s):  
Simon Cruanes
Keyword(s):  
Higher Order ◽  
Order Logic ◽  
Higher Order Logic ◽  
Research Directions ◽  
First Order ◽  
Theorem Provers ◽  
Smt Solvers ◽  
Automatic Theorem Provers ◽  
Automatic Theorem

We argue that automatic theorem provers should become more versatile and should be able to tackle problems expressed in richer input formats. Salient research directions include (i) developing tight combinations of SMT solvers and first-order provers; (ii) adding better handling of theories in first-order provers; (iii) adding support for inductive proving; (iv) adding support for user-defined theories and functions; and (v) bringing to the provers some basic abilities to deal with logics beyond first-order, such as higher-order logic.

scholarly journals Integer Induction in Saturation

2021 ◽  
pp. 361-377
Author(s):  
Petra Hozzová ◽  
Laura Kovács ◽  
Andrei Voronkov
Keyword(s):  
Theorem Proving ◽  
Program Analysis ◽  
State Of The Art ◽  
Inductive Reasoning ◽  
Theorem Prover ◽  
Inference Rules ◽  
First Order ◽  
Theorem Provers ◽  
Mathematical Properties

AbstractIntegers are ubiquitous in programming and therefore also in applications of program analysis and verification. Such applications often require some sort of inductive reasoning. In this paper we analyze the challenge of automating inductive reasoning with integers. We introduce inference rules for integer induction within the saturation framework of first-order theorem proving. We implemented these rules in the theorem prover Vampire and evaluated our work against other state-of-the-art theorem provers. Our results demonstrate the strength of our approach by solving new problems coming from program analysis and mathematical properties of integers.

scholarly journals Leo-III Version 1.1 (System description)

10.29007/grmx ◽  
2018 ◽  
Author(s):  
Christoph Benzmüller ◽  
Alexander Steen ◽  
Max Wisniewski
Keyword(s):  
Higher Order ◽  
Order Logic ◽  
Theorem Prover ◽  
Higher Order Logic ◽  
System Description ◽  
Agent Based ◽  
First Order ◽  
Theorem Provers ◽  
Automated Theorem Prover ◽  
Ordering Constraints

Leo-III is an automated theorem prover for (polymorphic) higher-order logic which supports all common TPTP dialects, including THF, TFF and FOF as well as their rank-1 polymorphic derivatives. It is based on a paramodulation calculus with ordering constraints and, in tradition of its predecessor LEO-II, heavily relies on cooperation with external first-order theorem provers.Unlike LEO-II, asynchronous cooperation with typed first-order provers and an agent-based internal cooperation scheme is supported. In this paper, we sketch Leo-III's underlying calculus, survey implementation details and give examples of use.

scholarly journals Aiming for the Goal with SInE

10.29007/q4pt ◽  
2020 ◽  
Author(s):  
Martin Suda
Keyword(s):  
Experimental Evaluation ◽  
Main Idea ◽  
Inference Engine ◽  
Theorem Prover ◽  
Selection Algorithm ◽  
First Order ◽  
Theorem Provers ◽  
The Individual ◽  
Goal Distance ◽  
Automatic Theorem

The Sumo INference Engine (SInE) is a well-established premise selection algorithm for first-order theorem provers, routinely used, especially on large theory problems. The main idea of SInE is to start from the goal formula and to iteratively add other formulas to those already added that are related by sharing signature symbols. This implicitly defines a certain heuristical distance of the individual formulas and symbols from the goal.In this paper, we show how this distance can be successfully used for other purposes than just premise selection. In particular, biasing clause selection to postpone introduction of input clauses further from the goal helps to solve more problems. Moreover, a precedence which respects such goal distance of symbols gives rise to a goal sensitive simplification ordering. We implemented both ideas in the automatic theorem prover Vampire and present their experimental evaluation on the TPTP benchmark.

scholarly journals Implementing Polymorphism in Zenon

10.29007/87vl ◽  
2018 ◽  
Author(s):  
Guillaume Bury ◽  
Raphaël Cauderlier ◽  
Pierre Halmagrand
Keyword(s):  
Search Algorithm ◽  
Order Logic ◽  
Theorem Prover ◽  
First Order Logic ◽  
First Order ◽  
Proof Search ◽  
Theorem Provers ◽  
Automated Theorem Prover

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.

scholarly journals Local proofs and AVATAR

10.29007/qgdk ◽  
2018 ◽  
Author(s):  
Giles Reger ◽  
Martin Suda
Keyword(s):  
Experimental Results ◽  
Theorem Prover ◽  
First Order ◽  
Theorem Provers

With first-order interpolation as the application in mind, we study the problem of gen- erating local proofs in theorem provers employing the AVATAR architecture. The theory is complemented by experimental results based on our implementation of the techniques in theorem prover Vampire.

scholarly journals AVATAR Modulo Theories

10.29007/k6tp ◽  
2018 ◽  
Author(s):  
Giles Reger ◽  
Nikolaj Bjorner ◽  
Martin Suda ◽  
Andrei Voronkov
Keyword(s):  
Theorem Proving ◽  
Satisfiability Modulo Theories ◽  
Theorem Prover ◽  
Ground Theory ◽  
Smt Solver ◽  
First Order ◽  
Proof Search ◽  
Smt Solvers ◽  
Sat Solver ◽  
A New Technique

This paper introduces a new technique for reasoning with quantifiers and theories. Traditionally, first-order theorem provers (ATPs) are well suited to reasoning with first-order problems containing many quantifiers and satisfiability modulo theories (SMT) solvers are well suited to reasoning with first-order problems in ground theories such as arithmetic. A recent development in first-order theorem proving has been the AVATAR architecture which uses a SAT solver to guide proof search based on a propositional abstraction of the first-order clause space. The approach turns a single proof search into a sequence of proof searches on (much) smaller sub-problems. This work extends the AVATAR approach to use a SMT solver in place of the SAT solver, with the effect that the first-order solver only needs to consider ground-theory-consistent sub-problems. The new architecture has been implemented using the Vampire theorem prover and Z3 SMT solver. Our experimental results, and the results of recent competitions, show that such a combination can be highly effective.

scholarly journals Global Subsumption Revisited (Briefly)

10.29007/qcd7 ◽  
2018 ◽  
Author(s):  
Giles Reger ◽  
Martin Suda
Keyword(s):  
Initial Problem ◽  
Theorem Prover ◽  
General Idea ◽  
Practical Implementation ◽  
Global Approach ◽  
Smt Solver ◽  
First Order ◽  
Theorem Provers ◽  
Sat Solver

Global subsumption is an existing simplification technique for saturation-based first-order theorem provers. The general idea is that we can replace a clause C by its subclause D if D follows from the initial problem as D will subsume C. The effectiveness of the technique comes from a cheap, global approach for (incompletely) checking whether D is a consequence of the initial problem. The idea is to produce and maintain a set S of ground clauses that follow from the input (e.g. grounded versions of all derived clauses) and to check whether a grounding of D follows from this set. As this is now a propositional problem this check can be performed by a SAT solver, making it efficient. In this paper we review the global subsumption technique and pose a number of questions related to the practical implementation of global subsumption and possible variations of the approach. We consider, for example, which groundings to place in S, how to select the subclause(s) D to check, how to integrate this technique with the AVATAR approach and whether it makes sense to replace the SAT solver with an SMT solver. This discussion takes place within the context of the Vampire theorem prover.

scholarly journals Superposition with First-class Booleans and Inprocessing Clausification

2021 ◽  
pp. 378-395
Author(s):  
Visa Nummelin ◽  
Alexander Bentkamp ◽  
Sophie Tourret ◽  
Petar Vukmirović
Keyword(s):  
State Of The Art ◽  
Higher Order ◽  
The State ◽  
Order Logic ◽  
Theorem Prover ◽  
First Order Logic ◽  
Higher Order Logic ◽  
First Order ◽  
Superposition Calculus

AbstractWe present a complete superposition calculus for first-order logic with an interpreted Boolean type. Our motivation is to lay the foundation for refutationally complete calculi in more expressive logics with Booleans, such as higher-order logic, and to make superposition work efficiently on problems that would be obfuscated when using clausification as preprocessing. Working directly on formulas, our calculus avoids the costly axiomatic encoding of the theory of Booleans into first-order logic and offers various ways to interleave clausification with other derivation steps. We evaluate our calculus using the Zipperposition theorem prover, and observe that, with no tuning of parameters, our approach is on a par with the state-of-the-art approach.

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