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Inductive Prover Based on Equality Saturation for a Lazy Functional Language [chapter]

Sergei Grechanik
2015 Lecture Notes in Computer Science  
The present paper shows how the idea of equality saturation can be used to build an inductive prover for a non-total first-order lazy functional language.  ...  We adapt equality saturation approach to a functional language by using transformations borrowed from supercompilation.  ...  On order of transformation Our experimental implementation of an equivalence prover for a first-order lazy language based on equality saturation is written in Scala and can be found on GitHub [1] .  ... 
doi:10.1007/978-3-662-46823-4_11 fatcat:kddscajkbzchvb5jbxhitpt3vu

Equational Logic and Rewriting [chapter]

Claude Kirchner, Hélène Kirchner
2014 Handbook of the History of Logic  
specification language based on first-order logic [6] that also supports partial functions and subsorting.  ...  of rule-based programming languages.  ... 
doi:10.1016/b978-0-444-51624-4.50006-x fatcat:eqtvr66m65eula7otl7fr6ppaa

Superposition with Structural Induction [chapter]

Simon Cruanes
2017 Lecture Notes in Computer Science  
We present a new extension of typed superposition that can perform structural induction. Several inductive goals can be attempted within a single saturation loop, by leveraging AVATAR [1] .  ...  Superposition-based provers have been successfully used to discharge proof obligations stemming from proof assistants. However, many such obligations require induction to be proved.  ...  Acknowledgments The author would like to thank Jasmin Blanchette, Gilles Dowek, Guillaume Burel, Pascal Fontaine, and reviewers of previous versions of this paper (one of them, in particular, for pointing  ... 
doi:10.1007/978-3-319-66167-4_10 fatcat:jzgkbftdsvdsthbtkxg4ipmeqm

A verified prover based on ordered resolution

Anders Schlichtkrull, Jasmin Christian Blanchette, Dmitriy Traytel
2019 Proceedings of the 8th ACM SIGPLAN International Conference on Certified Programs and Proofs - CPP 2019  
functional Standard ML code that constitutes a semidecision procedure for first-order logic.  ...  As a step towards verifying modern provers, we specify, using Isabelle/HOL, a purely functional first-order ordered resolution prover and establish its soundness and refutational completeness.  ...  Acknowledgments Johannes Hölzl gave us some useful advice on how to specify and reason about possibly nonterminating functions in Isabelle/HOL.  ... 
doi:10.1145/3293880.3294100 dblp:conf/cpp/SchlichtkrullBT19 fatcat:ftlcitpp7rbsbevdkf6nl6r25y

On theorem proving for program checking

Maria Paola Bonacina
2010 Proceedings of the 12th international ACM SIGPLAN symposium on Principles and practice of declarative programming - PPDP '10  
This article is a survey of recent results, related works and new challenges in automated theorem proving for program checking.  ...  , rewrite-based, completion-based, superposition-based, paramodulation-based, contraction-based, saturation-based or ordering-based, to emphasize one aspect or the other (e.g., [7, 19, 24, 65, 92, 103  ...  As in [96] , and in the CEGAR paradigm, the abstraction involves a map, called abstraction function, from the language of first-order formulae to a simpler language.  ... 
doi:10.1145/1836089.1836090 dblp:conf/ppdp/Bonacina10 fatcat:mh3666vskfcxvgnisxp7hnwvi4

Decompositions of Natural Numbers: From a Case Study in Mathematical Theory Exploration

Adrian Craciun, Madalina Hodorog
2007 Ninth International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC 2007)  
In the context of a scheme based exploration model proposed by Bruno Buchberger, we investigate the idea of decomposition, applied in the exploration of natural numbers.  ...  of domain elements into irreducible ones, and also the problem of irreducible decomposition which we then solve.  ...  In fact, the same happens to the substitutivity axioms for equality. • IR N consists of first order predicate logic calculus, equality reasoning, and the induction inference rule.  ... 
doi:10.1109/synasc.2007.55 dblp:conf/synasc/CraciunH07 fatcat:buicci7bz5edndjko6bsbostty

From Search to Computation: Redundancy Criteria and Simplification at Work [chapter]

Thomas Hillenbrand, Ruzica Piskac, Uwe Waldmann, Christoph Weidenbach
2013 Lecture Notes in Computer Science  
When executed by a theorem prover, the inference rules of these calculi usually generate a tremendously growing search space.  ...  For a number of subclasses of firstorder logic appropriate redundancy and simplification concepts even turn the superposition calculus into a decision procedure.  ...  Lazy CNF Transformation Practically all automated theorem provers in use today are based on clausal logic.  ... 
doi:10.1007/978-3-642-37651-1_7 fatcat:fr4uqtwvubbtvgjepamaamojeu

Formalizing Bachmair and Ganzinger's Ordered Resolution Prover [chapter]

Anders Schlichtkrull, Jasmin Christian Blanchette, Dmitriy Traytel, Uwe Waldmann
2018 Lecture Notes in Computer Science  
We present an Isabelle/HOL formalization of the first half of Bachmair and Ganzinger's chapter on resolution theorem proving, culminating with a refutationally complete first-order prover based on ordered  ...  We developed general infrastructure and methodology that can form the basis of completeness proofs for related calculi, including superposition.  ...  We perform an induction on k. The base case is trivial. For the induction step, if k is beyond the end of the list, then k i=0 N i = k−1 i=0 N i and we can apply the induction hypothesis directly.  ... 
doi:10.1007/978-3-319-94205-6_7 fatcat:hmjoecj5obe2ji6ldifhh3yfbu

A metaprogramming framework for formal verification

Gabriel Ebner, Sebastian Ullrich, Jared Roesch, Jeremy Avigad, Leonardo de Moura
2017 Proceedings of the ACM on Programming Languages  
We describe the metaprogramming framework currently used in Lean, an interactive theorem prover based on dependent type theory.  ...  This framework extends Lean's object language with an API to some of Lean's internal structures and procedures, and provides ways of reflecting object-level expressions into the metalanguage.  ...  We are very grateful to David Christiansen, Johannes Hölzl, Greg Malecha, and the anonymous referees for extensive comments, corrections, and advice.  ... 
doi:10.1145/3110278 dblp:journals/pacmpl/EbnerURAM17 fatcat:i2akdiqjezgmfklo3wuql7cnsu

Summaries for Context-Free Games [article]

Lukáš Holík and Roland Meyer and Sebastian Muskalla
2016 arXiv   pre-print
It is based on a novel representation of all plays starting in a non-terminal. The representation uses the domain of Boolean formulas over the transition monoid of the target automaton.  ...  The elements of the monoid are essentially procedure summaries, and our approach can be seen as the first summary-based algorithm for the synthesis of recursive programs.  ...  Acknowledgements We thank Olivier Serre, Matthew Hague, Georg Zetzsche, and Emanuele D'Osualdo for helpful discussions. We thank the reviewers for their feedback.  ... 
arXiv:1603.07256v4 fatcat:oios3y6jxveozp46xqrb3st33u

Summaries for Context-Free Games

Lukás Holík, Roland Meyer, Sebastian Muskalla, Marc Herbstritt
2016 Foundations of Software Technology and Theoretical Computer Science  
In this paper 1 , we fill in the empty spot in the picture and propose a solver and synthesis method for context-free inclusion games based on summaries.  ...  It is based on a novel representation of all plays starting in a non-terminal. The representation uses the domain of Boolean formulas over the transition monoid of the target automaton.  ...  We thank Olivier Serre, Matthew Hague, Georg Zetzsche, and Emanuele D'Osualdo for helpful discussions. We thank the reviewers for their feedback.  ... 
doi:10.4230/lipics.fsttcs.2016.41 dblp:conf/fsttcs/HolikMM16 fatcat:aulh53vhzregbmgaajbvitcuny

A Focusing Inverse Method Theorem Prover for First-Order Linear Logic [chapter]

Kaustuv Chaudhuri, Frank Pfenning
2005 Lecture Notes in Computer Science  
We present the theory and implementation of a theorem prover for first-order intuitionistic linear logic based on the inverse method.  ...  The central proof-theoretic insights underlying the prover concern resource management and focused derivations, both of which are traditionally understood in the domain of backward reasoning systems such  ...  Conclusion We have presented a theorem prover for first-order intuitionistic linear logic based on the inverse method which is already practical for a range of examples and significantly improves on prior  ... 
doi:10.1007/11532231_6 fatcat:pzh3dysrnzhl5eeufkl3lckcxe

Abstraction Refinement for Quantified Array Assertions [chapter]

Mohamed Nassim Seghir, Andreas Podelski, Thomas Wies
2009 Lecture Notes in Computer Science  
We present an abstraction refinement technique for the verification of universally quantified array assertions such as "all elements in the array are sorted".  ...  We implemented our technique in the ACSAR software model checker and successfully verified quantified array assertions for both text book examples and real-life examples taken from the Linux operating  ...  Our results indicate that, at least for quantified assertions over arrays, the use of sophisticated techniques for dealing with quantified assertions can often be avoided if one instead carefully adapts  ... 
doi:10.1007/978-3-642-03237-0_3 fatcat:udd7ww5wijek3cmlz2e6gj4sfa

Zap: Automated Theorem Proving for Software Analysis [chapter]

Thomas Ball, Shuvendu K. Lahiri, Madanlal Musuvathi
2005 Lecture Notes in Computer Science  
Automated theorem provers (ATPs) are a key component that many software verification and program analysis tools rely on.  ...  We believe that program analysis clients would benefit greatly if theorem provers were to provide a richer set of operations.  ...  The basic architecture of Zap is based on a lazy proof-explicating architecture for deciding first-order formulas [ABC + 02, BDS02, FJOS03] .  ... 
doi:10.1007/11591191_2 fatcat:uczhvijn6vew3bmwpaqhaxrkzy

Some Thoughts About FOL-Translations in Vampire

Giles Reger
2018 Conference on Automated Deduction  
One of the reasons for this is that first-order ATPs have reached a good level of maturity after decades of development.  ...  It is a common approach when faced with a reasoning problem to translate that problem into first-order logic and utilise a first-order automated theorem prover (ATP).  ...  Saturation-Based Proof Search After a clause set has been produced, Vampire attempts to saturate this set with respect to some inference system I.  ... 
dblp:conf/cade/Reger18 fatcat:ae5wocwmfjbm3adf3ddqdffwc4
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