DOCSLIB.ORG

Automated Reasoning

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Automated Reasoning CONTRIBUTI SCIENTIFICI AUTOMATED REASONING MARIA PAOLA BONACINA Dipartimento di Infor matica - Università degli Studi di Ver ona ALBERTO MARTELLI Dipartimento di Infor matica - Università degli Studi di Torino 1 Introduction counter-example,orcounter-model, that is, a model of S ∪{¬ϕ}. This branch of automated reasoning is called A central problem in automated reasoning is to determine automated model building. whether a conjecture ϕ, that represents a property to be In classical rst-order logic, deductive theorem proving veried, is a logical consequence of a set S of assump- is semi-decidable, while inductive theorem proving and tions, which express properties of the object of study (e.g., model building are not even semi-decidable.Itissignif- a system, a circuit, a program, a data type, a communica- icant that while books in theorem proving date from the tion protocol, a mathematical structure). early seventies [22, 48, 16, 27, 77, 44, 70], the rst book A conjoint problem is that of knowledge representation, on model building appeared only recently [21]. Most ap- or nding suitable formalisms for S and ϕ to represent as- proaches to automated model building belong to one of the pects of the real world, such as action, space, time, men- following three classes, or combine their principles: tal events and commonsense reasoning. While classical logic has been the principal formalism in automated rea- 1. Enumeration methods generate interpretations and soning, and many proof techniques have been studied and test whether they are models of the given set of for- implemented, non-classical logics, such as modal, tempo- mul; ral, description or nonmonotonic logics, have been widely 2. Saturation methods extract models from the nite set investigated to represent knowledge. of formul generated by a failed refutation attempt; and 2Automated reasoning in classical logic 3. Simultaneous methods search simultaneously for a Given the above central problem, one can try to answer af- refutation or a model of the given set of formul. ϕ S rmatively, by nding a proof of from . This problem In higher-order logics, that allow universal and existen- and the methods to approach it are called theorem prov- tial statements, not only on individuals, but also on func- ing. Theorem proving comprises both deductive theorem tions and predicates, even deductive theorem proving is no proving, which is concerned precisely with the entailment longer semi-decidable. Clearly, fully automated theorem S |= ϕ problem as stated above (in symbols: ), and induc- proving focuses on deductive theorem proving, while in- tive theorem proving, where the problem is to determine duction, model generation and reasoning in higher-order S ϕ whether entails all ground instances of (in symbols: logics resort to a larger extent to interactive theorem prov- S |= ϕσ σ , for all ground substitutions ). ing. Since the most important feature of higher-order logic In (fully) automated theorem proving, the machine alone for computer science are higher-order functions, that are is expected to nd a proof. In interactive theorem proving, a staple of functional programming languages, an inter- a proof is born out of the interaction between human and mediate solution is to develop a rst-order system, with machine. Since it is too difcult to nd a proof ignoring a functional programming language, used simultaneously the conjecture, the vast majority of theorem-proving meth- as programming language and as logical language [20, 43]. ods work refutationally, that is, they prove that ϕ follows logically from S,byshowing that S ∪{¬ϕ} generates a 2.1 Fully automated theorem proving contradiction, or is inconsistent. Otherwise, given assumptions S and conjecture ϕ, one Semi-decidability means that no algorithm is guaranteed to can try to answer negatively, disproving ϕ,bynding a halt, and return a proof, whenever S∪{¬ϕ} is inconsistent, 14 Anno III, N° 1/2, Marzo-Giugno 2006 CONTRIBUTI SCIENTIFICI or a model, whenever S ∪{¬ϕ} is consistent. The best one 1. the logic, can have is a semi-decision procedure, that is guaranteed to halt and return a proof, if S ∪{¬ϕ} is inconsistent. If 2. the form of admissible formulae for S or ϕ,or it halts without a proof, we can conclude that S ∪{¬ϕ} 3. the theory presented by the assumptions in S. is consistent, and try to extract a model from its output. However, if S ∪{¬ϕ} is consistent, the procedure is not An example of Case 1 is the guarded fragment of rst- guaranteed to halt. order logic, which propositional modal logic can be re- Intuitively, proofs of inconsistency of a given problem duced to. The most prominent instance is propositional S ∪{¬ϕ} are nite, if they exist, but there is an innite logic, whose decidable satisability problem is known as search space of logical consequences where to look for a SAT. Many problems in computer science can be encoded contradiction. A machine can explore only a nite part in propositional logic, reduced to SAT and submitted to of this innite space, and the challenge is to nd a proof SAT solvers.Asautomated reasoning is concerned primar- using as little resources as possible. A fundamental insight ily with complete SAT solvers, the dominating paradigm was the recognition that the ability to detect and discard is the DPLL procedure [25, 24, 79], implemented, among redundant formul is as crucial as the ability to generate others, in [78, 52]. consequences of given formul. In addition to standard As an example of Case 2, the Bernays-Schonnkel¬ class expansion inference rules of the form admits only sentences in the form A1 ... An (1) ∃x1,...xn.∀y1,...ym.P [x1,...xn,y1,...ym] B1 ... Bm where P is quantier-free. Decidable classes based on syn- which add inferred formul B1,...,Bm to the set of tactic restrictions are surveyed in [21]. known theorems, that already includes the premises Case 3 includes Presburger arithmetic or theories of data A1,...,An, contemporary inference systems feature con- structures, such as lists or arrays. For the latter, the typical traction rules, that delete or simplify already-inferred the- approach is to build a little engine of proof for each the- orems. The double-ruled inference rule form ory [66], by building the theorys axioms into a congruence A1 ... An closure algorithm to handle ground equalities [67, 54, 9]. (2) B1 ... Bm Little engines are combined to handle combinations of the- ories [53, 68, 31]. However, also generic theorem-proving means that the formul (Ai) above the rule are replaced by methods proved competitive on these problems [5]. those below (Bj). It is a deletion rule if the consequences Decidable does not mean pratical, and the decidable rea- are a proper subset of the premises; otherwise, it is a sim- soning problems are typically NP-complete. Since auto- plication rule. mated reasoning problems range from decidable, but NP- An expansion rule is sound if what is generated is complete,tosemi-decidable, or not even semi-decidable, logical consequence of the premises ({A1 ... An}|= automated reasoning relies pretty much universally on the {B1 ...Bm}). Classical examples are resolution and articial intelligence paradigm of search. paramodulation.Acontraction rule is sound if what is removed is logical consequence of what is left or added 2.3 Automated reasoning as a search problem ({B1 ...Bm}|= {A1 ... An}). Classical examples are subsumption and equational simplication from Knuth- Automated reasoning methods are strategies, composed Bendix completion.Aninference system is sound if all its of an inference system and a search plan. The inference rules are, and it is refutationally complete,ifitallows us to system is a non-deterministic set of inference rules, that derive a contradiction, whenever the initial set of formul denes a search space containing all possible inferences. is inconsistent. The challenge is dealing with contraction Describing formally the search space of a reasoning prob- without endangering completeness [36, 7, 8, 18]: a key lem is not obvious, and can be approached through differ- ingredient is to order the data (terms, literals, clauses, for- ent formalisms that capture different levels of abstraction mul, proofs) according to well-founded orderings. Infer- [62, 19]. The search plan guides the search and determines ence systems of this nature were applied successfully also the unique derivation to inductive theorem proving as in inductionless induction or proof by the lack of inconsistency [37, 40, 18]. S0 S1 ...Si Si+1 ... 2.2 Decision procedures and SAT solvers that the strategy computes from a given input S0 = S ∪ {¬ϕ}.Itisthe addition of the search plan that turns a non- Decidable instances of reasoning problems do exist. For deterministic inference system into a deterministic proof these problems, the search space is nite and decision pro- procedure. cedures are known. Decidability may stem from imposing The search plan decides, at each step, which inference restrictions on rule to apply to which data. If it selects an expansion rule, Anno III, N° 1/2, Marzo-Giugno 2006 15 CONTRIBUTI SCIENTIFICI the set of formul is expanded: guage understanding, symbolic computation, such as con- straint problem solving and computer algebra, compu- S tational logic, such as declarative programming and de- S ⊂ S S ductive databases, and mathematics, as witnessed by the If it selects a contraction rule, the set of formul is con- existence of databases of computer-checked mathematics tracted: [51]. Theorem provers are capable of proving non-trivial S mathematical theorems in theories such as Boolean alge- S ⊆ S S ≺mul S S bras, rings, groups, quasigroups and many-valued logic [3, 2, 41, 49, 75]. Last, the study of mechanical forms where ≺mul is the multiset extension of a well-founded of logical reasoning is part of the fundamental quest about ordering on clauses.
Load more

Recommended publications
  • Deduction Systems Based on Resolution, We Limit the Following Considerations to Transition Systems Based on Analytic Calculi
    Author’s Address Norbert Eisinger European Computer–Industry Research Centre, ECRC Arabellastr. 17 D-8000 M¨unchen 81 F. R. Germany [email protected] and Hans J¨urgen Ohlbach Max–Planck–Institut f¨ur Informatik Im Stadtwald D-6600 Saarbr¨ucken 11 F. R. Germany [email protected] Publication Notes This report appears as chapter 4 in Dov Gabbay (ed.): ‘Handbook of Logic in Artificial Intelligence and Logic Programming, Volume I: Logical Foundations’. It will be published by Oxford University Press, 1992. Fragments of the material already appeared in chapter two of Bl¨asis & B¨urckert: Deduction Systems in Artificial Intelligence, Ellis Horwood Series in Artificial Intelligence, 1989. A draft version has been published as SEKI Report SR-90-12. The report is also published as an internal technical report of ECRC, Munich. Acknowledgements The writing of the chapters for the handbook has been a highly coordinated effort of all the people involved. We want to express our gratitude for their many helpful contributions, which unfortunately are impossible to list exhaustively. Special thanks for reading earlier drafts and giving us detailed feedback, go to our second reader, Bob Kowalski, and to Wolfgang Bibel, Elmar Eder, Melvin Fitting, Donald W. Loveland, David Plaisted, and J¨org Siekmann. Work on this chapter started when both of us were members of the Markgraf Karl group at the Universit¨at Kaiserslautern, Germany. With our former colleagues there we had countless fruitful discus- sions, which, again, cannot be credited in detail. During that time this research was supported by the “Sonderforschungsbereich 314, K¨unstliche Intelligenz” of the Deutsche Forschungsgemeinschaft (DFG).
    [Show full text]
  • Mutation COS 326 Speaker: Andrew Appel Princeton University
    Mutation COS 326 Speaker: Andrew Appel Princeton University slides copyright 2020 David Walker and Andrew Appel permission granted to reuse these slides for non-commercial educational purposes C structures are mutable, ML structures are immutable C program OCaml program let fst(x:int,y:int) = x struct foo {int x; int y} *p; let p: int*int = ... in int a,b,u; let a = fst p in a = p->x; let u = f p in u = f(p); let b = fst p in b = p->x; xxx (* does a==b? Yes! *) /* does a==b? maybe */ 2 Reasoning about Mutable State is Hard mutable set immutable set insert i s1; let s1 = insert i s0 in f x; f x; member i s1 member i s1 Is member i s1 == true? … – When s1 is mutable, one must look at f to determine if it modifies s1. – Worse, one must often solve the aliasing problem. – Worse, in a concurrent setting, one must look at every other function that any other thread may be executing to see if it modifies s1. 3 Thus far… We have considered the (almost) purely functional subset of OCaml. – We’ve had a few side effects: printing & raising exceptions. Two reasons for this emphasis: – Reasoning about functional code is easier. • Both formal reasoning – equationally, using the substitution model – and informal reasoning • Data structures are persistent. – They don’t change – we build new ones and let the garbage collector reclaim the unused old ones. • Hence, any invariant you prove true stays true. – e.g., 3 is a member of set S.
    [Show full text]
  • The Method of Variable Splitting
    The Method of Variable Splitting Roger Antonsen Thesis submitted to the Faculty of Mathematics and Natural Sciences at the University of Oslo for the degree of Philosophiae Doctor (PhD) Date of submission: June 30, 2008 Date of public defense: October 3, 2008 Supervisors Prof. Dr. Arild Waaler, University of Oslo Prof. Dr. Reiner H¨ahnle, Chalmers University of Technology Adjudication committee Prof. Dr. Matthias Baaz, Vienna University of Technology Prof. emer. Dr. Wolfgang Bibel, Darmstadt University of Technology Dr. Martin Giese, University of Oslo Copyright © 2008 Roger Antonsen til mine foreldre, Karin og Ingvald Table of Contents Acknowledgments vii Chapter 1 Introduction 1 1.1 Influential Ideas in Automated Reasoning2 1.2 Perspectives on Variable Splitting4 1.3 A Short History of Variable Splitting7 1.4 Delimitations and Applicability 10 1.5 Scientific Contribution 10 1.6 A Few Words of Introduction 11 1.7 Notational Conventions and Basics 12 1.8 The Contents of the Thesis 14 Chapter 2 A Tour of Rules and Inferences 15 2.1 Ground Sequent Calculus 16 2.2 Free-variable Sequent Calculus 17 2.3 Another Type of Redundancy 19 2.4 Variable-Sharing Calculi and Variable Splitting 20 Chapter 3 Preliminaries 23 3.1 Indexing 23 3.2 Indices and Indexed Formulas 24 3.3 The -relation 28 3.4 The Basic Variable-Sharing Calculus 28 3.5 Unifiers and Provability 30 3.6 Permutations 31 3.7 Conformity and Proof Invariance 35 3.8 Semantics 37 3.9 Soundness and Completeness 40 Chapter 4 Variable Splitting 41 4.1 Introductory Examples 41 4.2 Branch Names
    [Show full text]
  • Constraint Microkanren in the Clp Scheme
    CONSTRAINT MICROKANREN IN THE CLP SCHEME Jason Hemann Submitted to the faculty of the University Graduate School in partial fulfillment of the requirements for the degree Doctor of Philosophy in the Department of School of Informatics, Computing, and Engineering Indiana University January 2020 Accepted by the Graduate Faculty, Indiana University, in partial fulfillment of the require- ments for the degree of Doctor of Philosophy. Daniel P. Friedman, Ph.D. Amr Sabry, Ph.D. Sam Tobin-Hochstadt, Ph.D. Lawrence Moss, Ph.D. December 20, 2019 ii Copyright 2020 Jason Hemann ALL RIGHTS RESERVED iii To Mom and Dad. iv Acknowledgements I want to thank all my housemates and friends from 1017 over the years for their care and support. I’m so glad to have you all, and to have you all over the world. Who would have thought that an old house in Bloomington could beget so many great memories. While I’m thinking of it, thanks to Lisa Kamen and Bryan Rental for helping to keep a roof over our head for so many years. Me encantan mis salseros y salseras. I know what happens in the rueda stays in the rueda, so let me just say there’s nothing better for taking a break from right-brain activity. Thanks to Kosta Papanicolau for his early inspiration in math, critical thinking, and research, and to Profs. Mary Flagg and Michael Larsen for subsequent inspiration and training that helped prepare me for this work. Learning, eh?—who knew? I want to thank also my astounding undergraduate mentors including Profs.
    [Show full text]
  • Development of Logic Programming: What Went Wrong, What Was Done About It, and What It Might Mean for the Future
    Development of Logic Programming: What went wrong, What was done about it, and What it might mean for the future Carl Hewitt at alum.mit.edu try carlhewitt Abstract follows: A class of entities called terms is defined and a term is an expression. A sequence of expressions is an Logic Programming can be broadly defined as “using logic expression. These expressions are represented in the to deduce computational steps from existing propositions” machine by list structures [Newell and Simon 1957]. (although this is somewhat controversial). The focus of 2. Certain of these expressions may be regarded as this paper is on the development of this idea. declarative sentences in a certain logical system which Consequently, it does not treat any other associated topics will be analogous to a universal Post canonical system. related to Logic Programming such as constraints, The particular system chosen will depend on abduction, etc. programming considerations but will probably have a The idea has a long development that went through single rule of inference which will combine substitution many twists in which important questions turned out to for variables with modus ponens. The purpose of the combination is to avoid choking the machine with special have surprising answers including the following: cases of general propositions already deduced. Is computation reducible to logic? 3. There is an immediate deduction routine which when Are the laws of thought consistent? given a set of premises will deduce a set of immediate This paper describes what went wrong at various points, conclusions. Initially, the immediate deduction routine what was done about it, and what it might mean for the will simply write down all one-step consequences of the future of Logic Programming.
    [Show full text]
  • Informal Proceedings of the 30Th International Workshop on Unification (UNIF 2016)
    Silvio Ghilardi and Manfred Schmidt-Schauß (Eds.) Informal Proceedings of the 30th International Workshop on Unification (UNIF 2016) June 26, 2016 Porto, Portugal 1 Preface This volume contains the papers presented at UNIF 2016: The 30th International Workshop on Unification (UNIF 2016) held on June 26, 2016 in Porto. There were 10 submissions. Each submission was reviewed by at least 3, and on the average 3.3, program committee members. The committee decided to accept 8 papers for publication in these Proceedings. The program also includes 2 invited talks. The International Workshop on Unification was initiated in 1987 as a yearly forum for researchers in unification theory and related fields to meet old and new colleagues, to present recent (even unfinished) work, and to discuss new ideas and trends. It is also a good opportunity for young researchers and researchers working in related areas to get an overview of the current state of the art in unification theory. The list of previous meetings can be found at the UNIF web page: http://www.pps.univ-paris-diderot.fr/~treinen/unif/. Typically, the topics of interest include (but are not restricted to): Unification algorithms, calculi and implementations • Equational unification and unification modulo theories • Unification in modal, temporal and description logics • Admissibility of Inference Rules • Narrowing • Matching algorithms • Constraint solving • Combination problems • Disunification • Higher-Order Unification • Type checking and reconstruction • Typed unification • Complexity issues • Query answering • Implementation techniques • Applications of unification • Antiunification/Generalization • This years UNIF is a satellite event of the first International Conference on Formal Structures for Computation and Deduction (FSCD).
    [Show full text]
  • Automated Reasoning
    Automated Reasoning Maria Paola Bonacina Alberto Martelli Dipartimento di Informatica Dipartimento di Informatica Universita` degli Studi di Verona Universita` degli Studi di Torino A central problem in automated reasoning is to deter- automated model building. mine whether a conjecture ϕ, that represents a property to In classical first-order logic, deductive theorem proving be verified, is a logical consequence of a set S of assump- is semi-decidable, while inductive theorem proving and tions, which express properties of the object of study (e.g., model building are not even semi-decidable. It is signif- a system, a circuit, a program, a data type, a communica- icant that while books in theorem proving date from the tion protocol, a mathematical structure). early seventies [22, 48, 16, 27, 77, 44, 70], the first book A conjoint problem is that of knowledge representation, on model building appeared only recently [21]. Most ap- or finding suitable formalisms for S and ϕ to represent as- proaches to automated model building belong to one of the pects of the real world, such as action, space, time, men- following three classes, or combine their principles: tal events and commonsense reasoning. While classical logic has been the principal formalism in automated rea- 1. Enumeration methods generate interpretations and soning, and many proof techniques have been studied and test whether they are models of the given set of for- implemented, non-classical logics, such as modal, tempo- mulæ; ral, description or nonmonotonic logics, have been widely 2. Saturation methods extract models from the finite set investigated to represent knowledge.
    [Show full text]
  • Computational Logic: Memories of the Past and Challenges for the Future
    Computational Logic: Memories of the Past and Challenges for the Future John Alan Robinson Highland Institute, 96 Highland Avenue, Greenfield, MA 01301, USA. Abstract. The development of computational logic since the introduction of Frege's modern logic in 1879 is presented in some detail. The rapid growth of the field and its proliferation into a wide variety of subfields is noted and is attributed to a proliferation of subject matter rather than to a proliferation of logic itself. Logic is stable and universal, and is identified with classical first order logic. Other logics are here considered to be first order theories, syntactically sugared in notationally convenient forms. From this point of view higher order logic is essentially first order set theory. The paper ends by presenting several challenging problems which the computational logic community now faces and whose solution will shape the future of the field. 1 Introduction Although logic and computing are each very old subjects, going back very much more than a mere hundred years, it is only during the past century that they have merged and become essentially one subject. As the logican Quine wrote in 1960, "The basic notions of proof theory converge with those of machine computing. ... The utterly pure theory of mathematical proof and the utterly technological theory of machine computation are thus at bottom one, and the basic insights of each are henceforth insights of the other" ([1], p. 41). The aptly-named subject of computational logic - no pun intended - subsumes a wide variety of interests and research activities. All have something to do with both logic and computation.
    [Show full text]
  • After Math: (Re)Configuring Minds, Proof, and Computing in the Postwar United States
    After Math: (Re)configuring Minds, Proof, and Computing in the Postwar United States The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Dick, Stephanie Aleen. 2015. After Math: (Re)configuring Minds, Proof, and Computing in the Postwar United States. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:14226096 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA After Math (Re)configuring Minds, Proof, and Computing in the Postwar United States Adissertationpresented by Stephanie Aleen Dick to The Department of the History of Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of the History of Science Harvard University Cambridge, Massachusetts November 2014 © 2014 Stephanie Aleen Dick. All rights reserved. Dissertation Advisor: Professor Peter Galison Stephanie Aleen Dick After Math (Re)configuring Minds, Proof, and Computing in the Postwar United States Abstract This dissertation examines the history of three early computer programs that were designed to prove mathematical theorems: The Logic Theory Machine, the Program P, and the Automated Reasoning Assistant, all developed between 1955 and 1975. I use these programs as an opportunity to explore ways in which mathematical knowledge and practice were transformed by the introduction of modern computing. The prospect of automation generated disagreement about the character of human mathematical faculties like intuition, reasoning, and understanding and whether computers could be made to possess or participate in them.
    [Show full text]
  • Extensional Paramodulation for Higher-Order
    Freie Universität Berlin Institut für Informatik Extensional Paramodulation for Higher-Order Logic and its Effective Implementation Leo-III – Preprint – See also the published version by AKA Verlag, as part of the DISKI series, volume 345, EAN/ISBN: 978-3-89838-739-2 ALEXANDER STEEN Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat) am Fachbereich Mathematik und Informatik der Freien Universität Berlin 2018 BERLIN,DEUTSCHLAND Title Extensional Paramodulation for Higher-Order Logic and its Effective Implementation Leo-III Author Alexander Steen School Freie Universität Berlin, Berlin, Germany Supervisor Prof. Dr. habil. Christoph Benzmüller Second Examiner Prof. Dr. Geoff Sutcliffe (University of Miami) This dissertation was submitted to the Department of Mathematics and Computer Science of Freie Universität Berlin on 07.06.2018 and defended on 11.07.2018. The doctoral commission consisted of Prof. Christoph Benzmüller (FU Berlin, Head of commission), Prof. Raul Rojas (FU Berlin), Prof. Geoff Sutcliffe (Uni- versity of Miami) and Dr. Klaus Kriegel (FU Berlin). The dissertation project was conducted within the ”Leo-III” project funded by the German Research Foundation (DFG) under grant BE 2501/11-1, and the project ”Consistent Rational Argumentation in Politics” funded by the Volkswagens- tiftung. All source code related to work presented in this thesis is publicly avail- able (under BSD-3 license) at https://github.com/leoprover/Leo-III. Supplemental material is available at https://alexandersteen.de/phd/. v Abstract Abstract In this thesis the theoretical foundations and the practical components for imple- menting an effective automated theorem proving system for higher-order logic are presented.
    [Show full text]
  • Learning Theorem Proving Components
    Learning Theorem Proving Components Karel Chvalovsk´y1[0000−0002−0541−3889], Jan Jakub˚uv1;2[0000−0002−8848−5537], Miroslav Olˇs´ak2[0000−0002−9361−1921], and Josef Urban1[0000−0002−1384−1613] 1 Czech Technical University in Prague, Prague, Czechia [email protected], [email protected] 2 University of Innsbruck, Innsbruck, Austria [email protected], [email protected] Abstract. Saturation-style automated theorem provers (ATPs) based on the given clause procedure are today the strongest general reasoners for classical first-order logic. The clause selection heuristics in such sys- tems are, however, often evaluating clauses in isolation, ignoring other clauses. This has changed recently by equipping the E/ENIGMA system with a graph neural network (GNN) that chooses the next given clause based on its evaluation in the context of previously selected clauses. In this work, we describe several algorithms and experiments with ENIGMA, advancing the idea of contextual evaluation based on learning important components of the graph of clauses. Keywords: Automated theorem proving · Machine Learning · Neural Networks · Decision Trees · Saturation-Style Proving 1 Introduction: Clause Selection and Context Clause selection is a crucial part of saturation-style [29] automated theorem provers (ATPs) such as E [32], Vampire [20], and Prover9 [22]. These systems, implementing the given-clause [21] algorithm, provide the strongest methods for proving lemmas in large interactive theorem prover (ITP) libraries [4], and occasionally prove open conjectures in specialized parts of mathematics [19]. Clause selection heuristics have a long history of research, going back to a number of experiments done with the Otter system [24].
    [Show full text]
  • History of Logic Programming
    LOGIC PROGRAMMING Robert Kowalski 1 INTRODUCTION The driving force behind logic programming is the idea that a single formalism suffices for both logic and computation, and that logic subsumes computation. But logic, as this series of volumes proves, is a broad church, with many denomi- nations and communities, coexisting in varying degrees of harmony. Computing is, similarly, made up of many competing approaches and divided into largely disjoint areas, such as programming, databases, and artificial intelligence. On the surface, it might seem that both logic and computing suffer from a similar lack of cohesion. But logic is in better shape, with well-understood relationships between different formalisms. For example, first-order logic extends propositional logic, higher-order logic extends first-order logic, and modal logic extends classical logic. In contrast, in Computing, there is hardly any relationship between, for example, Turing machines as a model of computation and relational algebra as a model of database queries. Logic programming aims to remedy this deficiency and to unify different areas of computing by exploiting the greater generality of logic. It does so by building upon and extending one of the simplest, yet most powerful logics imaginable, namely the logic of Horn clauses. In this paper, which extends a shorter history of logic programming (LP) in the 1970s [Kowalski, 2013], I present a personal view of the history of LP, focusing on logical, rather than on technological issues. I assume that the reader has some background in logic, but not necessarily in LP. As a consequence, this paper might also serve a secondary function, as a survey of some of the main developments in the logic of LP.
    [Show full text]