Constraint Microkanren in the Clp Scheme
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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). -
Typed Embedding of a Relational Language in Ocaml
Typed Embedding of a Relational Language in OCaml Dmitrii Kosarev Dmitry Boulytchev Saint Petersburg State University Saint Petersburg State University Saint Petersburg, Russia Saint Petersburg, Russia [email protected] [email protected] We present an implementation of the relational programming language miniKanren as a set of combi- nators and syntax extensions for OCaml. The key feature of our approach is polymorphic unification, which can be used to unify data structures of arbitrary types. In addition we provide a useful generic programming pattern to systematically develop relational specifications in a typed manner, and ad- dress the problem of integration of relational subsystems into functional applications. 1 Introduction Relational programming [11] is an attractive technique, based on the idea of constructing programs as relations. As a result, relational programs can be “queried” in various “directions”, making it possible, for example, to simulate reversed execution. Apart from being interesting from a purely theoretical standpoint, this approach may have a practical value: some problems look much simpler when considered as queries to some relational specification [5]. There are a number of appealing examples confirming this observation: a type checker for simply typed lambda calculus (and, at the same time, a type inferencer and solver for the inhabitation problem), an interpreter (capable of generating “quines” — programs producing themselves as a result) [7], list sorting (capable of producing all permutations), etc. Many logic programming languages, such as Prolog, Mercury [21], or Curry [13] to some extent can be considered relational. We have chosen miniKanren1 as a model language, because it was specifically designed as a relational DSL, embedded in Scheme/Racket. -
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. -
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 -
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. -
CMSC 330: Organization of Programming Languages
CMSC 330: Organization of Programming Languages Administrivia CMSC330 Spring 2020 1 Course Goals Understand why there are so many languages Describe and compare their main features Choose the right language for the job Write better code • Code that is shorter, more efficient, with fewer bugs In short: • Become a better programmer with a better understanding of your tools. CMSC330 Spring 2020 2 Course Activities Learn different types of languages Learn different language features and tradeoffs • Programming patterns repeat between languages Study how languages are specified • Syntax, Semantics — mathematical formalisms Study how languages are implemented • Parsing via regular expressions (automata theory) and context free grammars • Mechanisms such as closures, tail recursion, lazy evaluation, garbage collection, … Language impact on computer security CMSC330 Spring 2020 3 Syllabus Dynamic/ Scripting languages (Ruby) Functional programming (OCaml) Regular expressions & finite automata Context-free grammars & parsing Lambda Calculus Safe, “zero-cost abstraction” programming (Rust) Secure programming Scoping, type systems, parameter passing, Comparing language styles; other topics CMSC330 Spring 2020 4 Calendar / Course Overview Tests • 4 quizzes, 2 midterm exams, 1 final exam • Do not schedule your interviews on exam dates Clicker quizzes • In class, graded, during the lectures Projects • Project 1 – Ruby • Project 2-5 – OCaml (and parsing, automata) Ø P2 and P4 are split in two parts • Project 6 – Security CMSC330 Spring 2020 5 Clickers Turning Technology subscription is free. Physical clicker is preferred. • Clicker device: any of LCD, NXT, or QT2 models • Phone App: needs wifi CMSC330 Spring 2020 6 Quiz time! According to IEEE Spectrum Magazine which is the “top” programming language of 2018? A. -
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). -
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. -
Neural Guided Constraint Logic Programming for Program Synthesis
Neural Guided Constraint Logic Programming for Program Synthesis Lisa Zhang1,2, Gregory Rosenblatt4, Ethan Fetaya1,2, Renjie Liao1,2,3, William E. Byrd4, Matthew Might4, Raquel Urtasun1,2,3, Richard Zemel1,2 1University of Toronto, 2Vector Institute, 3Uber ATG, 4University of Alabama at Birmingham 1{lczhang,ethanf,rjliao,urtasun,zemel}@cs.toronto.edu 4{gregr,webyrd,might}@uab.edu Abstract Synthesizing programs using example input/outputs is a classic problem in artificial intelligence. We present a method for solving Programming By Example (PBE) problems by using a neural model to guide the search of a constraint logic program- ming system called miniKanren. Crucially, the neural model uses miniKanren’s internal representation as input; miniKanren represents a PBE problem as recursive constraints imposed by the provided examples. We explore Recurrent Neural Net- work and Graph Neural Network models. We contribute a modified miniKanren, drivable by an external agent, available at https://github.com/xuexue/neuralkanren. We show that our neural-guided approach using constraints can synthesize pro- grams faster in many cases, and importantly, can generalize to larger problems. 1 Introduction Program synthesis is a classic area of artificial intelligence that has captured the imagination of many computer scientists. Programming by Example (PBE) is one way to formulate program synthesis problems, where example input/output pairs specify a target program. In a sense, supervised learning can be considered program synthesis, but supervised learning via successful models like deep neural networks famously lacks interpretability. The clear interpretability of programs as code means that synthesized results can be compared, optimized, translated, and proved correct. -
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. -
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. -
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.