Algorithm = Logic + Control Robert Kowalski Imperial College, London

Total Page:16

File Type:pdf, Size:1020Kb

Algorithm = Logic + Control Robert Kowalski Imperial College, London S. 1 Programming J. J. Horning Languages Editor Algorithm = Logic + Control Robert Kowalski Imperial College, London An algorithm can be regarded as consisting of a . logic component, which specifies the knowledge to be used in solving problems, and a control component, which determines the problem-solving strategies by means of which that knowledge is used. The logic component determines the meaning of the algorithm whereas the control component only affects its effkiency. The effkiency of an algorithm can often be improved by improving the control component without changing the logic of the algorithm. We argue that computer programs would be more often correct and more easily improved and modified if their logic and control aspects were identified and separated in the program text. Key Words and Phrases: control language, logic programming, nonprocedural language, programming methodology, program specification, relational data structures CR Categories: 3.64, 4.20, 4.30, 5.21, 5.24 Introduction Predicate logic is a high level, human-oriented lanl guage for describing problems and problem-solving methods to computers. In this paper, we are concerned not with the use of predicate logic as a programming language in its own right, but with its use as a tool for the analysis of algorithms. Our main aim will be to study ways in which logical analysis can contribute to improv- ing the structure and efficiency of algorithms. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct Eommer&al advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. This research was supported- - by- a grant from the Science Research Council. Author’s address: R.A. Kowalski, Dept. of Computing and Con* trol, Imperial College of Science and Technology, 180 Queens Gate, London SW7 2BZ, England. 0 1979 ACM OOOl-0782/79/0700-0424 $00.75. Communications July 1979 of Volume 22 the ACM Number 7 The notion that computation = controlled deduction presses the knowledge which can be used in solving was first proposed by Pay Hayes [19] and more recently problems and the control component determines the way by Bibel [2] and Vaughn-Pratt [31]. A similar thesis that in which that knowledge can be used. The distinction database systems should be regarded as consisting of a between logic and control is not wholly unambiguous. relational component, which defines the logic of the data, The same algorithm A can often be analyzed in different and a control component, which stores and retrieves it, ways. has been successfully argued by Codd [10]. Hewitt's A = L~ + C~. argument [20] for the programming language PLAN- A = Lz + C2. NER, though generally regarded as an argument against logic, can also be regarded as an argument for the thesis One analysis might include in the logic component what that algorithms be regarded as consisting of both logic another analysis includes in the control component. In and control components. In this paper we shall explore general, we prefer an analysis which places the greatest some of the useful consequences of that thesis. burden for achieving efficiency on the control compo- We represent the analysis of an algorithm A into a nent. Such an analysis has two advantages: (1) the logic logic component L, which defines the logic of the algo- component can afford to be a clearer and more obviously rithm, and a control component C, which specifies the correct statement of the problem and the knowledge manner in which the definitions are used, symbolically which can be used in its solution and (2) the control by the equation component assumes greater responsibility for the effi- ciency of the algorithm, which consequently can be more A=L+C. readily improved by upgrading the efficiency of the control. Algorithms for computing factorials are a simple exam- It is the intention that this paper should be self- ple. The definition of factorial constitutes the logic com- contained. The first part, accordingly, introduces the ponent of the algorithms: clausal form of predicate logic and defines the top-down 1 is the factorial of 0; and bottom-up interpretations of Horn clauses. The body u is the factorial of x + 1 ~ v is the factorial of x and u is v times of the paper investigates the following decomposition of x+l. algorithms into their various components. The definition can be used bottom-up to derive a se- quence of assertions about factorial or it can be used top- down to reduce the problem of computing the factorial cLOgic ./ //•ithm\ of x + 1 to the subproblems of computing the factorial of x and multiplying the result by x + 1. Different ways of using the same definition give rise to different algo- Abstract Oefinitk)ns of ~,~Control component rithms. Bottom-up use of the definition behaves like procedure data -structures definitions represented by // ~ iteration. Top-down use behaves like recursive evalua- tion. Direction Strategy for Strategy for Scheme for (e.g. top-down execution of investigating storing data The manner in which the logic component is used to or bottorn-up) i)rocedure alternative represented calls (e.g. procedures relationally solve problems constitutes the control component. As a sequential or first approximation, we restrict the control component C parallel ) to general-purpose problem-solving strategies which do We study the affect of altering each of the above com- not affect the meaning of the algorithm as it is deter- ponents of an algorithm. The final section of the paper mined by the logic component L. Thus different algo- introduces a graphical notation for expressing, more rithms A1 and Az, obtained by applying different meth- formally than in the rest of the paper, certain kinds of ods of control Cx and C2 to the same logic definitions L, control information. Much of the material in this paper are equivalent in the sense that they solve the same has been abstracted from lecture notes [23] prepared for problems with the same results. Symbolically, if A1 = the advanced summer school on foundations of comput- L + C1 and Az = L + C2, then A1 and Az are equivalent. ing held at the Mathematical Centre in Amsterdam in The relationship of equivalence between algorithms, be- May 1974. cause they have the same logic, is the basis for using logical analysis to improve the efficiency of an algorithm Notation by retaining its logic but improving the way it is used. In particular, replacing bottom-up by top-down control We use the clausal form of predicate logic. Simple often (but not always) improves efficiency, whereas re- assertions are expressed by clauses: placing top-down sequential solution of subproblems by Father (Zeus, Ares) ~-- top-down parallel solution seems never to decrease effi- Mother (Hera, Ares) Father (Ares, Harmonia) ~- ciency. Mother (Semele, Dionisius) Both the logic and the control components of an Father (Zeus, Dionisius) algorithm affect efficiency. The logic component ex- etc. 425 Communications July 1979 of Volume 22 the ACM Number 7 Here Father (x, y) states that x is the father of y and .... An are conditions of the clause and the atoms B1 .... , Mother (x, y) states that x is the mother ofy. Bm are alternative conclusions of the clause. If the clause Clauses can also express general conditional propo- contains the variables Xl ..... Xk then interpret it as stating sitions: that Female (x) *- Mother (x, y) for all xl ..... xk Male (x) ,--- Father (x, y) B1 or ... or Bm if A1 and ... and An. Parent (x, y) ~ Mother (x, y) Parent (x, y) ~-- Father (x, y). If n = 0, then interpret it as stating unconditionally that for all x~, .... xk These state that B~ or ... or Bin. x is female/fx is mother of y, If m = 0, then interpret it as stating that x is male/fx is father of y, x is parent ofy tfx is mother of y, and for no x~, ..., xk x is parent ofy tfx is father ofy. A ~ and ... and A n. The arrow ~ is the logical connective "if"; "x" and "y" If m = n = O, then interpret the clause as a sentence are variables representing any individuals; "Zeus," which is always false. "Ares," etc. are constant symbols representing particular An atom (or atomic formula) is an expression of the individuals; "Father," "Mother," "Female," etc. are form predicate symbols representing relations among individ- P (h ..... tn) uals. Variables in different clauses are distinct even if they have the same names. where P is an n-place predicate symbol and t~, ..., t,, are A clause can have several joint conditions or several terms. Interpret the atom as asserting that the relation alternative conclusions. Thus called P holds among the individuals called t~..... tn. A term is a variable, a constant symbol, or an expres- Grandparent (x, y) ,--- Parent (x, z), Parent (z, y) Male (x), Female (x) ~-- Parent (x, y) sion of the form Ancestor (x, y) ,--- Parent (x, y) f (h ..... tn) Ancestor (x, y) ~ Ancestor (x, z), Ancestor (z, y) where f is an n-place function symbol and tl ..... tn are where x, y, and z are variables, state that for all x, y, and terms. g The sets of predicate symbols, function symbols, con- x is grandparent ofy/f x is parent of z and z is parent of y; stant symbols, and variables are any mutually disjoint x is male or x is female tfx is parent of y; sets.
Recommended publications
  • By Robert Kowalski
    Edinburgh Research Explorer Review of "Computational logic and human thinking: How to be artificially intelligent" by Robert Kowalski Citation for published version: Bundy, A 2012, 'Review of "Computational logic and human thinking: How to be artificially intelligent" by Robert Kowalski', Artificial Intelligence, vol. 191-192, pp. 96-97. https://doi.org/10.1016/j.artint.2012.05.006 Digital Object Identifier (DOI): 10.1016/j.artint.2012.05.006 Link: Link to publication record in Edinburgh Research Explorer Document Version: Early version, also known as pre-print Published In: Artificial Intelligence General rights Copyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s) and / or other copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated with these rights. Take down policy The University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorer content complies with UK legislation. If you believe that the public display of this file breaches copyright please contact [email protected] providing details, and we will remove access to the work immediately and investigate your claim. Download date: 27. Sep. 2021 Review of \Computational Logic and Human Thinking: How to be Artificially Intelligent" by Robert Kowalski Alan Bundy School of Informatics, University of Edinburgh, Informatics Forum, 10 Crichton St, Edinburgh, EH8 9AB, Scotland 1 August 2012 Abstract This is a review of the book \Computational Logic and Human Thinking: How to be Artificially Intelligent" by Robert Kowalski. Keywords: computational logic, human thinking, book review.
    [Show full text]
  • 7. Logic Programming in Prolog
    Artificial Intelligence 7. Logic Programming in Prolog Prof. Bojana Dalbelo Baˇsi´c Assoc. Prof. Jan Snajderˇ University of Zagreb Faculty of Electrical Engineering and Computing Academic Year 2019/2020 Creative Commons Attribution{NonCommercial{NoDerivs 3.0 v1.10 Dalbelo Baˇsi´c, Snajderˇ (UNIZG FER) AI { Logic programming AY 2019/2020 1 / 38 Outline 1 Logic programming and Prolog 2 Inference on Horn clauses 3 Programming in Prolog 4 Non-declarative aspects of Prolog Dalbelo Baˇsi´c, Snajderˇ (UNIZG FER) AI { Logic programming AY 2019/2020 2 / 38 Outline 1 Logic programming and Prolog 2 Inference on Horn clauses 3 Programming in Prolog 4 Non-declarative aspects of Prolog Dalbelo Baˇsi´c, Snajderˇ (UNIZG FER) AI { Logic programming AY 2019/2020 3 / 38 Logic programming Logic programming: use of logic inference as a way of programming Main idea: define the problem in terms of logic formulae, then let the computer do the problem solving (program execution = inference) This is a typical declarative programming approach: express the logic of computation, don't bother with the control flow We focus on the description of the problem (declarative), rather than on how the program is executed (procedural) However, we still need some flow control mechanism, thus: Algorithm = Logic + Control Different from automated theorem proving because: 1 explicit control flow is hard-wired into the program 2 not the full expressivity of FOL is supported Dalbelo Baˇsi´c, Snajderˇ (UNIZG FER) AI { Logic programming AY 2019/2020 4 / 38 Refresher: Declarative programming
    [Show full text]
  • Inconsistency Robustness for Logic Programs Carl Hewitt
    Inconsistency Robustness for Logic Programs Carl Hewitt To cite this version: Carl Hewitt. Inconsistency Robustness for Logic Programs. Inconsistency Robustness, 2015, 978-1- 84890-159. hal-01148496v6 HAL Id: hal-01148496 https://hal.archives-ouvertes.fr/hal-01148496v6 Submitted on 27 Apr 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Copyright Inconsistency Robustness for Logic Programs Carl Hewitt This article is dedicated to Alonzo Church and Stanisław Jaśkowski Abstract This article explores the role of Inconsistency Robustness in the history and theory of Logic Programs. Inconsistency Robustness has been a continually recurring issue in Logic Programs from the beginning including Church's system developed in the early 1930s based on partial functions (defined in the lambda calculus) that he thought would allow development of a general logic without the kind of paradoxes that had plagued earlier efforts by Frege, etc.1 Planner [Hewitt 1969, 1971] was a kind of hybrid between the procedural and logical paradigms in that it featured a procedural embedding of logical sentences in that an implication of the form (p implies q) can be procedurally embedded in the following ways: Forward chaining When asserted p, Assert q When asserted q, Assert p Backward chaining When goal q, SetGoal p When goal p, SetGoal q Developments by different research groups in the fall of 1972 gave rise to a controversy over Logic Programs that persists to this day in the form of following alternatives: 1.
    [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]
  • Logic for Problem Solving
    LOGIC FOR PROBLEM SOLVING Robert Kowalski Imperial College London 12 October 2014 PREFACE It has been fascinating to reflect and comment on the way the topics addressed in this book have developed over the past 40 years or so, since the first version of the book appeared as lecture notes in 1974. Many of the developments have had an impact, not only in computing, but also more widely in such fields as math- ematical, philosophical and informal logic. But just as interesting (and perhaps more important) some of these developments have taken different directions from the ones that I anticipated at the time. I have organised my comments, chapter by chapter, at the end of the book, leaving the 1979 text intact. This should make it easier to compare the state of the subject in 1979 with the subsequent developments that I refer to in my commentary. Although the main purpose of the commentary is to look back over the past 40 years or so, I hope that these reflections will also help to point the way to future directions of work. In particular, I remain confident that the logic-based approach to knowledge representation and problem solving that is the topic of this book will continue to contribute to the improvement of both computer and human intelligence for a considerable time into the future. SUMMARY When I was writing this book, I was clear about the problem-solving interpretation of Horn clauses, and I knew that Horn clauses needed to be extended. However, I did not know what extensions would be most important.
    [Show full text]
  • Copyright by Amelia J. Harrison 2017
    Copyright by Amelia J. Harrison 2017 The Dissertation Committee for Amelia J. Harrison certifies that this is the approved version of the following dissertation: Formal Methods for Answer Set Programming Committee: Vladimir Lifschitz, Supervisor Robert S. Boyer Isil Dillig Warren A. Hunt, Jr. Torsten Schaub Formal Methods for Answer Set Programming by Amelia J. Harrison Dissertation Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy The University of Texas at Austin December 2017 To Grandmother Betsy Acknowledgments It’s not an exaggeration to say that were it not for many people other than myself this document would not exist. This is by no means an exhaustive list. I’ve had the opportunity to collaborate with many wonderful and talented colleagues. Among them are Daniel Bailey, Martin Gebser, Yuliya Lierler, Roland Kaminski, Julian Michael, David Pearce, Dhananjay Raju, Mirek Truszczynski, Agustin Valverde, and Fangkai Yang. Without friends to help maintain balance, it’s entirely unclear if I would have been able to see this through. Backyard barbecues and Game of Thrones viewing parties have been essential. I feel especially lucky to have met early on, and shared much of this journey with Hannah Alpert-Abrams and Kate Mesh. My family has been a constant source of reassurance and has helped me nav- igate many of the inevitable bouts of self-doubt that arise in the process of a Ph.D. This includes my parents, Jack and Betsy Harrison, who have always unequivocally encouraged me to find my own path.
    [Show full text]
  • CSCI 334: Principles of Programming Languages Lecture 21: Logic
    Announcements CSCI 334: Principles of Programming Languages Be a TA! Lecture 21: Logic Programming Instructor: Dan Barowy Announcements Logic Programming • Logic programming began as a collaboration between AI researchers (e.g., John McCarthy) and logicians (e.g., John Alan Robinson) to solve problems in artificial intelligence. • Cordell Green built the first “question and answer” system using Robinson’s “unification algorithm,” demonstrating that it was practical to prove theorems automatically. Prolog Declarative Programming • Alain Colmerauer and Phillippe Roussel at Aix- Marseille University invented Prolog in 1972. • Declarative programming is a very different style of programming than • They were inspired by a particular formulation of you have seen to this point. logic, called “Horn clauses,” popularized by the • Mostly, you have seen imperative programs. logician Robert Kowalski. • In imperative-style programming, the programmer instructs the • Horn clauses have a “procedural interpretation,” computer how to compute the desired result. meaning that they suggest a simple procedure for • In declarative-style programming, the computer already knows how solving them, called “resolution.” to compute results. • John Alan Robinson’s unification algorithm is an • Instead, the programmer asks the computer what to compute. efficient algorithm for doing resolution, and this is essentially the algorithm used by Prolog. Declarative Programming Prolog • The goal of AI is to enable a computer to answer declarative queries. • Most of you have probably been CS majors for long enough that we • I.e., it already knows how to answer you. have sufficiently damaged your brain such that you do not recognize • Prolog was an attempt to solve this problem. the difference between these two concepts.
    [Show full text]
  • Database Updates in the Event Calculus
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector J. LOGIC PROGRAMMING 1992:12:121-146 121 DATABASE UPDATES IN THE EVENT CALCULUS ROBERT KOWALSKI D This paper investigates a special case of the event calculus, concerned with database updates. It discusses the way relational databases, historical databases, modal logic, the situation calculus, and case semantics deal with database updates and compares the event calculus with the situation calculus in detail. It argues that the event calculus can overcome the computational aspects of the frame problem in the situation calculus and that it can be implemented with an efficiency approaching that of destruc- tive assignment in relational databases. a INTRODUCTION The event calculus [91 was developed as a theory for reasoning about events in a logic-programming framework. It is based in part on the situation calculus [13,14], but focuses on the concept of event as highlighted in semantic network representa- tions of case semantics. Its main intended application is the representation of events in database updates and discourse representation. It is closely related to Allen’s interval temporal logic [l, 21 and Lee-Coelho-Cotta’s treatment of time in deductive databases [ll]. The relationship between the event calculus and the systems of Allen and Lee-Coelho-Cotta has been investigated by Sadri [15]. The event calculus is concerned with formalizing the effect of events on objects, their properties, and their relationships. Thus, for example, given a description of an event in which Bob gives Book1 to John, Address correspondence to Robert Kowalski, Department of Computing, Imperial College of Science and Technology, 180 Queen’s Gate, London SW7 2BZ, England.
    [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]
  • Imperial College Computing Student Workshop
    Andrew V. Jones (Ed.) Imperial College Computing Student Workshop Proceedings of ICCSW ‘11 September 29th–30th 2011, London UK Volume Editor Andrew V. Jones Verification of Autonomous Systems Group Department of Computing Imperial College London 180 Queen’s Gate, London, SW7 2AZ United Kingdom Email: [email protected] Published a technical report of the Department of Computing, Imperial College London. Technical Report Number: DTR11-9. Cite as: @proceedings{iccsw_2011, editor = {Andrew V. Jones}, title = {Imperial College Computing Student Workshop, 1st Student Workshop, ICCSW ‘11, London, UK, September 29-30, 2011. Proceedings}, booktitle = {ICCSW ‘11}, publisher = {Imperial College London}, series = {Department of Computing Technical Report}, volume = {DTR11-9}, year = {2011} } The copyright of the contents of this volume remains that of the original author(s). Preface This volume contains the proceedings of the inaugural Imperial College Student Workshop (ICCSW ‘11). The workshop took place on 29th–30th September 2011 in London UK. The event was hosted by Imperial College London. ICCSW ‘11 aimed to be a truly student-run workshop: all of the organisation was led by students and the majority of the submitting authors acted as reviewers for other papers. Both the quality and number of the submissions, as well as the high standard of reviews, exceeded the committee’s initial expectations. These proceedings contain 16 original contributions in various fields from across computer science, including both theoretical and applied papers. Both days of the workshop featured a keynote talk on a facet of computer science. The talks were titled: – Artificial Intelligence and Human Thinking, by Robert Kowalski (Imperial College London); and – Building and Operating Products at Google-scale, by Ollie Cook (Google Inc.) The workshop also featured two discussion panels.
    [Show full text]
  • Published in Arxiv
    Published in ArXiv http://arxiv.org/abs/0904.3036 Middle History of Logic Programming Resolution, Planner, Edinburg LCF, Prolog, Simula, and the Japanese Fifth Generation Project Carl Hewitt ©2013 http://carlhewitt.info This paper is dedicated to Alonzo Church and Stanislaw Jaśkowski. Logic Programming can be broadly defined as “using logic to infer computational steps from existing propositions” However, mathematical logic cannot always infer computational steps because computational systems make use of arbitration for determining which message is processed next by a recipient that is sent multiple messages concurrently. Since arrival orders are in general indeterminate, they cannot be inferred from prior information by mathematical logic alone. Therefore mathematical logic cannot in general implement computation. This conclusion is contrary to Robert Kowalski who stated “Looking back on our early discoveries, I value most the discovery that computation could be subsumed by deduction.” Nevertheless, logic programming (like functional programming) can be a useful programming idiom. Over the course of history, the term “functional programming” has grown more precise and technical as the field has matured. Logic Programming should be on a similar trajectory. Accordingly, “Logic Programming” should have a general precise characterization. Kowalski's approach has been to advocate limiting Logic Programming to backward-chaining only inference based on resolution. In contrast, our approach is explore Logic Programming using inconsistency-robust Natural Deduction that is contrary to requiring (usually undesirable) reduction to conjunctive normal form in order to use resolution. Because contemporary large software systems are pervasively inconsistent, it is not safe to reason about them using classical logic, e.g., using resolution theorem proving.
    [Show full text]
  • Logic Programming Robert Kowalski MIT Encyclopaedia of Cognitive
    Logic Programming Robert Kowalski MIT Encyclopaedia of Cognitive Science (eds. R A Wilson and F C Keil) MIT Press, 1999, pp. 484-486. Logic programming is the use of logic to represent programs and of deduction to execute programs in logical form. To this end, many different forms of logic and many varieties of deduction have been investigated. The simplest form of logic, which is the basis of most logic programming, is the Horn clause subset of logic, where programs consist of sets of implications: A0 if A1 and A2 and ... and An. Here each Ai is a simple (i.e. atomic) sentence. It is usual to write such implications backward. This is because deduction by backward reasoning interprets them as procedures: to solve the goal A0, solve the subgoals A1 and A2 and ... and An. The number of conditions, n, can be 0, in which case the implication is simply a fact, which behaves as a procedure which solves goals directly without introducing subgoals. The procedural interpretation of implications can be used for declarative programming, where the programmer describes information in logical form and the deduction mechanism uses backward reasoning to achieve problem solving behavior. Consider, for example, the implication X is a citizen of the USA if X was born in the USA. Backward reasoning treats the sentence as a procedure To show X is a citizen of the USA, show that X was born in the USA. In fact, backward reasoning can be used, not only to show a particular person is a citizen, but it can also be used to find people who are citizens by finding people who were born in the USA.
    [Show full text]