DOCSLIB.ORG

Belief Maintenance: an I Tapnty Management

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Belief Maintenance: an I Tapnty Management From: AAAI-88 Proceedings. Copyright ©1988, AAAI (www.aaai.org). All rights reserved. BELIEF MAINTENANCE: AN I TAPNTY MANAGEMENT Kathryn B. Laskey and Paul E. Lehner Decision Science Consortium, Inc. George Mason University 1895 Preston White Drive 4400 University Drive Reston, Virginia 22091 Fairfax, Virginia 22030 Abstract In the quantitative approach, uncertainty about a proposition's truth value is expressed Belief maintenance represents a unified by a number or numbers, typically in the range approach to assumption-based and between 0 and 1. Inference rules also have numerical uncertainty management. A numbers associated with them, indicating how formal equivalence is demonstrated strongly belief in the premises warrants belief between Shafer-Dempster belief theory in the conclusions. Various calculi have been and assumption-based truth maintenance proposed for propagating beliefs in chains of extended to incorporate a probability inference. calculus on assumptions. Belief propagation through truth maintenance We argue that a significant advantage may automatically and correctly accounts be had by combining symbolic and numeric for non-independencies among uncertainty management. Specific advantages propositions due to shared antecedents. include the following. Belief maintenance also incorporates an ability to represent and reason with Computational aspects of numeric reasoning. The defaults. The result is a framework numeric approach has been criticized on for non-monotonic reasoning about the efficiency grounds. The number of possible application of a quantitative combinations of truth values grows exponentially uncertainty calculus. with the number of propositions in a system. A numeric calculus maintains propositions, no matter how improbable, unless they are 1.0 INTRODUCTION explicitly proven false. Of course, sophisticated computational architectures and Automated reasoning systems must operate with control strategies can reduce inefficiency incomplete knowledge of the state of the world. (e.g. 9 by not exploring consequents of Much of the work of problem solving or inference improbable propositions). But additional lies in structuring exploration of the system's control strategies open up with the possibility world to reduce this uncertainty. Two general of applying non-monotonic qualitative reasoning approaches to uncertainty management have become to the application of an uncertainty calculus. popular. These approaches--symbolic truth For example, the control strategy might make an maintenance and numeric belief propagation--have assumption, which could be later retracted, been portrayed as rivals in a sometimes assigning a truth value of false to an acrimonious debate. Yet each has an important improbable proposition. Or the structure of an role to play in a comprehensive inference inference network could be subjected to strategy. qualitative reasoning by making retractable conditional independence assumptions. Symbolic truth maintenance [Doyle, 1979; deKleer, 19861 is based on the idea of extending Control of reasoning. An important role of ordinary truth-functional logic to allow the truth maintenance is to control reasoning--to incorporation of defaults or assumptions. A decide what to do next. Additional system based on such a logic can set default possibilities for control are opened up by values for uncertain propositions and reason as including a numeric uncertainty calculus. An if these values were known, but revise its example cited above is to avoid using inference defaults when they give rise to a contradiction. rules with improbable antecedents. Another This capability mimics the non-monotonic strategy involves searching for information that character of intelligent human reasoning. Truth will likely distinguish between two uncertain maintenance increases search efficiency by but competing hypotheses. permitting a control strategy that minimizes regeneration of previously considered search Conflict resolution. The two uncertainty paths. Uncertainty is represented entirely management traditions have very different qualitatively: what is known about propositions approaches to dealing with conflict [Cohen, is whether they are proven true, assumed true, Laskey and Ulvila, 19871. In the symbolic unknown, assumed false, or proven false in a tradition, conflicting conclusions indicate that given context. one of the lines of reasoning is faulty. 2 10 Automated Reasoning Defaults must be changed to restore consistency. reasoning system might incorporate a default By contrast, probabilistic and other numerical assumption that the rule is operating correctly systems regard conflict as an inevitable (i.e., the source is reliable). Receipt of consequence of an imperfect correlation or evidence against q would necessitate dropping causal link between evidence and conclusion. either this default or one of the assumptions Probabilistic reasoning regards inference as underlying the conclusion of wq. The belief weighing a balance of positive and negative calculus, on the other hand, simultaneously evidence; symbolic reasoning adjusts the set of maintains belief that the rule is and is not accepted defaults so that no simultaneous valid, using a number between zero and 1 to arguments exist for both a proposition and its encode the strength of belief in rule validity. complement. Intelligent reasoning, we feel, combines aspects of both these viewpoints. Simultaneously maintaining belief in must be able to entertain conflicting lines of inconsistent propositions is impossible in most argument, adjudicating between them based on the default reasoning systems. When beliefs become strength of each. On the other hand, when the inconsistent, these systems must explicitly conflict becomes too great, we begin to suspect change some of their defaults to regain a problem with one or the other argument, and consistency. Combining beliefs and defaults is examine our implicit beliefs for possible therefore difficult with traditional truth revision. maintenance. But the ATMS [deKleer, 19861 is explicitly designed to reason with multiple This paper represents a preliminary effort contexts. Each proposition is tagged with a toward unifying symbolic and numeric reasoning. label that represents the contexts in which it It has been implemented, but only on small-scale can be proven. Contexts are represented by problems. Much work remains in developing special symbols which deKleer calls assumptions. control strategies and algorithms to circumvent We depart from deKleer's terminology and use the the computational complexity associated with term tokens to refer to these special symbols, large belief networks. But the structure because we prefer the word assumption to retain described here represents an important step the connotation of a proposition that, although toward understanding the relationship between unproven, has been declared to be in the set of probabilistic and belief function models, truth believed propositions. The job of the ATMS is maintenance, and non-monotonic logics. to maintain a parsimonious representation of each proposition's dependence, either directly or through chains of inference, on tokens. Like 2.0 THEATMS AND SHAFER-D STER BELIEFS deKleer, we define &q environment to be a set of tokens and a conteXt to be the entire set of This section introduces an approach to propositions derivable in a particular uncertainty management that unifies the symbolic environment. A proposition's label contains a and numeric approaches. In particular, it is list of environments in which it can be derived. demonstrated that extending assumption-based There may be proofs for a proposition -under truth maintenance [deKleer, 19861 to allow a several mutually inconsistent environments; probability calculus on assumptions leads to a similarly, proofs for a proposition and its belief calculus that is formally equivalent to negation in different environments may be Shafer-Dempster theory. By appropriate entertained simultaneously. construction of inference rules, probabilistic models emerge as a special case. An advantage Consider again the inference from report r of the framework presented here is that to the truth of the reported proposition q. In nonindependencies are automatically and the ATMS, the "noisy" inference rule may be correctly accounted for in the belief encoded as follows. The token V is introduced computations. to represent rule validity, i.e., the proposition that the source is reporting Eet us introduce the theory with an example reliably. (Following deKleer, we use uppercase used by Shafer [1987] to illustrate belief letters to represent tokens and lowercase function theory. Suppose we receive a report r letters to represent other propositions). The that a proposition q is the case. There is some original noisy rule is replaced by the rule probability, say .6, that the source is rAV -, q. When this rule fires, the token V is reporting reliably; otherwise, with probability added to the label of the ATMS node associated .4, the report bears no relation to the truth of with q, indicating that q is valid in any 4. In the first case, the report implies q; in context in which V is true. the second case, both q and its negation are consistent with the report. For Shafer, belief If this token is treated as a default in a proposition is defined as the probability assumption, then the proposition q is‘believed that the evidence implies its truth. Thus, the until this default becomes inconsistent with the above evidence
Load more

Recommended publications
  • Graph-Based Reasoning in Collaborative Knowledge Management for Industrial Maintenance Bernard Kamsu-Foguem, Daniel Noyes
    Graph-based reasoning in collaborative knowledge management for industrial maintenance Bernard Kamsu-Foguem, Daniel Noyes To cite this version: Bernard Kamsu-Foguem, Daniel Noyes. Graph-based reasoning in collaborative knowledge manage- ment for industrial maintenance. Computers in Industry, Elsevier, 2013, Vol. 64, pp. 998-1013. 10.1016/j.compind.2013.06.013. hal-00881052 HAL Id: hal-00881052 https://hal.archives-ouvertes.fr/hal-00881052 Submitted on 7 Nov 2013 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. Open Archive Toulouse Archive Ouverte (OATAO) OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in: http://oatao.univ-toulouse.fr/ Eprints ID: 9587 To link to this article: doi.org/10.1016/j.compind.2013.06.013 http://www.sciencedirect.com/science/article/pii/S0166361513001279 To cite this version: Kamsu Foguem, Bernard and Noyes, Daniel Graph-based reasoning in collaborative knowledge management for industrial
    [Show full text]
  • Next Generation Logic Programming Systems Gopal Gupta
    Next Generation Logic Programming Systems Gopal Gupta In the last 30 years logic programming (LP), with Prolog as the most representative logic programming language, has emerged as a powerful paradigm for intelligent reasoning and deductive problem solving. In the last 15 years several powerful and highly successful extensions of logic programming have been proposed and researched. These include constraint logic programming (CLP), tabled logic programming, inductive logic programming, concurrent/parallel logic programming, Andorra Prolog and adaptations for non- monotonic reasoning such as answer set programming. These extensions have led to very powerful applications in reasoning, for example, CLP has been used in industry for intelligently and efficiently solving large scale scheduling and resource allocation problems, tabled logic programming has been used for intelligently solving large verification problems as well as non- monotonic reasoning problems, inductive logic programming for new drug discoveries, and answer set programming for practical planning problems, etc. However, all these powerful extensions of logic programming have been developed by extending standard logic programming (Prolog) systems, and each one has been developed in isolation from others. While each extension has resulted in very powerful applications in reasoning, considerably more powerful applications will become possible if all these extensions were combined into a single system. This power will come about due to search space being dramatically pruned, reducing the time taken to find a solution given a reasoning problem coded as a logic program. Given the current state-of-the-art, even two extensions are not available together in a single system. Realizing multiple or all extensions in a single framework has been rendered difficult by the enormous complexity of implementing reasoning systems in general and logic programming systems in particular.
    [Show full text]
  • Logic-Based Technologies for Intelligent Systems: State of the Art and Perspectives
    information Article Logic-Based Technologies for Intelligent Systems: State of the Art and Perspectives Roberta Calegari 1,* , Giovanni Ciatto 2 , Enrico Denti 3 and Andrea Omicini 2 1 Alma AI—Alma Mater Research Institute for Human-Centered Artificial Intelligence, Alma Mater Studiorum–Università di Bologna, 40121 Bologna, Italy 2 Dipartimento di Informatica–Scienza e Ingegneria (DISI), Alma Mater Studiorum–Università di Bologna, 47522 Cesena, Italy; [email protected] (G.C.); [email protected] (A.O.) 3 Dipartimento di Informatica–Scienza e Ingegneria (DISI), Alma Mater Studiorum–Università di Bologna, 40136 Bologna, Italy; [email protected] * Correspondence: [email protected] Received: 25 February 2020; Accepted: 18 March 2020; Published: 22 March 2020 Abstract: Together with the disruptive development of modern sub-symbolic approaches to artificial intelligence (AI), symbolic approaches to classical AI are re-gaining momentum, as more and more researchers exploit their potential to make AI more comprehensible, explainable, and therefore trustworthy. Since logic-based approaches lay at the core of symbolic AI, summarizing their state of the art is of paramount importance now more than ever, in order to identify trends, benefits, key features, gaps, and limitations of the techniques proposed so far, as well as to identify promising research perspectives. Along this line, this paper provides an overview of logic-based approaches and technologies by sketching their evolution and pointing out their main application areas. Future perspectives for exploitation of logic-based technologies are discussed as well, in order to identify those research fields that deserve more attention, considering the areas that already exploit logic-based approaches as well as those that are more likely to adopt logic-based approaches in the future.
    [Show full text]
  • Compiling a Default Reasoning System Into Prolog
    Compiling A Default Reasoning System into Prolog David Poole Department of Computer Science University of British Columbia Vancouver BC Canada VT W p o olecsub cca July Abstract Articial intelligence researchers have b een designing representa tion systems for default and ab ductive reasoning Logic Programming researchers have b een working on techniques to improve the eciency of Horn Clause deduction systems This pap er describ es how one such default and ab ductive reasoning system namely Theorist can b e translated into Horn clauses with negation as failure so that we can use the clarity of ab ductive reasoning systems and the eciency of Horn clause deduction systems Wethus showhowadvances in expressivepower that articial intelligence workers are working on can directly utilise advances in eciency that logic programming re searchers are working on Actual co de from a running system is given Intro duction Many p eople in Articial Intelligence have b een working on default reasoning and ab ductive diagnosis systems The systems implemented so far eg are only prototyp es or have b een develop ed in A Theorist to Prolog Compiler ways that cannot take full advantage in the advances of logic programming implementation technology Many p eople are working on making logic programming systems more ecient These systems however usually assume that the input is in the form of Horn clauses with negation as failure This pap er shows howto implement the default reasoning system Theorist by compiling its input into Horn clauses
    [Show full text]
  • Mathematical Reasoning with Diagrams Mateja Jamnik
    Mathematical Reasoning with Diagrams Mateja Jamnik ISBN: 1-57586-324-3 Copyright notice: Excerpted from Mathematical Reasoning with Diagrams by Mateja Jamnik, published by CSLI Publications. ©2001 by CSLI Publications. All rights reserved. This text may be used and shared in accordance with the fair-use provisions of U.S. copyright law, and it may be archived and redistributed in electronic form, provided that this entire notice, including copyright information, is carried and provided that CSLI Publications is notified and no fee is charged for access. Archiving, redistribution, or republication of this text on other terms, in any medium, requires the consent of CSLI Publications. Contents Foreword vii Preface xi 1 Introduction 1 2 The History of Diagrammatic Systems 11 3 Diagrammatic Theorems and the Problem Domain 27 4 The Constructive ω-Rule and Schematic Proofs 49 5 Designing a Diagrammatic Reasoning System 71 6 Diagrammatic Operations 89 7 The Construction of Schematic Proofs 103 8 The Verification of Schematic Proofs 123 9 Diamond in Action 149 10 Complete Automation 163 Appendix A: More Examples of Diagrammatic Theorems 175 Appendix B: The ω-Rule 181 Glossary 185 References 190 Index 199 v Foreword The advent of the modern computer in the nineteen-fifties immediately suggested a new research challenge: to seek ways of programming these versatile machines which would make them behave as much like intel- ligent human beings as possible. After fifty years or so, this quest has produced some intriguing results, but until now progress has been dis- appointingly slow. This book is a welcome and encouraging sign that things may at last be about to change.
    [Show full text]
  • Ontology and Information Systems
    Ontology and Information Systems 1 Barry Smith Philosophical Ontology Ontology as a branch of philosophy is the science of what is, of the kinds and structures of objects, properties, events, processes and relations in every area of reality. ‘Ontology’ is often used by philosophers as a synonym for ‘metaphysics’ (literally: ‘what comes after the Physics’), a term which was used by early students of Aristotle to refer to what Aristotle himself called ‘first philosophy’.2 The term ‘ontology’ (or ontologia) was itself coined in 1613, independently, by two philosophers, Rudolf Göckel (Goclenius), in his Lexicon philosophicum and Jacob Lorhard (Lorhardus), in his Theatrum philosophicum. The first occurrence in English recorded by the OED appears in Bailey’s dictionary of 1721, which defines ontology as ‘an Account of being in the Abstract’. Methods and Goals of Philosophical Ontology The methods of philosophical ontology are the methods of philosophy in general. They include the development of theories of wider or narrower scope and the testing and refinement of such theories by measuring them up, either against difficult 1 This paper is based upon work supported by the National Science Foundation under Grant No. BCS-9975557 (“Ontology and Geographic Categories”) and by the Alexander von Humboldt Foundation under the auspices of its Wolfgang Paul Program. Thanks go to Thomas Bittner, Olivier Bodenreider, Anita Burgun, Charles Dement, Andrew Frank, Angelika Franzke, Wolfgang Grassl, Pierre Grenon, Nicola Guarino, Patrick Hayes, Kathleen Hornsby, Ingvar Johansson, Fritz Lehmann, Chris Menzel, Kevin Mulligan, Chris Partridge, David W. Smith, William Rapaport, Daniel von Wachter, Chris Welty and Graham White for helpful comments.
    [Show full text]
  • The Logic Reasoning System a Brief Introduction   Franz Wotawa
    The Logic Reasoning System A Brief Introduction Franz Wotawa Technische Universität Graz Institute for Software Technology Inffeldgasse 16b/2, 8010 Graz, Austria [email protected] http://www.ist.tugraz.at/ The Logic Reasoning System – A brief introduction Franz Wotawa1 Graz University of Technology Institute for Software Technology Inffeldgasse 16b/2, A-8010 Graz, Austria [email protected] Abstract. In this article we describe how to use the Logic Reason- ing System that is based on an Assumption-Based Truth Maintenance system. In particular we show by example how models following the consistency-based reasoning and the abductive reasoning methodology can be formulated. We use one example from the boolean circuit domain and give the two different models for the circuit, which can be used for diagnosis. Moreover, we show how the system can be used to compute diagnoses using consistency-based diagnosis and abductive diagnosis re- spectively. Keywords: Logic-based reasoning systems, ATMS, Diagnosis 1 Introduction The Logic Reasoning System (LRS) can be obtained from the author of this article. The system is implemented in Java 1.5 and ships out as an executable JAR file. To install the system the JAR file and the example models should be copied to a directory on your computer. You can open LRS by simple executing the JAR file if the Java runtime engine 1.5 or higher is installed. The system comes with no warranty. Before discussing modeling we briefly recall the related literature behind LRS. The implementation is based on DeKleer’s Assumption-based Truth Main- tenance System (ATMS) [1, 2].
    [Show full text]
  • A Classification Approach for Automated Reasoning Systems--A Case Study in Graph Theory Rong Lin Old Dominion University
    Old Dominion University ODU Digital Commons Computer Science Theses & Dissertations Computer Science Spring 1989 A Classification Approach for Automated Reasoning Systems--A Case Study in Graph Theory Rong Lin Old Dominion University Follow this and additional works at: https://digitalcommons.odu.edu/computerscience_etds Part of the Artificial Intelligence and Robotics Commons Recommended Citation Lin, Rong. "A Classification Approach for Automated Reasoning Systems--A Case Study in Graph Theory" (1989). Doctor of Philosophy (PhD), dissertation, Computer Science, Old Dominion University, DOI: 10.25777/5qk4-1w09 https://digitalcommons.odu.edu/computerscience_etds/115 This Dissertation is brought to you for free and open access by the Computer Science at ODU Digital Commons. It has been accepted for inclusion in Computer Science Theses & Dissertations by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. A CLASSIFICATION APPROACH FOR AUTOMATED REASONING SYSTEMS - A CASE STUDY IN GRAPH THEORY By Rong Lin A THESIS SUBMITTED TO THE FACULTY OF OLD DOMINION UNIVERSITY N PARTIAL FULFILLMENT OF TIIE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY COMPUTER SCIENCE Norfolk, Virginia April, 1989 Approved by: Shunichi Toida (Director) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT Reasoning systems which create classifications of structured objects face the problem of how object descriptions can be used to reflect their components as well as relations among these com­ ponents. Current reasoning sysLcms on grapli theory do not adequately provide models to discover complex relations among mathematical concepts (eg: relations involving subgraphs) mainly due to the inability to solve this problem.
    [Show full text]
  • The Fragmentation of Reasoning
    PUBLISHED IN: Pablo Quintanilla, Carla Mantilla, Paola Cépeda (eds.), Cognición social y lenguaje. La intersubjetividad en la evolución de la especie y en el desarrollo del niño, Lima: Pontificia Universidad Católica del Perú, 2014. ISBN: 978-612-4146-80-0 THE FRAGMENTATION OF REASONING Peter Carruthers University of Maryland 1. INTRODUCTION Scientists who study human reasoning across a range of cognitive domains have increasingly converged on the idea that there are two distinct systems (or types of system) involved. These domains include learning (Berry & Dienes, 1993; Reber, 1993), conditional and probabilistic reasoning (Evans & Over, 1996; Sloman, 1996 and 2002; Stanovich, 1999), decision making (Kahneman & Frederick, 2002; Kahneman, 2003), and social cognition of various sorts (Petty & Cacioppo, 1986; Chaike, and others, 1989; Wilson, and others, 2000). Although terminology has differed, many now use the labels «System 1» and «System 2» to mark the intended distinction. System 1 is supposed to be fast and unconscious in its operations, issuing in intuitively compelling answers to learning or reasoning problems in ways that subjects themselves have no access to. System 2, in contrast, is supposed to be slow and conscious in its operations, and is engaged whenever we are induced to tackle reasoning tasks in a reflective manner. Many theorists now accept that System 1 is really a set of systems, arranged in parallel, while believing that System 2 is a single serially-operating ability. Moreover, System 1 is generally thought to be unchangeable in its basic operations, to be universal amongst humans, and to be shared (at least in significant part) with other species of animal.
    [Show full text]
  • Case-Based Reasoning Is a Methodology Not a Technologyଝ
    Knowledge-Based Systems 12 (1999) 303–308 Case-based reasoning is a methodology not a technologyq I. Watson AI-CBR, University of Salford, Salford M5 4WT, UK Received 1 December 1998; accepted 17 March 1999 Abstract This paper asks whether case-based reasoning is an artificial intelligence (AI) technology like rule-based reasoning, neural networks or genetic algorithms or whether it is better described as a methodology for problem solving, that may use any appropriate technology. By describing four applications of case-based reasoning (CBR), that variously use: nearest neighbour, induction, fuzzy logic and SQL, the author shows that CBR is a methodology and not a technology. The implications of this are discussed. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Case-based reasoning; Artificial intelligence technology; Problem solving methodolgy 1. Introduction 2. Case-based reasoning Artificial intelligence (AI) is often described in terms of CBR arose out of research into cognitive science, most the various technologies developed over the last three or prominently that of Roger Schank and his students at Yale four decades. Technologies such as logic programming, University [1–4]. It is relevant to the argument presented in rule-based reasoning, neural networks, genetic algorithms, this paper that CBR’s origins were stimulated by a desire to fuzzy logic, constraint-based programming and others. understand how people remember information and are in These technologies are characterised by specific program- turn reminded of information; and that subsequently it ming languages or environments (e.g. Prolog or rule-based was recognised that people commonly solve problems by shells) or by specific algorithms and techniques (e.g.
    [Show full text]
  • Logic Programming and Knowledge Representation—The A-Prolog Perspective ✩
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Artificial Intelligence 138 (2002) 3–38 www.elsevier.com/locate/artint Logic programming and knowledge representation—The A-Prolog perspective ✩ Michael Gelfond a,∗, Nicola Leone b a Department of Computer Science, Texas Tech University, Lubbock, TX 79409-3104, USA b Department of Mathematics, University of Calabria, 87030 Rende (CS), Italy Abstract In this paper we give a short introduction to logic programming approach to knowledge representation and reasoning. The intention is to help the reader to develop a ‘feel’ for the field’s history and some of its recent developments. The discussion is mainly limited to logic programs under the answer set semantics. For understanding of approaches to logic programming built on well-founded semantics, general theories of argumentation, abductive reasoning, etc., the reader is referred to other publications. 2002 Elsevier Science B.V. All rights reserved. Keywords: Logic programming; Nonmonotonic reasoning; Default reasoning; Answer set programming 1. Introduction If we want to design an entity (a machine or a program) capable of behaving intelligently in some environment, then we need to supply this entity with sufficient knowledge about this environment. To do that, we need an unambiguous language capable of expressing this knowledge, together with some precise and well understood way of manipulating sets of sentences of the language which will allow us to draw inferences, answer queries, and update both the knowledge base and the desired program behavior. A good knowledge representation language should allow construction of elaboration tolerant knowledge bases, i.e., bases in which small modifications of the informal body of knowledge correspond to small modifications of the formal base representing this knowledge.
    [Show full text]
  • 4 Combinatorics and Probability
    CHAPTER 4 Combinatorics ✦ ✦ ✦ ✦ and Probability In computer science we frequently need to count things and measure the likelihood of events. The science of counting is captured by a branch of mathematics called combinatorics. The concepts that surround attempts to measure the likelihood of events are embodied in a field called probability theory. This chapter introduces the rudiments of these two fields. We shall learn how to answer questions such as how many execution paths are there in a program, or what is the likelihood of occurrence of a given path? ✦ ✦ ✦ ✦ 4.1 What This Chapter Is About We shall study combinatorics, or “counting,” by presenting a sequence of increas- ingly more complex situations, each of which is represented by a simple paradigm problem. For each problem, we derive a formula that lets us determine the number of possible outcomes. The problems we study are: ✦ Counting assignments (Section 4.2). The paradigm problem is how many ways can we paint a row of n houses, each in any of k colors. ✦ Counting permutations (Section 4.3). The paradigm problem here is to deter- mine the number of different orderings for n distinct items. ✦ Counting ordered selections (Section 4.4), that is, the number of ways to pick k things out of n and arrange the k things in order. The paradigm problem is counting the number of ways different horses can win, place, and show in a horse race. ✦ Counting the combinations of m things out of n (Section 4.5), that is, the selection of m from n distinct objects, without regard to the order of the selected objects.
    [Show full text]