Science of Computational Logic Working Material 1 Steffen H - PDF document

Science of Computational Logic — Working Material 1 — Steffen H¨ olldobler International Center for Computational Logic Technische Universit¨ at Dresden D–01062 Dresden sh@iccl.tu-dresden.de January 16, 2012 1 The working material is incomplete and may contain errors. Any suggestions are greatly appreciated.

Contents 1 Description Logic 3 1.1 Terminologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Assertions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Subsumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Unsatisfiability Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Equational Logic 11 2.1 Equational Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Paramodulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Term Rewriting Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.1 Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.2 Confluence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.3 Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4 Unification Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4.1 Unification under Equality . . . . . . . . . . . . . . . . . . . . . . . 28 2.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.4.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.4.4 Multisets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.5 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3 Actions and Causality 41 3.1 Conjunctive Planning Problems . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2 Blocks World . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2.1 A Fluent Calculus Implementation . . . . . . . . . . . . . . . . . . . 44 3.2.2 SLDE-Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2.3 Solving Conjunctive Planning Problems . . . . . . . . . . . . . . . . 46 3.2.4 Solving the Frame Problem . . . . . . . . . . . . . . . . . . . . . . . 46 iii

iv CONTENTS 3.2.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4 Deduction, Abduction, and Induction 51 4.1 Deduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.1.1 Sorts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.2 Abduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2.1 Abduction in Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.2.2 Knowledge Assimilation . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2.3 Theory Revision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2.4 Abduction and Model Generation . . . . . . . . . . . . . . . . . . . 60 4.2.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.3 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5 Non-Monotonic Reasoning 65

Notation In this book we will make the following notational conventions: a constant b constant C unary relation symbol denoting a concept C set of concept formulas D non-empty domain of an interpretaion E set of equations E ⌈ s ⌉ expression containing an occurrence of the term s E ⌈ s/t ⌉ expression where an occurrence of the term s has been replaced by t E R equational system obtained from the term rewriting system R E ≈ axioms of equality ε empty substitution g function symbol f function symbol F formula F set of formulas g function symbol G formula H formula I interpretation K a set of formulas often called knowledge base l term; left-hand side of an equation or rewrite rule L literal p relation symbol r term; the right-hand side of equations or rewrite rules R binary relation symbol denoting a role R term rewriting system s term t term θ substitution u term U variable V variable W variable X variable Y variable Z variable In addition, we will consider the following precedence hierarchy among connectives: {∀ , ∃} ≻ ¬ ≻ {∧ , ∨} ≻ → ≻ ↔ .

2 CONTENTS

Chapter 1 Description Logic In the late 1960s and early 1970s, it was recognized that knowledge representation and reasoning is at the heart of any intelligent system. Heavily influenced by the work of Quillian on so-called semantic networks [Qui68] and the work of Minsky on so-called frames [Min75] simple graphs and structured objects were used to represent knowledge and many algorithms were developed which manipulated these data structures. At first sight, these systems were quite attractive because they apparently admitted an intuitive semantics, which was easy to understand. For example, a graph like the one shown in Figure 1.1 seems to represent the following short story. Dogs, cats and mice are mammals. Dogs dislike cats and, in particular, the dog Rudi, which is a German shepherd, has bitten the cat Tom while Tom was chasing the mouse Jerry. Simple algorithms operating on this graph can be applied to conclude that, for example, German shepherds are mammals, Rudi dislikes Tom, etc. Shortly afterwards, however, it was recognized that systems based on these techniques lack a formal semantics (see e.g. [Woo75]). What precisely is denoted by a link? What precisely is denoted by a vertex? It was also observed that the algorithms which operated on these data structures did not always yield the intended results. This led to a formal reconstruction of semantic networks as well as frame systems within logic (see e.g. [Sch76, Hay79]). At around the same time, Brachman developed the idea that formally defined concepts should be interrelated and organized in networks such that the structure of these networks allows reasoning about possible conclusions [Bra78]. This line of research led to the knowledge representation and reasoning system KlOne [BS85], which is the ancester of a whole family of systems. Such systems have been used in a wide range of practical applications including financial management systems, computer configuration systems, software information systems and database interfaces. KlOne has also led to a thorough investigation of the semantics of the representations used in these systems and the development of correct and complete algorithms for computing with these representations. Today the field is called description logic and this chapter gives an introduction into such logics. Description logics focus on descriptions of concepts and their interrelationships in cer- tain domains. Based on so-called atomic concepts and relations between concepts, which are traditionally called roles , more complex concepts are formed with the help of certain 3

4 CHAPTER 1. DESCRIPTION LOGIC mammals are are are dogs cats mice dislike are german shephards is a is a jerry rudi tom has bitten was chasing Figure 1.1: A simple semantic network with apparently obvious intended meaning. operators. Furthermore, assertions about certain aspects of the world can be made. For example, a certain individual may be an instance of a certain concept or two individuals are connected via a certain role. The basic inference tasks provided by description logics are subsumption and unsatisfiability testing . Subsumption is used to check whether a cat- egory is a subset of another category. As we shall see in the next paragraph, description logics do not allow the specification of subsumption hierarchies explicitly but these hier- archies depend on the definitions of the concepts. The unsatisfiability check allows the determination of whether an individual belongs to a certain concept. A formal account of these notions will be developed in the following sections. 1.1 Terminologies We consider an alphabet with constant symbols, the variables X, Y, . . . , the connectives ¬ , ∧ , ∨ , → , ↔ , the quantifiers ∀ and ∃ , and the special symbols ( , , , ) . For notational convenicence, C shall denote a unary relation symbols and R a binary relation symbol C R in the sequel. Informally, C denotes a concept whereas R denotes a role. Terms are defined as usual, ie., the set of terms is the union of the set of constant symbols and the set of variables. The set of role formulas consists of all strings of the form R ( X, Y ). The set of atomic concept formulas consists of all strings of the form C ( X ). As we will see shortly, each concept formula contains precisely one free variable. Hence, concept formulas will be denoted by F ( X ) and G ( X ), where X is the only free variable occurring in F and G . The set of concept formulas is the smallest set C concept formula satisfying the following conditions: 1. All atomic formulas are in C . 2. If F ( X ) is in C , so is ¬ F ( X ). 3. If F ( X ) and G ( X ) are in C , so are F ( X ) ∧ G ( X ) and F ( X ) ∨ G ( X ).