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chap2 - Knowledge Representation and Reasoning Logics for...

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Knowledge Representation and Reasoning Logics for Artificial Intelligence Stuart C. Shapiro Department of Computer Science and Engineering and Center for Cognitive Science University at Buffalo, The State University of New York Buffalo, NY 14260-2000 [email protected] copyright c 1995, 2004–2010 by Stuart C. Shapiro Page 1
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Contents Part I 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. Propositional Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3. Predicate Logic Over Finite Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 4. Full First-Order Predicate Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 5. Summary of Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 Part II 6. Prolog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 7. A Potpourri of Subdomains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 8. SNePS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 9. Belief Revision/Truth Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 10. The Situation Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 11. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596
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Part III 12. Production Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609 13. Description Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 14. Abduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 Page 3
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2 Propositional Logic Logics that do not analyze information below the level of the proposition. 2.1 What is a Proposition? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 CarPool World: A Motivational “Micro-World” . . . . . . . . . . . . . . 23 2.3 The “Standard” Propositional Logic . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4 Important Properties of Logical Systems . . . . . . . . . . . . . . . . . . . 133 2.5 Clause Form Propositional Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Page 19
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2.1 What is a Proposition? An expression in some language that is true or false whose negation makes sense that can be believed or not whose negation can be believed or not that can be put in the frame “I believe that it is not the case that .” Page 20
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Examples Of propositions Betty is the driver of the car. Barack Obama is sitting down or standing up. If Opus is a penguin, then Opus doesn’t fly. Of non-propositions Barack Obama how to ride a bicycle If the fire alarm rings, leave the building. Page 21
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Sentences vs. Propositions A sentence is an expression of a (written) language that begins with a capital letter and ends with a period, question mark, or exclamation point. Some sentences do not contain a proposition: “Hi!”, “Why?”, “Pass the salt!” Some sentences do not express a proposition, but contain one: “Is Betty driving the car?” Some sentences contain more than one proposition: If Opus is a penguin, then Opus doesn’t fly. Page 22
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2.2 CarPool World: A Motivational “Micro-World” Tom and Betty carpool to work. On any day, either Tom drives Betty or Betty drives Tom. In the former case, Tom is the driver and Betty is the passenger. In the latter case, Betty is the driver and Tom is the passenger. Propositions: Betty drives Tom. Tom drives Betty. Betty is the driver. Tom is the driver. Betty is the passenger. Tom is the passenger. Page 23
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2.3 The “Standard” Propositional Logic 1. Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2. Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3. Proof Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Page 24
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2.3.1 Syntax of the “Standard” Propositional Logic Atomic Propositions Any letter of the alphabet, e.g.: P Any letter of the alphabet with a numerical subscript, e.g.: Q 3 Any alphanumeric string, e.g.: Tom is the driver is an atomic proposition.
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