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© 2002 Franz J. Kurfess Logic and Reasoning 1 CPE/CSC 481: Knowledge-Based Systems Dr. Franz J. Kurfess Computer Science Department Cal Poly

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© 2002 Franz J. Kurfess Logic and Reasoning 2 Course Overview u Introduction u Knowledge Representation u Semantic Nets, Frames, Logic u Reasoning and Inference u Predicate Logic, Inference Methods, Resolution u Reasoning with Uncertainty u Probability, Bayesian Decision Making u Expert System Design u ES Life Cycle u CLIPS Overview u Concepts, Notation, Usage u Pattern Matching u Variables, Functions, Expressions, Constraints u Expert System Implementation u Salience, Rete Algorithm u Expert System Examples u Conclusions and Outlook

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© 2002 Franz J. Kurfess Logic and Reasoning 3 Overview Logic and Reasoning u Motivation u Objectives u Knowledge and Reasoning u logic as prototypical reasoning system u syntax and semantics u validity and satisfiability u logic languages u Reasoning Methods u propositional and predicate calculus u inference methods u Knowledge Representation and Reasoning Methods u Production Rules u Semantic Nets u Schemata and Frames u Logic u Important Concepts and Terms u Chapter Summary

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© 2002 Franz J. Kurfess Logic and Reasoning 4 Logistics u Term Project u Lab and Homework Assignments u Exams u Grading

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© 2002 Franz J. Kurfess Logic and Reasoning 5 Bridge-In

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© 2002 Franz J. Kurfess Logic and Reasoning 6 Pre-Test

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© 2002 Franz J. Kurfess Logic and Reasoning 7 Motivation

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© 2002 Franz J. Kurfess Logic and Reasoning 8 Objectives

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© 2002 Franz J. Kurfess Logic and Reasoning 9 Evaluation Criteria

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© 2002 Franz J. Kurfess Logic and Reasoning 10 Chapter Introduction u Review of relevant concepts u Overview new topics u Terminology

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© 2002 Franz J. Kurfess Logic and Reasoning 11 Introduction to Logic expresses knowledge in a particular mathematical notation All birds have wings --> ¥x. Bird(x) -> HasWings(x) rules of inference guarantee that, given true facts or premises, the new facts or premises derived by applying the rules are also true All robins are birds --> ¥x Robin(x) -> Bird(x) given these two facts, application of an inference rule gives: ¥x Robin(x) -> HasWings(x)

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© 2002 Franz J. Kurfess Logic and Reasoning 12 Logic and Knowledge rules of inference act on the superficial structure or syntax of the first 2 formulas doesn't say anything about the meaning of birds and robins could have substituted mammals and elephants etc. major advantages of this approach deductions are guaranteed to be correct to an extent that other representation schemes have not yet reached easy to automate derivation of new facts problems computational efficiency uncertain, incomplete, imprecise knowledge

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© 2002 Franz J. Kurfess Logic and Reasoning 13 Validity and Satisfiability a sentence is valid or necessarily true if and only if it is true under all possible interpretations in all possible worlds also called a tautology IsBird(Robin) V ~IsBird(Robin) Stench[1,1] V ~Stench[1,1] OpenArea[square in front of me] V Wall[square in front of me] is NOT a tautology! assumes every square has either a wall or an open area, so not true for all worlds "If every square has either a wall or an open area in it, then OpenArea[square in front of me] V Wall[square in front of me]" is a tautology... a sentence is satisfiable iff there is some interpretation in some world for which it is true a sentence that is not satisfiable is unsatisfiable (also known as a contradiction): It is raining and it is not raining.

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© 2002 Franz J. Kurfess Logic and Reasoning 14 Summary of Logic Languages propositional logic facts true/false/unknown first-order logic facts, objects, relations true/false/unknown temporal logic facts, objects, relations, times true/false/unknown probability theory facts degree of belief [0..1] fuzzy logic degree of truth degree of belief [0..1]

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© 2002 Franz J. Kurfess Logic and Reasoning 15 Propositional Logic u Syntax u Semantics u Validity and Inference u Models u Inference Rules u Complexity

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© 2002 Franz J. Kurfess Logic and Reasoning 16 Syntax u symbols logical constants True, False propositional symbols P, Q, … u logical connectives u conjunction , disjunction , u negation , u implication , equivalence u parentheses , u sentences u constructed from simple sentences u conjunction, disjunction, implication, equivalence, negation

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© 2002 Franz J. Kurfess Logic and Reasoning 17 BNF Grammar Propositional Logic Sentence AtomicSentence | ComplexSentence AtomicSentence True | False | P | Q | R |... ComplexSentence (Sentence ) | Sentence Connective Sentence | Sentence Connective | | | ambiguities are resolved through precedence or parentheses e.g. P Q R S is equivalent to ( P) (Q R)) S

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© 2002 Franz J. Kurfess Logic and Reasoning 18 Semantics u interpretation of the propositional symbols and constants u symbols can be any arbitrary fact u sentences consisting of only a propositional symbols are satisfiable, but not valid the constants True and False have a fixed interpretation True indicates that the world is as stated False indicates that the world is not as stated u specification of the logical connectives u frequently explicitly via truth tables

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© 2002 Franz J. Kurfess Logic and Reasoning 19 Truth Tables for Connectives P True True False False P Q False True P Q False True P Q True False True P Q True False True Q False True False True P False True

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© 2002 Franz J. Kurfess Logic and Reasoning 20 Validity and Inference u truth tables can be used to test sentences for validity u one row for each possible combination of truth values for the symbols in the sentence the final value must be True for every sentence

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© 2002 Franz J. Kurfess Logic and Reasoning 21 Propositional Calculus properly formed statements that are either True or False syntax logical constants, True and False proposition symbols such as P and Q logical connectives: and ^, or V, equivalence, implies => and not ~ parentheses to indicate complex sentences sentences in this language are created through application of the following rules True and False are each (atomic) sentences Propositional symbols such as P or Q are each (atomic) sentences Enclosing symbols and connective in parentheses yields (complex) sentences, e.g., (P ^ Q)

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© 2002 Franz J. Kurfess Logic and Reasoning 22 Complex Sentences Combining simpler sentences with logical connectives yields complex sentences conjunction sentence whose main connective is and: P ^ (Q V R) disjunction sentence whose main connective is or: A V (P ^ Q) implication (conditional) sentence such as (P ^ Q) => R the left hand side is called the premise or antecedent the right hand side is called the conclusion or consequent implications are also known as rules or if-then statements equivalence (biconditional) (P ^ Q) (Q ^ P) negation the only unary connective (operates only on one sentence) e.g., ~P

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© 2002 Franz J. Kurfess Logic and Reasoning 23 Syntax of Propositional Logic A BNF (Backus-Naur Form) grammar of sentences in propositional logic Sentence -> AtomicSentence | ComplexSentence AtomicSentence -> True | False | P | Q | R |... ComplexSentence -> (Sentence) | Sentence Connective Sentence | ~Sentence Connective -> ^ | V | | =>

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© 2002 Franz J. Kurfess Logic and Reasoning 24 Semantics propositions can be interpreted as any facts you want e.g., P means "robins are birds", Q means "the wumpus is dead", etc. meaning of complex sentences is derived from the meaning of its parts one method is to use a truth table all are easy except P => Q this says that if P is true, then I claim that Q is true; otherwise I make no claim; P is true and Q is true, then P => Q is true P is true and Q is false, then P => Q is false P is false and Q is true, then P => Q is true P is false and Q is false, then P => Q is true

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© 2002 Franz J. Kurfess Logic and Reasoning 25 Exercise Semantics and Truth Tables Use a truth table to prove the following: P represents the fact "Wally is in location [1, 3]" - W[1,3] H represents the fact "Wally is in location [2, 2]" - W[2,2] We know that Wally is either in [1,3] or [2,2]: (P V H) We learn that Wally is not in [2,2]: ~H Can we prove that Wally is in [1,3]: ((P V H) ^ ~H) => P This says that if the agent has some premises, and a possible conclusion, it can determine if the conclusion is true (i.e., all the rows of the truth table are true)

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© 2002 Franz J. Kurfess Logic and Reasoning 26 Inference Rules more efficient than truth tables

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© 2002 Franz J. Kurfess Logic and Reasoning 27 Modus Ponens eliminates => (X => Y), X ______________ Y If it rains, then the streets will be wet. It is raining. Infer the conclusion: The streets will be wet. (affirms the antecedent)

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© 2002 Franz J. Kurfess Logic and Reasoning 28 Modus tollens (X => Y), ~Y _______________ ¬ X If it rains, then the streets will be wet. The streets are not wet. Infer the conclusion: It is not raining. NOTE: Avoid the fallacy of affirming the consequent: If it rains, then the streets will be wet. The streets are wet. cannot conclude that it is raining. If Bacon wrote Hamlet, then Bacon was a great writer. Bacon was a great writer. cannot conclude that Bacon wrote Hamlet.

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© 2002 Franz J. Kurfess Logic and Reasoning 29 Syllogism chain implications to deduce a conclusion) (X => Y), (Y => Z) _____________________ (X => Z)

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© 2002 Franz J. Kurfess Logic and Reasoning 30 More Inference Rules and-elimination and-introduction or-introduction double-negation elimination unit resolution

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© 2002 Franz J. Kurfess Logic and Reasoning 31 Resolution (X v Y), (~Y v Z) _________________ (X v Z) basis for the inference mechanism in the Prolog language and some theorem provers

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© 2002 Franz J. Kurfess Logic and Reasoning 32 Complexity issues truth table enumerates 2 n rows of the table for any proof involving n symbol it is complete computation time is exponential in n checking a set of sentences for satisfiability is NP-complete but there are some circumstances where the proof only involves a small subset of the KB, so can do some of the work in polynomial time if a KB is monotonic (i.e., even if we add new sentences to a KB, all the sentences entailed by the original KB are still entailed by the new larger KB), then you can apply an inference rule locally (i.e., don't have to go checking the entire KB)

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© 2002 Franz J. Kurfess Logic and Reasoning 33 Horn clauses or sentences class of sentences for which a polynomial-time inference procedure exists P1 ^ P2 ^...^Pn => Q where Pi and Q are non-negated atoms not every knowledge base can be written as a collection of Horn sentences

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© 2002 Franz J. Kurfess Logic and Reasoning 34 Reasoning in Knowledge-Based Systems shallow and deep reasoning forward and backward chaining alternative inference methods metaknowledge

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© 2002 Franz J. Kurfess Logic and Reasoning 35 Shallow and Deep Reasoning shallow reasoning also called experiential reasoning aims at describing aspects of the world heuristically short inference chains possibly complex rules deep reasoning also called causal reasoning aims at building a model of the world that behaves like the “real thing” long inference chains often simple rules that describe cause and effect relationships

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© 2002 Franz J. Kurfess Logic and Reasoning 36 Examples Shallow and Deep Reasoning shallow reasoning deep reasoning IF a car has a good battery good spark plugs gas good tires THEN the car can move IF the battery is good THEN there is electricity IF there is electricity ANDgood spark plugs THEN the spark plugs will fire IF the spark plugs fire AND there is gas THEN the engine will run IF the engine runs AND there are good tires THEN the car can move

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© 2002 Franz J. Kurfess Logic and Reasoning 37 Forward Chaining given a set of basic facts, we try to derive a conclusion from these facts example: What can we conjecture about Clyde? IF elephant(x) THEN mammal(x) IF mammal(x) THEN animal(x) elephant (Clyde) modus ponens: IF p THEN q p q unification: find compatible values for variables

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© 2002 Franz J. Kurfess Logic and Reasoning 38 Forward Chaining Example IF elephant(x) THEN mammal(x) IF mammal(x) THEN animal(x) elephant(Clyde) modus ponens: IF p THEN q p q elephant (Clyde) IF elephant( x ) THEN mammal( x ) unification: find compatible values for variables

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© 2002 Franz J. Kurfess Logic and Reasoning 39 Forward Chaining Example IF elephant(x) THEN mammal(x) IF mammal(x) THEN animal(x) elephant(Clyde) modus ponens: IF p THEN q p q elephant (Clyde) IF elephant(Clyde) THEN mammal(Clyde) unification: find compatible values for variables

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© 2002 Franz J. Kurfess Logic and Reasoning 40 Forward Chaining Example IF elephant(x) THEN mammal(x) IF mammal(x) THEN animal(x) elephant(Clyde) modus ponens: IF p THEN q p q elephant (Clyde) IF elephant(Clyde) THEN mammal(Clyde) IF mammal( x ) THEN animal( x ) unification: find compatible values for variables

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© 2002 Franz J. Kurfess Logic and Reasoning 41 Forward Chaining Example IF elephant(x) THEN mammal(x) IF mammal(x) THEN animal(x) elephant(Clyde) modus ponens: IF p THEN q p q elephant (Clyde) IF elephant(Clyde) THEN mammal(Clyde) IF mammal(Clyde) THEN animal(Clyde) unification: find compatible values for variables

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© 2002 Franz J. Kurfess Logic and Reasoning 42 Forward Chaining Example IF elephant(x) THEN mammal(x) IF mammal(x) THEN animal(x) elephant(Clyde) modus ponens: IF p THEN q p q elephant (Clyde) IF elephant(Clyde) THEN mammal(Clyde) IF mammal(Clyde) THEN animal(Clyde) animal( x ) unification: find compatible values for variables

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© 2002 Franz J. Kurfess Logic and Reasoning 43 Forward Chaining Example IF elephant(x) THEN mammal(x) IF mammal(x) THEN animal(x) elephant(Clyde) modus ponens: IF p THEN q p q elephant (Clyde) IF elephant(Clyde) THEN mammal(Clyde) IF mammal(Clyde) THEN animal(Clyde) animal(Clyde) unification: find compatible values for variables

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© 2002 Franz J. Kurfess Logic and Reasoning 44 Backward Chaining

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© 2002 Franz J. Kurfess Logic and Reasoning 45 Backward Chaining try to find supportive evidence (i.e. facts) for a hypothesis example: Is there evidence that Clyde is an animal? IF elephant(x) THEN mammal(x) IF mammal(x) THEN animal(x) elephant (Clyde) modus ponens: IF p THEN q p q unification: find compatible values for variables

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© 2002 Franz J. Kurfess Logic and Reasoning 46 Backward Chaining Example IF elephant(x) THEN mammal(x) IF mammal(x) THEN animal(x) elephant(Clyde) modus ponens: IF p THEN q p q IF mammal( x ) THEN animal( x ) animal(Clyde) unification: find compatible values for variables ?

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© 2002 Franz J. Kurfess Logic and Reasoning 47 Backward Chaining Example IF elephant(x) THEN mammal(x) IF mammal(x) THEN animal(x) elephant(Clyde) modus ponens: IF p THEN q p q IF mammal(Clyde) THEN animal(Clyde) animal(Clyde) unification: find compatible values for variables ?

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© 2002 Franz J. Kurfess Logic and Reasoning 48 Backward Chaining Example IF elephant(x) THEN mammal(x) IF mammal(x) THEN animal(x) elephant(Clyde) modus ponens: IF p THEN q p q IF elephant( x ) THEN mammal( x ) IF mammal(Clyde) THEN animal(Clyde) animal(Clyde) unification: find compatible values for variables ? ?

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© 2002 Franz J. Kurfess Logic and Reasoning 49 Backward Chaining Example IF elephant(x) THEN mammal(x) IF mammal(x) THEN animal(x) elephant(Clyde) modus ponens: IF p THEN q p q IF elephant(Clyde) THEN mammal(Clyde) IF mammal(Clyde) THEN animal(Clyde) animal(Clyde) unification: find compatible values for variables ? ?

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© 2002 Franz J. Kurfess Logic and Reasoning 50 Backward Chaining Example IF elephant(x) THEN mammal(x) IF mammal(x) THEN animal(x) elephant(Clyde) modus ponens: IF p THEN q p q elephant ( x ) IF elephant(Clyde) THEN mammal(Clyde) IF mammal(Clyde) THEN animal(Clyde) animal(Clyde) unification: find compatible values for variables ? ? ?

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© 2002 Franz J. Kurfess Logic and Reasoning 51 Backward Chaining Example IF elephant(x) THEN mammal(x) IF mammal(x) THEN animal(x) elephant(Clyde) modus ponens: IF p THEN q p q elephant (Clyde) IF elephant(Clyde) THEN mammal(Clyde) IF mammal(Clyde) THEN animal(Clyde) animal(Clyde) unification: find compatible values for variables

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© 2002 Franz J. Kurfess Logic and Reasoning 52 Forward vs. Backward Chaining Forward ChainingBackward Chaining planning, controldiagnosis data-drivengoal-driven (hypothesis) bottom-up reasoningtop-down reasoning find possible conclusions supported by given facts find facts that support a given hypothesis similar to breadth-first search similar to depth-first search antecedents (LHS) control evaluation consequents (RHS) control evaluation

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© 2002 Franz J. Kurfess Logic and Reasoning 53 Alternative Inference Methods

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© 2002 Franz J. Kurfess Logic and Reasoning 54 Metaknowledge

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© 2002 Franz J. Kurfess Logic and Reasoning 55 Post-Test

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© 2002 Franz J. Kurfess Logic and Reasoning 56 Evaluation u Criteria

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© 2002 Franz J. Kurfess Logic and Reasoning 57 Use of References [Giarratano & Riley 1998] [Giarratano & Riley 1998] [Russell & Norvig 1995] [Russell & Norvig 1995] [Jackson 1999] [Jackson 1999] [Durkin 1994] [Durkin 1994] [Giarratano & Riley 1998]

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© 2002 Franz J. Kurfess Logic and Reasoning 58 References [Altenkrüger & Büttner] Doris Altenkrüger and Winfried Büttner. Wissensbasierte Systems - Architektur, Enwicklung, Echtzeit-Anwendungen. Vieweg Verlag, 1992. [Awad 1996] Elias Awad. Building Expert Systems - Principles, Procedures, and Applications. West Publishing, Minneapolis/St. Paul, MN, 1996. [Bibel 1993] Wolfgang Bibel with Steffen Höldobler and Torsten Schaub. Wissensrepräsentation und Inferenz - Eine grundlegende Einführung. Vieweg Verlag, 1993. [Durkin 1994] John Durkin. Expert Systems - Design and Development. Prentice Hall, Englewood Cliffs, NJ, 1994. [Giarratano & Riley 1998] Joseph Giarratano and Gary Riley. Expert Systems - Principles and Programming. 3 rd ed., PWS Publishing, Boston, MA, 1998 [Jackson, 1999] Peter Jackson. Introduction to Expert Systems. 3 rd ed., Addison-Wesley, 1999. [Russell & Norvig 1995] Stuart Russell and Peter Norvig, Artificial Intelligence - A Modern Approach. Prentice Hall, 1995.

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© 2002 Franz J. Kurfess Logic and Reasoning 59 Important Concepts and Terms natural language processing neural network predicate logic propositional logic rational agent rationality Turing test agent automated reasoning belief network cognitive science computer science hidden Markov model intelligence knowledge representation linguistics Lisp logic machine learning microworlds

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© 2002 Franz J. Kurfess Logic and Reasoning 60 Summary Chapter-Topic

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© 2002 Franz J. Kurfess Logic and Reasoning 61

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