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Lecture07 - CS440/ECE448 Intro to Articial Intelligence...

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Lecture 7: Propositional logic Prof. Julia Hockenmaier [email protected] http://cs.illinois.edu/fa11/cs440 CS440/ECE448: Intro to Artificial Intelligence

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Thursday ʼ s key concepts Combining CSP search and inference: Ordering variables (minimum remaining value, degree heuristics) Ordering values (forward checking, MAC) Global constraints: Constraint hypergraph; auxiliary variables Continuous domains: bounds consistency 2 CS440/ECE448: Intro AI
Path consistency and arc consistency X is arc consistent with respect to Y if for every value of X there exists some value of Y such that C(X,Y) is satisfied. X and Y are path consistent with respect to Z if for every pair of values of X and Y that satisfy C(X, Y), there exists some value of Z such that C(X,Z) and C(Y,Z) is satisfied. 3 CS440/ECE448: Intro AI

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Global (n-ary) constraints: Constraint Hypergraph F T U W R O TWO + TWO = FOUR C 1000 C 100 C 10 CS440/ECE448: Intro AI 4
Propositional logic

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Propositional logic Syntax: What is the language of well-formed formulas of propositional logic? Semantics: What is the interpretation of a well-formed formula in propositional logic? Inference rules and algorithms: How can we reason with propositional logic?
Syntax: the building blocks Variables: p | q | r | … Constants: (true) , (false) Unary connectives: ¬ (negation) Binary connectives: (conjunction) (disjunction) (implication)

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Syntax: well-formed formulas WFF Atomic | Complex Atomic Constant | Variable WFF’ Atomic | (Complex) Complex ¬ WFF’ | WFF’ WFF’ | WFF’ WFF’ | WFF’ WFF’
Semantics: truth values The interpretation α v of a well-formed formula α under a model v is a truth value: α v {true, false}. A model (=valuation) v is a complete* assignment of truth values to variables: v(p) = true v(q) = false, … *each variable is either true or false With n variables, there are 2 n different models Models of α ( ʻ M( α )’ ): set of models where α is true

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Interpretation α v of α Interpretation of constants : ⟦⊤⟧ v = true, ⟦⊥⟧ v = false Interpretation of variables defined by v p v = v(p) Interpretation of connectives
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Lecture07 - CS440/ECE448 Intro to Articial Intelligence...

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