Lecture09HO - CS440/ECE448: Intro to Articial Intelligence!...

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Lecture 9: More on predicate logic Prof. Julia Hockenmaier [email protected] http://cs.illinois.edu/fa11/cs440 CS440/ECE448: Intro to ArtiFcial Intelligence Quick upgrade on quizzes Review: syntax of predicate logic The building blocks A (Fnite) set of variables VAR : VAR ={x, y, z,…} A (Fnite) set of constants CONST : CONST ={john, mary, tom,…} ±or n=1…N : A (Fnite) set of n -place function symbols FUNC FUNC 1 ={fatherOf, successor,…} A (Fnite) set of n -place predicate symbols PRED n : PRED 1 ={student, blue,…} PRED 2 ={friend, sisterOf,…} 4
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Putting everything together Terms : constants ( john ); variables ( x ); n-ary function symbols applied to n terms ( fatherOf(x) ) Ground terms contain no variables Formulas : n- ary predicate symbols applied to n terms ( likes(x,y) ); negated formulas ( ¬ fatherOf(x) ); conjunctions, disjunctions or implications of two formulas; quantiFed formulas Ground formulas (= sentences; propositions) contain no free variables Open formulas contain at least one free variable 5 CS440/ECE448: Intro AI Semantics of predicate logic Model M=(D,I) The domain D is a nonempty set of objects: D ={a1, b4, c8,…} The interpretation function I maps: - each constant c to an element c I of D: John I = a1 - each n -place function symbol f to an (total) n -ary function f I D n ! D : fatherOf I (a1) = b4 - each n -place predicate symbol p to an n -ary relation p I D n : child I ={a1,c8} likes I ={ a1, b4 , b4,a1 } 7 Interpretation of variables A variable assignment v over a domain D is a (partial) function from variables to D. The assignment v = [ a21/x, b13/y ] assigns object a21 to the variable x, and object b13 to variable y. We recursively manipulate variable assignments when interpreting quantiFed formulas. Notation: v [ b/z ] is just like v , but it also maps z to b. We will make sure that v is undeFned for z. 8
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Interpretation of terms Variables: x M,g = g(x) defned by the variable assignment Constants: c M,g = c I defned by the interpretation Function Functions: defned by the interpretation Function and recursion on the arguments f(t 1 ,….t n ) M,g = f I ( t 1 M,g ,…, t n M,g ) 9 Interpretation of formulas Atomic Formulas: P(t 1 ,...t n ) M,g =true iFF ⟨⟦ t 1 M,g ,... t n M,g P I Complex Formulas (connectives): ¬ ! M,g =true iFF ! M,g =false ! " M,g =true iFF ! M,g =true and " M,g =true ! v " M,g =true iFF ! M,g =true or " M,g =true ! # " M,g =true iFF ! M,g =false or " M,g =true 10 CS440/ECE448: Intro AI Interpretation of formulas: quanti±ers Universal quantifer: x ! M,g =true iFF ! M, g[u/x] =true For all u D Existential quantifer: x ! M,g =true iFF ! M ,g[u/x] =true For at least one u D 11 CS440/ECE448: Intro AI Satisfaction and entailment ! is satisfed in M ( M
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This note was uploaded on 10/13/2011 for the course CS 440 taught by Professor Levinson,s during the Spring '08 term at University of Illinois, Urbana Champaign.

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Lecture09HO - CS440/ECE448: Intro to Articial Intelligence!...

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