Lecture09 - CS440/ECE448: Intro to Articial Intelligence!...

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Lecture 9: More on predicate logic Prof. Julia Hockenmaier juliahmr@illinois.edu http://cs.illinois.edu/fa11/cs440 CS440/ECE448: Intro to ArtiFcial Intelligence
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Review: syntax of predicate logic
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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
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Semantics of predicate logic
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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
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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
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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): ¬ φ M,g =true iff φ M,g =false φ ψ M,g =true iff φ M,g =true and ψ M,g =true φ v ψ M,g =true iff φ M,g =true or ψ M,g =true φ ψ M,g =true iff φ M,g =false or ψ M,g =true 10 CS440/ECE448: Intro AI
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Interpretation of formulas: quantiFers Universal quantifer: x φ M,g =true iFF φ M, g[u/x] =true For all u D Existential quantifer: x φ M,g =true iFF φ M ,g[u/x] =true For at least one u D 11 CS440/ECE448: Intro AI
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