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Lectures 13-14 First-Order Logic

Explore the basics of first-order logic, including propositions, terms, interpretation, models, syntax, semantics, and inference. Dive into universal statements and learn about syntactic sugar, semantics, grounding rules, and unification in this introductory guide.

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Lectures 13-14 First-Order Logic

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  1. Lectures 13-14First-Order Logic CSE 573 Artificial Intelligence I Henry Kautz Fall 2001 CSE 573

  2. Axiom Schemas • Useful to allow schemas that stand for sets of sentences. • Blowup: (index range)nesting • for i, j in {1, 2, 3, 4} such that adjacent(i,j): • for c in {R, B, G}: • ( Xc,i  Xc,j) 3 4 2 1 CSE 573

  3. Representing Information about Different Objects • Suppose we want to talk about many different objects… • Where is truck #87? What about airplane #32? • Which vehicle carries package #806? CSE 573

  4. Complex Propositions • At( TRUCK87, UW ) • At( PLANE32, SEATAC ) • In( PACKAGE806, PLANE32 ) • Ingredients: • Example of a sentence: CSE 573

  5. Complex Terms • plus(1,3) • 4 = 1+3 • teacher_of( CSE573, AU2001 ) • Overworked(teacher_of(CSE573, AU2001 ))  Underpaid(teacher_of(CSE573, AU2001 ))) • teacher_of(CSE573, AU2001 ) = HENRY What is the obvious conclusion? CSE 573

  6. Models of Ground Sentences • What does entailment mean? • { Overworked(teacher_of(CSE573, AU2001)),HENRY = teacher_of(CSE573, AU2001)) } • Overworked(HENRY) CSE 573

  7. First Order Interpretation • Domain  of objects • : Constants   • : n-ary Function symbols  (n  ) • : Propositional symbols  {true, false} • : n-ary Predicate symbols  Subsets of n • ( “=”) = { (d, d) | d} CSE 573

  8. Do Obvious Recursive Thing! • (P(a)) = true iff (a) (P) • (Q(a,b)) = true iff ((a), (b)) (Q) • (S  T) = true iff (S) and (T) • (S  T) = true iff (S) or (T) • (S) = true iff (S) = false • (f(a)) = (f)[(a) ] • (f(a,b)) = (f)[(a), (b) ] CSE 573

  9. Example • Overworked(teacher_of(CSE573, AU2001))  HENRY = teacher_of(CSE573, AU2001)) CSE 573

  10. Not Done Yet… • So far we’ve provided a way to break propositions down into meaningful bits • Brings logic closer to the way we think about the world – objects, relationships… • Doesn’t fundamentally change the expressive or computational properties of propositional logic • “Naming convention” for propositions • Inference: same algorithms: Davis-Putnam, GSAT, Resolution CSE 573

  11. Universal Statements • Consider: • All men are mortal. • A parent’s parent is a grandparent. • No adjacent countries are colored the same color. CSE 573

  12. Universal Statements vs Propositional Schemas • Do not need to fully instantiate universal statements – more compact. • Can use universals in queries, because are real sentences. • Can use universals even if you do not have constants to uniquely identify all the objects in the domain. • Can use universals to talk about infinite sets CSE 573

  13. Syntax  x .  =  x .   x .  =  x .  • x . ( Man(x)  Mortal(x) ) •   x . (Man(x)  Mortal(x) ) CSE 573

  14. Some Syntactic Sugar • Vehicle = { TRUCK99, TRUCK33, PLANE66 } shorthand for • Vehicle(TRUCK99)  • Vehicle(TRUCK33)  • Vehicle(PLANE66 )  • x . (Vehicle(x)  (x= TRUCK99  x= TRUCK33  x= PLANE66 )) Predicate closure axioms Domain closure axiom • x . (x= TRUCK99  x= TRUCK33  x= PLANE66 ) CSE 573

  15. Semantics • A model  also maps variables to domain objects: •  : Variables   • x/d = the model that maps “x” to d, but is otherwise just like  • ( x .  ) = true iff for all d it is the case that x/d() is true CSE 573

  16. Inference in First-Order Logic • Proof theory: • Makes the leap from truth and modelsto symbol pushing • Consider a special case: • No function symbols • Closed domain, or quantify only over closed predicates CSE 573

  17. Grounding Out a First-Order Theory (Special Case) • Vehicle = { TRUCK99, TRUCK33, PLANE66 } • City = { SEATTLE, BOSTON } • x . ( Vehicle(x)  ( y . City(y)  Based(x, y) ) ) (Based(TRUCK99, SEATTLE)  Based(TRUCK99, BOSTON ))  (Based(TRUCK33, SEATTLE)  Based(TRUCK33, BOSTON ))  (Based(PLANE66 , SEATTLE)  Based(PLANE66 , BOSTON )) CSE 573

  18. Grounding Rules • Foo = { F1, F2, … } • x . ( Foo(x)  Bar(x)) ( Bar(F1)  Bar(F2)  … ) • x . ( Foo(x)  Bar(x) ) ( Bar(F1)  Bar(F2)  … ) CSE 573

  19. When Grounding is a Bad Idea • Everyone has friend, all of whose friends drink heavily. CSE 573

  20. Hardness of Full First-Order Logic • Can we always in principle propositionalize a theory? CSE 573

  21. Lifted Resolution • First-order clausal form • Begin with universal quantifiers • Rest is a clause • No existentials, but may include function symbols ( Man(x)Mortal(x)) (Mortal(y)Fallible(y)) (Man(z)Fallible(z)) CSE 573

  22. Unification • Match two literals if: • Same predicate, one positive, one negative • Match variable(s) to other vars, constants, or complex terms (function symbols) (Mortal(y)Fallible(y)) (Mortal(HENRY)) (Fallible(HENRY)) CSE 573

  23. Unification with Multiple Variables • You always hurt the ones you love. • Politicians love themselves. • Therefore, politicians hurt themselves. CSE 573

  24. Unification with Function Symbols • Say s(x) means the successor of x s(1) = 2, s(2)=3, etc. • A number is less than it’s successor. • “Less than” is transitive. • Therefore, a number is less than it’s successor’s successor. CSE 573

  25. Unification with Function Symbols (Less(a,s(a))) (Less(b,c) Less(c,d) Less(b,d)) (Less(s(a),d) Less(a,d)) rename variables: (Less(s(e),f) Less(e,f)) Less(e,s(s(e))) CSE 573

  26. Converting to Clausal Form: Skolem Functions • Everyone loves someone. • x . y . ( Loves(x,y) ) • x . ( Loves(x, f33(y)) ) There is somebody whom everyone loves.  y .  x . ( Loves(x,y) ) • x . ( Loves(x, F99) ) CSE 573

  27. Everyone Drinks? • Everyone has friend, all of whose friends drink heavily. • x . y .  x . (Friend(x,y)  ( Friend(y,z)  Drinks(z))) ( Friend(x,f(x) ) ( Friend(f(w),z)  Drinks(w) ) • Conclusion: Everyone drinks heavily. • x . ( Drinks(x) ) •  x . ( Drinks(x) ) • x . ( Drinks(x) )  Drinks(G) CSE 573

  28. The Case of the Missing Axiom ( Drinks(G)) ( Friend(f(w),z)  Drinks(z) )  Friend(f(w),G) Friend(x,f(x)) Cannot unifiy f(x) and G ! CSE 573

  29. Friend is Reflexive •  x ,y . ( Friend(x,y)  Friend(y,x) ) CSE 573

  30. FOL Refutation Proof ( Friend(a,b)  Friend(b,a) ) ( Friend(f(w),z)  Drinks(z) ) Friend(x,f(x))  Drinks(G)  Friend(f(x),x)  Friend(f(w),G) () CSE 573

  31. Next • Applications of Logic • Ontologies • Reasoning about Change • Planning CSE 573

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