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CS 2710, ISSP 2610

This text provides an introduction to First-Order Predicate Calculus (FOPC) and its relationship to Propositional Logic. It covers topics such as ontological commitment, syntax, quantifiers, and semantics.

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CS 2710, ISSP 2610

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  1. CS 2710, ISSP 2610 Chapter 8, Part 1 First Order Predicate Calculus FOPC

  2. Propositional Logic to FOPC B11  (P12 v P21) B23  (P32 v P 23 v P34 v P 43) … “Internal squares adjacent to pits are breezy”: All X Y (B(X,Y) ^ (X > 1) ^ (Y > 1) ^ (Y < 4) ^ (X < 4))  (P(X-1,Y) v P(X,Y-1) v P(X+1,Y) v (X,Y+1))

  3. FOPC Worlds • Rather than just T,F, now worlds contain: • Objects: the gold, the wumpus, people, ideas, … “the domain” • Predicates: holding, breezy, red, sisters • Functions: fatherOf, colorOf, plus Ontological commitment

  4. FOPC Syntax • Add variables and quantifiers to propositional logic

  5. Sentence  AtomicSentence | ComplexSentence AtomicSentence  Predicate(Term,…) | Term = Term Term  Function(Term,…) |Constant |Variable ComplexSentence  (Sentence) | ~Sentence | Sentence ^ Sentence | Sentence v Sentence | Sentence  Sentence | Sentence  Sentence| Quantifier Variable, … Sentence Quantifier  all, exists Constant  john, 1, … Variable  A, B, C, X Predicate  breezy, sunny, red Function  fatherOf, plus Knowledge engineering involves devising the constants, predicates, and functions for your problem

  6. Examples • Everyone likes chocolate • X (person(X)  likes(X, chocolate)) • Someone likes chocolate • X (person(X) ^ likes(X, chocolate)) • Everyone likes chocolate unless they are allergic to it • X (person(X)  (likes(X, chocolate)  allergic(X, chocolate))) • X ((person(X) ^ allergic (X, chocolate))  likes(X, chocolate)) [Messy sentence on board]

  7. Quantifiers • All X p(X) means that p holds for all elements in the domain • Exists X p(X) means that p holds for at least one element of the domain

  8. All • Usually used with implications • All X (student(X,CS2710)  smart(X)) • NOT usually: • All X (student(X,CS2710) ^ smart(X))

  9. Exists • Usually used with ^ to specify information about individuals • Exists X (student(X,CS2710) ^ smart (X) ^ speaks(X,klingon)) • NOT usually: • Exists X (student(X,CS2710)  (smart(X) ^ speaks(X,klingon))) • [Very weak statement! ]

  10. Quantification and Negation • ~(all X p(X)) equiv exists X ~p(X) • ~(exists X p(X)) equiv all X ~p(X)

  11. Nesting of Variables Put quantifiers in front of likes(P,F) Assume the domain of discourse of P is the set of people Assume the domain of discourse of F is the set of foods • Everyone likes some kind of food • There is a kind of food that everyone likes • Someone likes all kinds of food • Every food has someone who likes it

  12. Answers(DOD of P is people and F is food) Everyone likes some kind of food All P Exists F likes(P,F) There is a kind of food that everyone likes Exists F All P likes(P,F) Someone likes all kinds of food Exists P All F likes(P,F) Every food has someone who likes it All F Exists P likes(P,F)

  13. Full Answers (without Domain of Discourse Assumptions) Everyone likes some kind of food All P (person(P)  Exists F (food(F) and likes(P,F))) There is a kind of food that everyone likes Exists F (food(F) and (All P (person(P)  likes(P,F)))) Someone likes all kinds of food Exists P (person(P) and (All F (food(F)  likes(P,F)))) Every food has someone who likes it All F (food (F)  Exists P (person(P) and likes(P,F)))

  14. Semantics of FOPC • Interpretation: assignment of elements from the world to elements of the language • The world consists of a domain ofobjects D, a set of predicates, and a set of functions

  15. Semantics • Each constant is assigned an element of the domain • Each N-ary predicate is assigned a set of N-tuples [which are?] • Each N-ary function symbol is assigned a set of N+1 tuples that does not include any pair of tuples that have the first N elements but different (n+1)st elements

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