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

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))

Ontological commitment 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

FOPC Syntax Add variables and quantifiers to propositional logic

Knowledge engineering involves devising the 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

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]

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

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

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! ]

Quantification and Negation ~(all X p(X)) equiv exists X ~p(X) ~(exists X p(X)) equiv all X ~p(X)

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

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)

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)))

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

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