1. Chapter 8, Part 1 First Order Predicate Calculus FOPC
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. 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

4. FOPC Syntax
Add variables and quantifiers to propositional logic

5. 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

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 of objects 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