1 First-Order Logic Propositional logic does not represent and cannot handle objects. First-order logic is to represent and reason on objects and their relations High-order logic is to deal with relations of relations

2 First-Order Logic Syntax Constant symbols: a, b, c, John, … to represent primitive objects Variable symbols: x, y, z, … to represent unknown objects Predicate symbols: safe, married, love, … to represent relations married(John) love(John, Mary)

3 First-Order Logic Syntax Function symbols: square, father, … to represent simple objects safe(square(1, 2)) love(father(John), mother(John)) Terms: to represent complex objects –Constant symbols –If f is a function symbol, and t 1, t 2, …, t n are terms, then so is f(t 1, t 2, …, t n ) love(mother(father(John)), John)

4 First-Order Logic Syntax Logical connectives: , , , ,  Universal quantifier:  x p(x)  x love(father(x), mother(x)) Existential quantifier:  x p(x)   x  p(x)  x  married(x)

5 First-Order Logic Syntax Equality: father(John) = Henry  x, y brother(x, Richard)  brother(y, Richard)   (x = y)

6 First-Order Logic Syntax Sentences: –Atomic sentences: p(t 1, t 2, …, t n ) –If  is a sentence, then so are  and (  ) –If  and  are sentences, then so are  , and  –If  is a sentence, then so are  and 

7 First-Order Logic Semantics Truth value with respect to an interpretation (possible world) Possible world: –Set of objects –Constants  objects –Functions  mapping from objects to objects –Predicates  relations on objects

8 First-Order Logic Semantics    x p(f(x), x) W 1 : Set of objects: {John, Tom, Mary} Functions: f  mother, mother(John) = Mary, mother(Tom) = Mary Predicates: p  love, {love(Mary, John), love(Mary, Tom)} W 2 : Set of objects: {John, Tom, Mary} Functions: f  mother, mother(John) = Mary, mother(Tom) = Mary Predicates: p  love, {love(Mary, John)}  is true with respect to W 1 (?), but not W 2

9 FOL: Kinship Domain  m, c mother(c) = m  female(m)  parent(m, c)  w, h husband(h, w)  male(h)  spouse(h, w)  x male(x)   female(x)  p, c parent(p, c)  child(c, p)  g, c grand-parent(g, c)   p parent(g, p)  parent(p, c)  x, y sibling(x, y)  x  y   p parent(p, x)  parent(p, y)

10 FOL: Number Domain nat-num(0)  n nat-num(n)  nat-num(S(n))  n 0  S(n)  m nat-num(m)  +(0, m) = m  m, n nat-num(m)  nat-num(n)  +(S(m), n) = S(+(m, n))

11 FOL: Set Domain  s set(s)  (s = {})  (  x, s 2 set(s 2 )  s = {x|s 2 })  x, s {x|s} = {}  x, s x  s  s = {x|s}  x, s x  s   y, s 2 (s = {y|s 2 }  (x = y  x  s 2 ))  s 1, s 2 s 1  s 2  (  x x  s 1  x  s 2 )  s 1, s 2 s 1 = s 2  (s 1  s 2  s 2  s 1 )

12 FOL: Wumpus Game Environment: square(1, 2)  [1, 2] percept([stench, breeze, glitter, none, none], 5)  x, y, a, b adjacent([x, y], [a, b])  [a, b]  {[x+1, y], [x-1, y], [x, y+1], [x, y-1]}  s, t at(Agent, s, t)  breeze(t)  breezy(s)

13 FOL: Wumpus Game Diagnostic rules:  s breezy(s)   r adjacent(r, s)  pit(r)  s  r adjacent(r, s)  pit(r)   breezy(s)  s breezy(s)   r adjacent(r, s)  pit(r)

14 FOL: Wumpus Game Causal rules:  r pit(r)  (  s adjacent(r, s)  breezy(s))  s [  r adjacent(r, s)   pit(r)]   breezy(s)

15 Homework In Russell & Norvig’s AIMA (2 nd ed.): Exercises of Chapter 8.