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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

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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)

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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 t1, t2, …, tn are terms,

then so is f(t1, t2, …, tn)

love(mother(father(John)), John)

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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)

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First-Order Logic Syntax• Equality:

father(John) = Henry

x, y brother(x, Richard) brother(y, Richard) (x = y)

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First-Order Logic Syntax• Sentences:

– Atomic sentences: p(t1, t2, …, tn)

– If is a sentence, then so are and ()

– If and are sentences, then so are , and

– If is a sentence, then so are and

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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

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First-Order Logic Semantics x p(f(x), x)

W1:Set of objects: {John, Tom, Mary}Functions: f mother, mother(John) = Mary, mother(Tom) = MaryPredicates: p love, {love(Mary, John), love(Mary, Tom)}

W2:Set of objects: {John, Tom, Mary}Functions: f mother, mother(John) = Mary, mother(Tom) = MaryPredicates: p love, {love(Mary, John)}

is true with respect to W1 (?), but not W2

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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)

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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))

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FOL: Set Domain

s set(s) (s = {}) (x, s2 set(s2) s = {x|s2})

x, s {x|s} = {}

x, s x s s = {x|s}

x, s x s y, s2 (s = {y|s2} (x = y x s2))

s1, s2 s1 s2 (x x s1 x s2)

s1, s2 s1 = s2 (s1 s2 s2 s1)

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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)

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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)

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FOL: Wumpus Game• Causal rules:

r pit(r) (s adjacent(r, s) breezy(s))

s [r adjacent(r, s) pit(r)] breezy(s)

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Homework• In Russell & Norvig’s AIMA (2nd ed.):

Exercises of Chapter 8.