Transcript COSC 4117 Artificial Intelligence

Normal Forms and Resolution
in FOL
automated reasoning with predicates
Conjunctive normal form
 KB must be in CNF for resolution
 conjunction of disjunctions of predicates
or negations
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1a. eliminate <=>
A <=> B
A => B ^ B => A
1b. eliminate =>
A => B
~A  B
reduction to CNF
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eliminate => and <=>
move negatives inward to predicates
Standardize apart variables
Skolemize (eliminate existentials)
Drop universals (assumed for variables)
Distribute conjunction over disjunction
Standardize apart again if necessary
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example reduction
(Luger and Stubblefiled, 1993)
 All people who are smart and not
poor are happy. Those people who
read are not stupid. Happy people
have exciting lives. John can read and
is wealthy. Can anyone be found who
has an exciting life?
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 All people who are smart and not poor are
happy. Those people who read are not stupid.
Happy people have exciting lives. John can
read and is wealthy. Can anyone be found who
has an exciting life?
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x (Smart(x)  ~Poor(x)) => Happy(x)
x Read(x) => ~Stupid(x)
x Happy(x) => Exciting(x)
Read(John)  Wealthy(John)
x Exciting(x)
 also need general knowledge
 ?
 All people who are smart and not poor are
happy. Those people who read are not stupid.
Happy people have exciting lives. John can
read and is wealthy. Can anyone be found who
has an exciting life?

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x (Smart(x)  ~Poor(x)) => Happy(x)
x Read(x) => ~Stupid(x)
x Happy(x) => Exciting(x)
Read(John)  Wealthy(John)
x Exciting(x)
 also need general knowledge
 x Wealthy(x) <=> ~Poor(x)
 x Smart(x) <=> ~Stupid(x)
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x (Smart(x)  ~Poor(x)) => Happy(x)
x Read(x) => ~Stupid(x)
x Happy(x) => Exciting(x)
Read(John)  Wealthy(John)
x Exciting(x)
x Wealthy(x) <=> ~Poor(x)
x Smart(x) <=> ~Stupid(x)
1. eliminate => and <=>
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x ~(Smart(x)  ~Poor(x))  Happy(x)
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x (Smart(x)  ~Poor(x)) => Happy(x)
x Read(x) => ~Stupid(x)
x Happy(x) => Exciting(x)
Read(John)  Wealthy(John)
x Exciting(x)
x Wealthy(x) <=> ~Poor(x)
x Smart(x) <=> ~Stupid(x)
1. eliminate => and <=>
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x ~(Smart(x)  ~Poor(x))  Happy(x)
x ~Read(x)  ~Stupid(x)
x ~Happy(x)  Exciting(x)
Read(John)  Wealthy(John)
x Exciting(x)
x ~Wealthy(x)  ~Poor(x)
x Wealthy(x)  Poor(x)
x Smart(x)  Stupid(x)
x ~Smart(x)  ~Stupid(x)
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x ~(Smart(x)  ~Poor(x))  Happy(x)
x ~Smart(x)  Poor(x)  Happy(x)
x ~Read(x)  ~Stupid(x)
x ~Happy(x)  Exciting(x)
Read(John)  Wealthy(John)
x Exciting(x)
x ~Wealthy(x)  ~Poor(x)
x Wealthy(x)  Poor(x)
x Smart(x)  Stupid(x)
x ~Smart(x)  ~Stupid(x)
3. standardize apart variables
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x ~Smart(x)  Poor(x)  Happy(x)
y ~Read(y)  ~Stupid(y)
z ~Happy(z)  Exciting(z)
Read(John)  Wealthy(John)
v Exciting(v)
u ~Wealthy(u)  ~Poor(u)
t Wealthy(t)  Poor(t)
s Smart(s)  Stupid(s)
r ~Smart(r)  ~Stupid(r)
(2. ~ inwards)
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x ~Smart(x)  Poor(x)  Happy(x)
y ~Read(y)  ~Stupid(y)
z ~Happy(z)  Exciting(z)
Read(John)  Wealthy(John)
v Exciting(v)
(4. Skolemize)
Exciting(K)
u ~Wealthy(u)  ~Poor(u)
t Wealthy(t)  Poor(t)
s Smart(s)  Stupid(s)
r ~Smart(r)  ~Stupid(r)
5. Drop universals
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~Smart(x)  Poor(x)  Happy(x)
~Read(y)  ~Stupid(y)
~Happy(z)  Exciting(z)
Read(John)  Wealthy(John)
Exciting(K)
~Wealthy(u)  ~Poor(u)
Wealthy(t)  Poor(t)
Smart(s)  Stupid(s)
~Smart(r)  ~Stupid(r)
6. Distrubute  over  (none), 7.restandardize apart (none)
Conjunctive normal form
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~Smart(x)  Poor(x)  Happy(x)
~Read(y)  ~Stupid(y)
~Happy(z)  Exciting(z)
Read(John)
Wealthy(John)
Exciting(K)
~Wealthy(u)  ~Poor(u)
Wealthy(t)  Poor(t)
Smart(s)  Stupid(s)
~Smart(r)  ~Stupid(r)
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Resolution with FOL
 Predicates can be resolved (positive
with negative) only if they can be
unified
 P(x)  ~Q(x), R(y)  ~P(y)
Subst({x/y}) ~Q(y)  R(y)
(in effect, need to establish that general
knowledge applies to facts)
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example resolution (continued)
(Luger and Stubblefiled, 1993)
 All people who are smart and not
poor are happy. Those people who
read are not stupid. Happy people
have exciting lives. John can read and
is wealthy. Can anyone be found who
has an exciting life?
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Question in CNF
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~Smart(x)  Poor(x)  Happy(x)
~Read(y)  ~Stupid(y)
~Happy(z)  Exciting(z)
Read(John)
Wealthy(John)
~Wealthy(u)  ~Poor(u)
Wealthy(t)  Poor(t)
Smart(s)  Stupid(s)
~Smart(r)  ~Stupid(r)
 Show by resolution refutation that:

Exciting(K)
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Strategy of Resolution refutation
 convert to CNF
 assume inverse of question and add
to KB
 ~x Exciting(x) or x ~Exciting(x)
 show by resolution that a
contradiction results
 refute assumption
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Resolution refutation (1)
 Assume inverse, add to KB
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~Smart(x)  Poor(x)  Happy(x)
~Read(y)  ~Stupid(y)
~Happy(z)  Exciting(z)
Read(John)
Wealthy(John)
~Wealthy(u)  ~Poor(u)
Wealthy(t)  Poor(t)
Smart(s)  Stupid(s)
~Smart(r)  ~Stupid(r)
~Exciting(k)
// assumption – note variable k
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Resolution
refutation (2)
~Exciting(k)
~Happy(z)  Exciting(z)
{k/z}
~Happy(k) ~Smart(x)  Poor(x)  Happy(x)
~Smart(x)  Poor(x)  Happy(x)
{k/x}
~Read(y)  ~Stupid(y)
~Smart(k)  Poor(k)
~Happy(z)  Exciting(z)
Read(John)
~Read(y)  ~Stupid(y)
Wealthy(John)
{y/s}
~Wealthy(u)  ~Poor(u)
Smart(s)  Stupid(s)
Wealthy(t)  Poor(t)
~Read(y)  Smart(y)
Smart(s)  Stupid(s)
{k/y}
~Smart(r)  ~Stupid(r)
~Read(k)  Poor(k)
~Exciting(k)
~Wealthy(u)  ~Poor(u)
{John/u}
{John/k}
~Poor(John)
~Read(John)
{}
nil
Read(John)
Wealthy(John)
Resolution refutation (3)
 conflict Read(John) , ~Read(John) => nil
 initial assumption of
 so
~x Exciting(x)
is wrong
x Exciting(x)
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