Artificial Intelligence
Lecture 4
Faculty of Computer Science
International Islamic University Islamabad. Pakistan
Predicate Calculus
Predicate Calculus
 An extension to propositional calculus.
 Predicates to describe relationships
 Allow variables in expressions
 Example "I am in Paris"
 propositional symbol P to denote the sentence
 predicate place describes the relationship of I and Paris:
place(I, paris)
 predicate place(X,paris) denotes whether X is in Paris
 place(I,Y) denotes whether I am at Y
Using Inference Rules to Produce
Predicate Calculus Expressions
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The ability to infer new correct expressions from a set of true
assertions is an important feature of the predicate calculus.
These new expressions are correct in that they are ‘consistent’
with all previous interpretations of the original set of
expressions.
S: a set of predicate calculus sentences
"X ( human(X) a mortal(X) )
I: An interpretation
human(bill_gates)
X: an expression logically from S
mortal(bill_gates)
Exercise
The letters ‘B’ and ‘C’ represent proper names ‘Bob’ and ‘Cathy’.
‘m’ is one-place predicate meaning ‘is a mechanic’.
‘n’ is one-place predicate meaning ‘is a nurse’.
‘t’ is two-place predicate meaning ‘is taller than’.
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Either Cathy is a mechanic or a nurse (or both).
a
m(C) v n(C).
If Cathy is a mechanic then she is not a nurse.
a
m(C) a ~n(C).
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Cathy is taller than Bob.
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t(C,B).
For any three objects, if the first is taller than the second and the second is taller
than the third, then the first is taller than the third.
a
a
"X "Y "Z ( t(X, Y) ^ t(Y, Z) a t(X, Z) ).
Some Predicate Calculus
Equivalences
First Order Predicate Calculus
First Order Predicate Calculus
 First-order predicate calculus: universally and
existentially quantified variables refer only to objects
(constants), not to predicates or functions
 valid expressions: V X likes(kin,X) and E X V Y
likes(X,Y)
 - invalid expression: V (likes) likes(kin,ice_cream)
 this is valid in second order predicate calculus
Inference Rules
Inference Rules
 Ability of producing new, correct predicate calculus
sentences from an original set of predicate calculus
sentences
Modus Ponens:
 given: P=>Q and P are true
 conclusion: Q is true
 Example:
 P=>Q: “If I am in Paris then I am in France”
 P: “I am in Paris”
 Q: “I am in France” added to P=>Q and P
Inference Rules
Modus Tolens:
 given: P=>Q is true, and Q is false
 conclusion: P is false
 Example:
 P=>Q: “If I am in Paris then I am in France”
 ~Q: “I am not in France”
 ~P: “I am not in Paris” added to P=>Q and
~Q
Inference Rules
(Resolution)
Inference Rules
 Resolution:
 given: PvQ and ~QvR are true
 conclusion: PvR is true
Universal instantiation:
 given: V X p(X) is true, and constant x is from the
domain of X
 conclusion: p(x) is true
 example: all birds have feathers and a penguin is a bird
 conclusion: penguin has feathers
Unification
Unification
 The process of finding the substitutions needed to make two
predicate calculus expressions match.
 Example: {foo(X,a,goo(Y), foo(fred, a, goo(Z))}
 - Substitution {fred/X} yields {foo(fred,a,goo(Y), foo(fred, a,
goo(Z))}
 - substitution: {fred/X, Z/Y} yields {foo(fred,a,goo(Z),
foo(fred, a, goo(Z))}
 - here ‘fred/X’ indicates that ‘fred’ is substituted for the variable
X or the variable X is said to be bound to the value ‘fred’
Unification Rules
 a variable may be replaced by a constant: c/X
 a variable may be replaced by a variable: Y/X
 a variable may be replaced by a function expression as long as
the function expression does not contain the variable: p(Y)/X
 once a variable has been bound, future unification and inferences
must take this substitution into account
Unification
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Unification is an algorithm for determining the substitutions needed to
make two predicate calculus expressions match.
Unification Substitutions
alpha( X, a, beta(Y) ).
Is to be unified with
1.
alpha( fred, a, beta(Z) ).
2.
alpha( W, a, beta(jack) ).
3.
alpha( Z, a, beta( beta(Z) ) ).
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In this example, the unifications that would make the original
expression identical to each of the other three are written as:
1.
{ fred / X, Z / Y }
2.
{ W / X, jack / Y }
3.
{ Z / X, beta(Z) / Y}
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The notation X / Y indicates that X is substituted for the variable Y.
Substitutions are also called ‘bindings’.
Unification
Composition of Unification Substitutions
Consider the following sequence of substitutions:
{ X / Y, W / Z }, { V / X }, { a / V, f(b) / W }
These are equivalent to the single substitution
{a / Y, f(b) / Z }
Steps involved were
1.
Compose { X / Y, W / Z } with { V / X } to give { V / Y, W / Z }
2.
Compose { V / Y, W / Z } with {a / V, f(b) / W } to give { a / Y, f(b) / Z }
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If S and S’ are two substitution sets, then composition of S and S’ (written
as S’S) is obtained by applying S to the elements of S’ and adding the
results to S.
Examples
Emma is a Doberman pinscher and a good dog:
Googdog(emma) ^ isa(emma,doberman)
 every dog is an animal:
V X (dog(X) => animal (X))
 every boy has a bicycle:
 V X (E Y (boy(X) => (bicycle(Y) ^ own(X,Y))))
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Few More Examples….
Application: A Logic-Based Financial
Advisor
Application: A Logic-Based Financial Advisor
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1. People with inadequate savings should increase their savings, regardless of
income.
2. People with adequate savings and an adequate income should consider a
riskier but potentially more profitable investment in the stock market.
3. People with a lower income who already have adequate savings should split
their investment between savings and stocks, to increase the cushion in
savings while attempting to increase their income through stocks.
4. Adequate savings is to have more than $5,000 in the bank for each
dependent.
5. Adequate income is steady, more than $15,000 plus $4,000 for each
dependent.
Application: A Logic-Based Financial
Advisor
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The above set of logical sentences describes the
problem domain. The assertions are numbered so that
they may be referenced in the following trace:
Using unification and modus ponens, a correct
investment strategy for this individual may be inferred
as a logical consequence of the above descriptions. A
first step would be to unify the conjunctions of 11 and
13 with the first two components of the premise of 7
i.e,
earnings(25000,steady) ^ dependents(3) unifies with
earnings(X,steady) ^ dependents(Y)
Under the substitutions {25000/X, 3/Y}. This
substitution yields the new implication.
Application: A Logic-Based Financial
Advisor
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earnings(25000,steady)^dependents(3)^~greater(25000,
minincome(3))=>income (inadequate).
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earnings(25000,steady)^dependents(3)^~greater(25000,
27000)=> income (inadequate).
All three components of the premise are individually
true, by 3, 11, and the mathematical definition of
greater, their conjunction is true and the entire premise
is true. Now, Modus ponens may therefore be applied,
yielding the conclusion income (inadequate). This is
added as a new assertion.
14. income (inadequate)
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Application: A Logic-Based Financial
Advisor
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Similarly, amount_saved(22000)^dependents(3) unifies
with the first two elements of the premise of assertion 4
under the substitution {22000/X,3/Y} yielding the
implication:
amount_saved(22000)^dependents(3)^greater(22000,min
savings(3))=>savings_account(adequate).
Here, evaluating the function minsavings(3) yields:
amount_saved(22000)^dependents(3)^greater(22000,150
00)=>savings_account(adequate).
It yields the conclusion:
15. savings_account(adequate). So after applying Modus
ponens on 14, 15 & 3
Application: A Logic-Based Financial
Advisor
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3.savings_account(adequate)^income(inadequate)
=> investment (combination).
This will be the suggested investment of this
individual.
The above application shows how predicate
calculus may be used to reason about a realistic
problem, drawing correct conclusions by applying
inference rules to the initial problem description.
In the later lectures, we will discuss, how these
things can be implemented on a computer.
Home Work
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Read chapter 2 of George F. Luger along with the
handout on Predicate Calculus.
Try to convert as many sentences as you can in
Predicate Calculus.
Try to implement Unification and unify the predicate
expressions
We will do Resolution Theorem Proving in the next
lecture ref: ch.12 Luger……………..it includes
predicate, unification, modus ponens, refutation along
with other stuff covered so far…..
Assignment No 1
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Implement the algorithm verifiy_sentence from
chapter no 2 of the book by George F. Luger in
any computer language of your choice.
Due date October 25th, 2010.
Submit the code (.cpp) and (.exe) files in the in
assign1_roll_number folder with your
registration # as your file names. 04-123.cpp
Add as many comments as possible in your code
file, so that documentation won’t be required.