1. Encoding Knowledge with First Order Predicate Logic
Outline:
Motivation
Predicate calculus terminology
Expressing knowledge in logic
An example proof by resolution
Formal interpretations and satisfiability
CSE (c) S. Tanimoto, Predicate Logic

2. Motivation: More Accurate and General Representations of Knowledge
“If Willy is a bird, then Willie can fly.”
Propositional calculus:
P: Willy is a bird
Q: Willy can fly
P -> Q
“Any bird can fly.”
Predicate calculus
Forall x (Bird(x) -> CanFly(x) )
CSE (c) S. Tanimoto, Predicate Logic

3. CSE 415 -- (c) S. Tanimoto, 2002 Predicate Logic
Another Example
“John’s grandfather was a blacksmith, and his mother was a seamstress.”
Propositional calculus:
P1: John’s grandfather was a blacksmith.
P2: John’s grandmother was a seamstress.
P & Q
Predicate calculus:
Q1(x): x is a blacksmith.
Q2(x): x is a seamstress.
f(x): grandfather of x.
Q1(f(x)) & Q2(m(x))
CSE (c) S. Tanimoto, Predicate Logic

4. Terminology of the Predicate Calculus
Predicate symbols:
P, Q, R, P1, P2, ... , Q1, Q2, ..., R1, R2, ...
If P takes n arguments, we call P an n-ary predicate
P( term1, term2, , termn)
n >= 0
Function symbols:
f, g, h, f1, f2, , g1, g2, , h1, h2, . . .
A term is either:
a constant: e.g., a, b, c, a1, a2, , b1, b2, , c1, c2, . . .
a variable: e.g., x, y, z, x1, x2, , y1, y2, , z1, z2, . . .
an n-ary function symbol, followed by n terms in parentheses.
Examples of terms: x, a, f(x), g1(a, y), h(f(x), g(y, b))
CSE (c) S. Tanimoto, Predicate Logic

5. CSE 415 -- (c) S. Tanimoto, 2002 Predicate Logic
Well-Formed Formulas
Atomic formula: an n-ary predicate symbol followed by n terms in parentheses:
e.g., P(x, f(x, a, y))
If F is an atomic formula, then it is a well-formed formula (WFF).
Compound formula: a statement formed from one or more WFFs using logical connectives.
If F is a compound formula, then it is a WFF.
e.g., P(x, f(x, a, y)) v (Q -> R(x))
Quantified formula: a statement formed from a WFFs by prefixing it with a quantifier and a variable.
e.g., (Forall x) (P(x, f(x, a, y)) v (Q -> R(x))) (Exists y) Q(y)
If F is a quantified formula, then it is a WFF.
CSE (c) S. Tanimoto, Predicate Logic

6. A Syllogism for Predicate Logic
Coloured flowers are always scented.
I dislike flowers that are not grown in the open air.
No flowers grown in the open air are colourless.
Therefore, I dislike all flowers that are not scented.
Lewis Carroll
CSE (c) S. Tanimoto, Predicate Logic

7. Encoding the Syllogism
Predicate symbols:
C(x) = x is coloured.
S(x) = x is scented.
D(x) = I dislike x.
A(x) = x is grown in the open air.
CSE (c) S. Tanimoto, Predicate Logic

8. Encoding the Syllogism
Coloured flowers are always scented.
(forall x) ( C(x) -> S(x) )
I dislike flowers that are not grown in the open air.
(forall x) ( ~A(x) -> D(x) )
No flowers grown in the open air are colourless.
(forall x) ( A(x) -> ~~C(x) )
Therefore, I dislike all flowers that are not scented.
(forall x) (~S(x) -> D(x) )
CSE (c) S. Tanimoto, Predicate Logic

9. Proof by Resolution: (1) Negate the conclusion
Coloured flowers are always scented.
(forall x) ( C(x) -> S(x) )
I dislike flowers that are not grown in the open air.
(forall x) ( ~A(x) -> D(x) )
No flowers grown in the open air are colourless.
(forall x) ( A(x) -> ~~C(x) )
It is not the case that I dislike all flowers that are not scented.
~(forall x) (~S(x) -> D(x) )
CSE (c) S. Tanimoto, Predicate Logic

10. Proof by Resolution: (2) Obtain Clause Form
Coloured flowers are always scented.
(forall x) ( C(x) -> S(x) )
(forall x) (~C(x) v S(x) ) Rewrite -> using ~ and v.
~C(x) v S(x) Drop leading universal quantifier.
The other clauses are:
A(x) v D(x)
~A(x) v C(x)
(exists x) ~(~S(x) -> D(x) ) Move negation inward
(exists x) (~S(x) & ~D(x) ) DeMorgan’s law
~S(a) & ~D(a) Introduce Skolem constant
~S(a), ~D(a) Break into 2 clauses.
CSE (c) S. Tanimoto, Predicate Logic

11. Proof by Resolution: (3) Resolve
P1: ~C(x) v S(x)
P2: A(y) v D(y) variables are standardized apart.
P3: ~A(z) v C(z)
P4: ~S(a)
P5: ~D(a)
P6: A(a) P2, P5 { a/y }
P7: C(a) P3, P6 { a/z }
P8: S(a) P1, P7 { a/x }
P9: [] P4, P8 { }
CSE (c) S. Tanimoto, Predicate Logic

12. Formal Interpretations
An interpretation for a given formula W consists of the following:
A domain D of elements that can be referred to by terms. E.g., D = (0, 1, 2, }
An assignment that maps each constant symbol of W to an element of D.
For each n-ary function symbol of W a mapping from n-tuples of elements of D to single elements of D.
For each n-ary predicate symbol of W a mapping from n-tuples of elements of D to {T, F}.
CSE (c) S. Tanimoto, Predicate Logic

13. Example of Interpretation
Let W = (forall x) (P(a) & Q(x, a) )
An interpretation for W:
D = { apple, peach }
{ a = apple } mapping for constants
{ } mappings for functions
{ P(apple) = T, P(peach) = T; mappings for predicates
Q(apple,apple) = T, Q(apple, peach) = T,
Q(peach,apple) = T, Q(peach, peach) = T }
CSE (c) S. Tanimoto, Predicate Logic

14. CSE 415 -- (c) S. Tanimoto, 2002 Predicate Logic
Satisfiability
A formula W is satisfiable iff there exists some interpretation of W that make W true.
A formula W is unsatisfiable iff there does NOT exist any interpretation of W that make W true.
Then W is inconsistent. W is a contradiction.
If every interpretation of W satisfies W (makes W true), then W is a tautology.
If I is an interpretation for W that satisfies W , then I is a model for W.
CSE (c) S. Tanimoto, Predicate Logic

15. Another Interpretation
Let W = (forall x) (P(a) & Q(x, a) )
An interpretation for W:
D = { 0, 1, 2, }
{ a = 0 } mapping for constants
{ } mappings for functions
{ P(n) iff n = 0; mappings for predicates
Q(m,n) iff n > m. }
Note that ~Q(x, x).
This interpretation fails to satisfy W. It’s not a model for W.
CSE (c) S. Tanimoto, Predicate Logic

16. The Basis of Proof by Resolution
Resolution is used to prove that a given set of clauses is inconsistent. (i.e., it cannot have a model).
Each time a resolvent is formed, it is added to the set of clauses. This does not change the consistency of the set.
However, if the null clause is ever added to the set, then it becomes very obvious that the set is inconsistent.
CSE (c) S. Tanimoto, Predicate Logic