All rights reservedL. Manevitz Lecture 41 Artificial Intelligence Logic L. Manevitz.
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Transcript of All rights reservedL. Manevitz Lecture 41 Artificial Intelligence Logic L. Manevitz.
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All rights reserved L. Manevitz Lecture 4 1
Artificial IntelligenceLogic
L. Manevitz
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All rights reserved L. Manevitz Lecture 4 2
Logic - Advantages
• Pretty universal – (need to choose the correct language).
• Clear semantics– Interpretation.– Intended Interpretation.
• Uniform Method of Manipulating– Theorem proving via Resolution.– Green’s Trick for answer production.
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All rights reserved L. Manevitz Lecture 4 3
Representing Information
Formulas Formulas
Semanticconnection
Formal connection
RepresentationInterpretation
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All rights reserved L. Manevitz Lecture 4 4
Basic Concept
Let be a set of formulas.
Let φ be a specific formula.
When does φ logically follow from ?
Meaning : in every possible interpretation where is true then φ is true.
Notationally : φ
( “semantically implies φ).
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All rights reserved L. Manevitz Lecture 4 5
Syntax
• Language:– Logical symbols.– Relation symbols.– Function symbols.– Constant symbols.
• Terms – recursive.• Formulas – recursive.• Sentences – no free variables.
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All rights reserved L. Manevitz Lecture 4 6
Semantics - Interpretation
θ= <A|R1,R2,..,Ri,F1,F2,..,Fk,C1,C2,..,Cl>
Ri AxAx..xA
Fk : AxAx..xA A
Cl A
θ θ θ θ θ θ θ θ θ
θ
θ
θ
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All rights reserved L. Manevitz Lecture 4 7
Semantics
• To handle variables, we extend the notion of satisfaction to include variables
(explain on blackboard)
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All rights reserved L. Manevitz Lecture 4 8
How to work with
• Basic semantic equivalents :– (e.g. De Morgan’s Laws, Assoc. Laws,
Distr. Laws).
• Fundamental Problem :– Find mechanical procedure to test if
φ or not.
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All rights reserved L. Manevitz Lecture 4 9
Abstraction of Proof - φ
• Intuitive: φ1
φ2
φn = φ
• Example: “Acceptable Rule”
φ ψ (MP)
Where φi or
φi follows from {φj | j<i}
By “acceptable” rule
φψ
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All rights reserved L. Manevitz Lecture 4 10
Definitions
• A system S is consistent if whenever φ then φ.
• A system S is complete if whenever φ then φ.
s
s
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All rights reserved L. Manevitz Lecture 4 11
Theorem
• Completeness Theorem:
There is a system S
s.t. φ φ
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All rights reserved L. Manevitz Lecture 4 12
Example
• Trivially complete
where A is any formula.
• Sound Rule: (MP)
A
A BAB
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All rights reserved L. Manevitz Lecture 4 13
Example cont.
[M | s] A B iff [M | s] A or [M | s] B by def.
If [M | s] A then [M | s] A by def.
It follows that [M | s] B.
A BAB
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All rights reserved L. Manevitz Lecture 4 14
What kind of systems ?
• Mimic People (“Natural Deduction”).
• Mimic Mathematical Proof.
• Convenient for Computers.
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All rights reserved L. Manevitz Lecture 4 15
Proof Systems
“Natural Deduction”
Formal Computationale.g. Resolution
Completeness Theorem
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All rights reserved L. Manevitz Lecture 4 16
Control
DATA
Set of formulas
Rules of Deduction
What rule to apply to what ?
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All rights reserved L. Manevitz Lecture 4 17
Resolution System (Syntax)Sound Rule
• Basic Idea:
(φ1 … φn) ( φ1 ψ1 … ψm)
φ2 … φn ψ1 … ψm
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All rights reserved L. Manevitz Lecture 4 18
Semantic Equivalents
• We can check this, now that we have a definition of truth in interpretations.
• Example ~(~p) is equivalent to p
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All rights reserved L. Manevitz Lecture 4 19
Semantic Equivalents
• ( A) A
• A B A B
• A B B A
• A B (A B) (B A)
• Associative :– A (B C) (A B) C– A (B C) (A B) C
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All rights reserved L. Manevitz Lecture 4 20
Semantic Equivalents cont.
• Commutative:– A B B A– A B B A
• Distributive:– A (B C) (A B) (A C)– A (B C) (A B) (A C)
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All rights reserved L. Manevitz Lecture 4 21
Semantic Equivalents cont.
• De Morgan:– (A B) ( A) ( B)– (A B) ( A) ( B)
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All rights reserved L. Manevitz Lecture 4 22
Quantifier Equivalences
• De Morgan:• (( x)A(x)) ( x) A(x)• (( x)A(x)) ( x) A(x)
• x A(x) y A(y)• x A(x) y A(y)
• x [P(x) Q(x)] ( x P(x)) ( x Q(x))• x [P(x) Q(x)] ( x P(x)) ( x Q(x))
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All rights reserved L. Manevitz Lecture 4 23
Resolution (Predicate) S
• Convert to set of clauses.• Convert S to set of clauses.• Let Clauses := all the above clauses.• Repeat until NIL found :
– Select two clauses.
– Resolve (using Unification).
– Check result :• If result is NIL finish.
• Otherwise add result to Clauses
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All rights reserved L. Manevitz Lecture 4 24
Over all Procedure
1. Eliminate , .
2. Push to atomic.
3. Eliminate .
4. Rename variables in a formula.
5. Move quantifier to the left.
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All rights reserved L. Manevitz Lecture 4 25
Over all Procedure
6. Push down.
7. Eliminate conj. - By making separate formulas.
8. Rename variables in all formulas.
9. Eliminate - Convention.
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All rights reserved L. Manevitz Lecture 4 26
Example
• [ x (A(x) B(x)] [ y C(y)]
• [ x (A(x) B(x)] [ y C(y)]
• [ x (A(x) B(x)] [ y C(y)]
• [ x ( A(x) B(x)] [ y C(y)]
• A(a) B(a) y C(y)
• y [ A(a) B(a) C(y)]
• Get Clause [ A(a) B(a) C(y)]
To replace add a constant to language.
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All rights reserved L. Manevitz Lecture 4 27
- Elimination example
• x y Boss (y,x)– Problem:
• Constant not sufficient.
• y depends on x.
– Solution:• Add a new function symbol f().
• x Boss (f(x),x)
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All rights reserved L. Manevitz Lecture 4 28
Example
1. x [Brick(x) [ y[On(x,y) Pyramid(y)] y[On(x,y) On(y,x)]
y[ Brick(y) Equal(x,y)]]]
2. x [ Brick(x) [ y[On(x,y) Pyramid(y)] y[On(x,y) On(y,x)]
y[ ( Brick(y)) Equal(x,y)]]]
Eliminate
Push
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All rights reserved L. Manevitz Lecture 4 29
Example cont.
3. x [ Brick(x) [ y[On(x,y) Pyramid(y)] y[ On(x,y) On(y,x)] y[Brick(y) Equal(x,y)]]]
4. x [ Brick(x) [[On(x,support(x)) Pyramid(support(x))]
y[ On(x,y) On(y,x)] y[Brick(y)
Equal(x,y)]]]
Eliminate
Rename variables apart
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All rights reserved L. Manevitz Lecture 4 30
Example cont.
5. x [ Brick(x) [[On(x,support(x)) Pyramid(support(x))]
y[ On(x,y) On(y,x)] z[Brick(z) Equal(x,z)]]]
6. x y z [ Brick(x) [[On(x,support(x)) Pyramid(support(x))]
[ On(x,y) On(y,x)] [Brick(z)
Equal(x,z)]]]
Move to left
Push down
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All rights reserved L. Manevitz Lecture 4 31
Example cont.
7. x y z [ Brick(x) [On(x,support(x)) Pyramid(support(x))]]
[ Brick(x) On(x,y) On(y,x)] [ Brick(x) Brick(z) Equal(x,z)]]
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All rights reserved L. Manevitz Lecture 4 32
Example cont.
8. x y z [[ Brick(x) On(x,support(x))] [ Brick(x)
Pyramid(support(x))] [ Brick(x) On(x,y) On(y,x)]
[ Brick(x) Brick(z) Equal(x,z)]]Eliminate
conj.
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All rights reserved L. Manevitz Lecture 4 33
Example cont.
9. x [ Brick(x) On(x,support(x))] x [ Brick(x) Pyramid(support(x))] x y [ Brick(x) On(x,y) On(y,x)] x z [ Brick(x) Brick(z) Equal(x,z)]
Rename variables
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All rights reserved L. Manevitz Lecture 4 34
Example cont.
10. x [ Brick(x) On(x,support(x))] w [ Brick(w) Pyramid(support(w))] u y [ Brick(u) On(u,y) On(y,u)] v z [ Brick(v) Brick(z) Equal(v,z)]
Eliminate
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All rights reserved L. Manevitz Lecture 4 35
Example cont.
• Clause form:
Brick(x) On(x,support(x)) Brick(w)
Pyramid(support(w)) Brick(u) On(u,y) On(y,u) Brick(v) Brick(z) Equal(v,z)