CS621: Artificial Intelligence

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CS621: Artificial Intelligence. Pushpak Bhattacharyya CSE Dept., IIT Bombay Lecture–10: Soundness of Propositional Calculus 12 th August, 2010. Soundness, Completeness &amp; Consistency. Soundness. Semantic World ---------- Valuation, Tautology. Syntactic World ---------- Theorems,

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### CS621: Artificial Intelligence

Pushpak BhattacharyyaCSE Dept., IIT Bombay

Lecture–10: Soundness of Propositional Calculus

12th August, 2010

Soundness, Completeness &Consistency

Soundness

Semantic

World

----------

Valuation,

Tautology

Syntactic

World

----------

Theorems,

Proofs

Completeness

*

*

Soundness
• Provability Truth
• Completeness
• Truth Provability
Soundness:Correctness of the System
• Proved entities are indeed true/valid
• Completeness:Power of the System
• True things are indeed provable

TRUE

Expressions

Outside

Knowledge

System

Validation

Consistency

The System should not be able to

prove both P and ~P, i.e., should not be

able to derive

F

Examine the relation between

Soundness

&

Consistency

Soundness Consistency

If a System is inconsistent, i.e., can derive

F , it can prove any expression to be a

theorem. Because

F P is a theorem

InconsistencyUnsoundness

To show that

FP is a theorem

Observe that

F, PF ⊢ F By D.T.

F ⊢ (PF)F; A3

⊢ P

i.e. ⊢ FP

Thus, inconsistency implies unsoundness

UnsoundnessInconsistency
• Suppose we make the Hilbert System of propositional calculus unsound by introducing (A /\ B) as an axiom
• Now AND can be written as
• (A(BF ))F
• If we assign F to A, we have
• (F (BF )) F
• But (F (BF )) is an axiom (A1)
• Hence F is derived
Inconsistency is a Serious issue.

Informal Statement of Godel Theorem:

If a sufficiently powerful system is complete it is inconsistent.

Sufficiently powerful: Can capture at least Peano Arithmetic

Introduce Semantics in Propositional logic

Valuation Function V

Definition of V

V(F ) = F

Where F is called ‘false’ and is one of the two symbols (T, F)

Syntactic ‘false

Semantic ‘false’

V(F) = F

V(AB) is defined through what is called the truth table

V(A) V(B) V(AB)

T F F

T T T

F F T

F T T

Tautology

An expression ‘E’ is a tautology if

V(E) = T

for all valuations of constituent propositions

Each ‘valuation’ is called a ‘model’.

To see that

(FP) is a tautology

two models

V(P) = T

V(P) = F

V(FP) = T for both

FP is a theorem

FP is a tautology

Soundness

Completeness

If a system is Sound & Complete, it does not

matter how you “Prove” or “show the validity”

Take the Syntactic Path or the Semantic Path

Syntax vs. Semantics issue

Refers to

FORM VS. CONTENT

Tea

(Content)

Form

Form & Content

Godel, Escher, Bach

painter

musician

logician

Problem

(P Q)(P Q)

Semantic Proof

A B

P Q P Q P Q AB

T F F T T

T T T T T

F F F F T

F T F T T

To show syntactically

(P Q) (P Q)

i.e.

[(P (Q F )) F ]

[(P F ) Q]

If we can establish

(P (Q F )) F ,

(P F ), Q F ⊢ F

This is shown as

Q F hypothesis

(Q F ) (P (Q F)) A1

QF; hypothesis

(QF)(P(QF)); A1

P(QF); MP

F; MP

Thus we have a proof of the line we started with

Soundness Proof

Hilbert Formalization of Propositional

Calculus is sound.

“Whatever is provable is valid”

Statement

Given

A1, A2, … ,An|- B

V(B) is ‘T’ for all Vs for which V(Ai) = T

Proof

Case 1 B is an axiom

V(B) = T by actual observation

Statement is correct

Case 2 B is one of Ais

if V(Ai) = T, so is V(B)

statement is correct

Case 3 B is the result of MP on Ei & Ej

Ejis Ei B

Suppose V(B) = F

Then either V(Ei) = F or V(Ej) = F

.

.

.

Ei

.

.

.

Ej

.

.

.

B

Thus we progressively deal with shorter and shorter proof body.

Ultimately we hit an axiom/hypothesis.

Hence V(B) = T

Soundness proved

### A puzzle(Zohar Manna, Mathematical Theory of Computation, 1974)

From Propositional Calculus

Tourist in a country of truth-sayers and liers
• Facts and Rules: In a certain country, people either always speak the truth or always lie. A tourist T comes to a junction in the country and finds an inhabitant S of the country standing there. One of the roads at the junction leads to the capital of the country and the other does not. S can be asked only yes/no questions.
• Question: What single yes/no question can T ask of S, so that the direction of the capital is revealed?
Diagrammatic representation

Capital

S (either always says the truth

Or always lies)

T (tourist)

Deciding the Propositions: a very difficult step- needs human intelligence
• Q: S always speaks the truth
Meta Question: What question should the tourist ask
• The form of the question
• Very difficult: needs human intelligence