CS621: Artificial Intelligence

Pushpak BhattacharyyaCSE Dept., IIT Bombay

Lecture–10: Soundness of Propositional Calculus

12th August, 2010

Soundness, Completeness &Consistency

Soundness

Syntactic World

----------Theorems,

Proofs

SemanticWorld

----------Valuation,Tautology

Soundness

Completeness

* *

� Soundness

� Provability Truth� 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 Expression

s

System

OutsideKnowledge

Validation

Consistency

The System should not be able to

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

able to derive

F

Examine the relation between

SoundnessSoundness

&

Consistency

Soundness Consistency

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

F , it can prove any expression to be aF , it can prove any expression to be a

theorem. Because

F � P is a theorem

Inconsistency�UnsoundnessTo 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� 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: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 Syntactic ‘falseDefinition of V

V(F ) = F

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

Semantic ‘false’

V(F ) = F

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

V(A) V(B) V(A�B)

T F FT T TF F TF T T

Tautology

An expression ‘E’ is a tautology if

V(E) = TV(E) = T

for all valuations of constituent propositions

Each ‘valuation’ is called a ‘model’.

To see that

(F�P) is a tautology

two modelsV(P) = TV(P) = F

V(F�P) = T for both

F�P is a theorem

Soundness Completeness

F�P is a tautology

If a system is Sound & Complete, it does not

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

Take the Syntactic Path or the Semantic Path

Syntax vs. Semantics issue

Refers to

FORM VS. CONTENTFORM VS. CONTENT

Tea

(Content)Form

Form & Content

logician

paintermusician

Godel, Escher, Bach

By D. Hofstadter

logician

Problem

(P Q)�(P Q)

Semantic ProofSemantic ProofA 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 (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); MPP�(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

GivenGiven

A1, A2, … ,An |- B

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

Proof

Case 1 B is an axiomCase 1 B is an axiom

V(B) = T by actual observation

Statement is correct

Case 2 B is one of Ais

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

statement is correct

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

Ej is Ei B.

.Ej is Ei B

Suppose V(B) = F

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

.

.

Ei

.

.

.

Ej

.

.

.

B

i.e. Ei/Ej is result of MP of two expressions coming before them

Thus we progressively deal with shorter and shorter proof body. 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 alwayslie. 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 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 truthOr always lies)

T (tourist)

Deciding the Propositions: a very difficult step- needs human intelligence

� P: Left road leads to capital

� Q: S always speaks the truth

Meta Question: What question should the tourist ask

� The form of the question

� Very difficult: needs human intelligence

� The tourist should ask� The tourist should ask� Is R true?

� The answer is “yes” if and only if the left road leads to the capital

� The structure of R to be found as a function of P and Q

A more mechanical part: use of truth table

P Q S’s Answer

R

T T Yes TT T Yes T

T F Yes F

F T No F

F F No T

Get form of R: quite mechanical

� From the truth table

� R is of the form (P x-nor Q) or (P ≡ Q)

Get R in English/Hindi/Hebrew…

� Natural Language Generation: non-trivial

� The question the tourist will ask is

� Is it true that the left road leads to the Is it true that the left road leads to the capital if and only if you speak the truth?

� Exercise: A more well known form of this question asked by the tourist uses the X-OR operator instead of the X-Nor. What changes do you have to incorporate to the solution, to get that answer?