Transcript pptx

CS621: Artificial Intelligence
Pushpak Bhattacharyya
CSE Dept.,
IIT Bombay
Lecture–10: Soundness of Propositional
Calculus
12th August, 2010
Soundness, Completeness &
Consistency
Soundness
Semantic
World
----------
Syntactic
World
---------Theorems,
Proofs
*
Valuation,
Tautology
Completeness
*

Soundness


Provability
Truth
Completeness

Truth
Provability

Soundness:


Proved entities are indeed true/valid
Completeness:

Correctness of the System
Power of the System
True things are indeed provable
System
TRUE
Expression
s
Validation
Outside
Knowledge
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
Syntactic ‘false
Semantic ‘false’
Where F is called ‘false’ and is one of the two
symbols (T, F)
V(F ) = F
V(AB) is defined through what is called the
truth table
V(A)
T
T
F
F
V(B)
F
T
F
T
V(AB)
F
T
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
Soundness
Completeness
FP is a tautology
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
Form
(Content)
Form & Content
painter
logician
musician
Godel, Escher, Bach
By D. Hofstadter
Problem
(P
Q)(P
Q)
A
B
Semantic Proof
P
T
T
F
F
Q
F
T
F
T
P
Q
F
T
F
F
P
Q
T
T
F
T
AB
T
T
T
T
To show syntactically
(P Q) (P
i.e.
[(P
(Q
F ))
Q)
F ]
[(P
F )
Q]
If we can establish
(P
(Q
F ))
F,
(P
F ), Q
This is shown as
Q
F hypothesis
(Q F ) (P (Q
F ⊢F
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
.
.
.
Ei
.
.
.
Ej
.
.
.
B
B is the result of MP on Ei & Ej
Ej is Ei
B
Suppose V(B) = F
Then either V(Ei) = F or V(Ej) = F
i.e. Ei/Ej is result of MP of two expressions
coming before them
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 truthsayers and liers

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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


P: Left road leads to capital
Q: S always speaks the truth
Meta Question: What question
should the tourist ask
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The form of the question
Very difficult: needs human intelligence
The tourist should ask
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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
R
T
S’s
Answer
Yes
T
T
F
Yes
F
F
T
No
F
F
F
No
T
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…
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Natural Language Generation: non-trivial
The question the tourist will ask is
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
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?