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?