1. CS344 : Introduction to Artificial Intelligence
Pushpak Bhattacharyya CSE Dept., IIT Bombay
Lecture 12, 13, 14- Soundness and Completeness; proof of soundness; start of proof of completeness
Week of 2/2/09

2. Soundness, Completeness & Consistency
Semantic
World
Valuation,
Tautology
Syntactic
World
Theorems,
Proofs
Completeness * *

3. Soundness
Provability Truth
Completeness
Truth Provability

4. Soundness: Correctness of the System
Proved entities are indeed true/valid
Completeness: Power of the System
True things are indeed provable

5. TRUE
Expressions
Outside
Knowledge
System
Validation

6. Consistency F The System should not be able to
prove both P and ~P, i.e., should not be
able to derive F

7. Examine the relation between Soundness & Consistency

8. 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

9. 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

10. 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

11. 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’

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

13. Tautology An expression ‘E’ is a tautology if V(E) = T
for all valuations of constituent propositions
Each ‘valuation’ is called a ‘model’.

14. To see that
(FP) is a tautology
two models
V(P) = T
V(P) = F
V(FP) = T for both

15. FP is a theorem
FP is a tautology
Soundness
Completeness

16. 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

17. Syntax vs. Semantics issue
Refers to
FORM VS. CONTENT
Tea
(Content)
Form

18. Form & Content Godel, Escher, Bach By D. Hofstadter painter musician
logician

19. 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

20. To show syntactically
(P Q) (P Q)
i.e.
[(P (Q F )) F ]
[(P F ) Q]

21. 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

22. 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

23. “Whatever is provable is valid”
Soundness Proof
Hilbert Formalization of Propositional
Calculus is sound.
“Whatever is provable is valid”

24. Statement
Given
A1, A2, … ,An + B
V(B) is ‘T’ for all Vs for which V(Ai) = T

25. Proof
Case 1 B is an axiom
V(B) = T by actual observation
Statement is correct

26. Case 2 B is one of Ais
if V(Ai) = T, so is V(B)
statement is correct

27. Case 3 B is the result of MP on Ei & Ej Ei is Ei B Suppose V(B) = F
Then either V(Ei) = F or V(Ej) = F . Ei Ej B

28. 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

29. Towards Completeness Proof

30. Soundness: Correctness of the System
Proved entities are indeed true/valid
Completeness: Power of the System
True things are indeed provable

31. Tautology An expression ‘E’ is a tautology if V(E) = T
for all valuations of constituent propositions
Each ‘valuation’ is called a ‘model’.

32. Necessary results Statement: (pq)((~pq)q) Proof:
If we can show that
(pq), (~pq) |- q
Or,
(pq), (~pq), qF |- F
Then we are done.

33. Proof continued 1. (pq) H1 2. (~pq) H2 3. qF H3
4. (~pq) (~qp) theorem of contraposition
5. ~qp MP, 2, 4
6. P MP, 3,5
7. q MP, 6, 1
8. F MP,7,3
QED

34. How to prove contraposition
To show
(pq)(~q~p)
Proof:
pq, ~q, p |- F
Very obvious!

35. An example to illustrate the completeness proofq
p(p V q) T F

36. Running the completeness proof
For every row of the truth table set up a proof:
p, ~q |- p(p V q)
p, q |- p(p V q)
~p, q |- p(p V q)
~p, ~q |- p(p V q)

37. p, ~q |- p  (p V q)
i.e. p, ~q, p |- p V q
p, ~q, p, ~p |- q
p, ~q, p, ~p |- F
|- F  q
|- q

38. p, q |- p  (p V q)
i.e. p, q, p, ~p |- q
same as 1

39. ~p, q |- p  (p V q)
~p, q, p, ~p |- q
Same as 1, since F is derived
4. ~p, ~q |- p  (p V q)

40. Why all this? If we have shown p, q |- A and p, ~q |- A
then we can show that
p |- A

41. p |- (q  A)
also p |- (~q  A)
But (q  A)  ((~q  A)  A)
is a theorem
by MP twice
p |- A

42. General Statement of the completeness proof
If V(A) = T for all models
then
|- A

43. Elaborating,
If P1, P2, …, Pn are constituent propositions of A
and if V(A) = T for every model V(Pi) = T/F
then
|- A

44. We have a truth table with 2n rows
P1 P2 P Pn A
F F F F T
F F F T T .
T T T T T

45. If we can show
P1’, P2’, …, Pn’ |- A’
For every row where
Pi’ = Pi if V(Pi) = T
= ~Pi if V(Pi) = F
And A’ = A if V(A) = T
= ~A if V(A) = F

46. Lemma If row has P1’, P2’, …, Pn’, A’ Then P1’, P2’, …, Pn’ |- A’
A very critical result linking syntax with semantics