1. CS621: Artificial Intelligence
Pushpak Bhattacharyya CSE Dept., IIT Bombay
Lecture 11- Soundness and Completeness; proof of soundness; start of proof of completeness
12th august, 2010

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

3. 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’

4. 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

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

6. Soundness
Provability Validity
Completeness
Validity Provability

7. Soundness: Correctness of the System
Proved entities are indeed valid
Completeness: Power of the System
Valid things are indeed provable

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

9. Examine the relation between Soundness & Consistency

10. 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

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

12. 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

13. 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

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

15. 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

16. 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

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

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

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

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

21. Case 3 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 . Ei Ej B

22. 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

23. Towards Completeness Proof

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

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

26. 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.

27. 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

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

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

30. 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)

31. 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

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

33. ~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)

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

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

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

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

38. 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

39. 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

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