1. 09-09-05Prof. Pushpak Bhattacharyya, IIT Bombay 1 CS 621 Artificial Intelligence Lecture 16 – 09/09/05 Prof. Pushpak Bhattacharyya Soundness, Completeness, Consistency and AI

2. 09-09-05Prof. Pushpak Bhattacharyya, IIT Bombay 2 Intelligence It is important to see how much of intelligence is MECHANISABLE. ROUTINE + SOPHISTICATED how much

3. 09-09-05Prof. Pushpak Bhattacharyya, IIT Bombay 3 Mechanical Devices Some of them are - lathe - pulley - crane Augment the physical ability.

4. 09-09-05Prof. Pushpak Bhattacharyya, IIT Bombay 4 Mental Ability - Routine component ? Multiplication - Table Calculator performs - Routine calculation

5. 09-09-05Prof. Pushpak Bhattacharyya, IIT Bombay 5 Domains Numerical Domain - Machines take up routine tasks & outperform humans. Symbolic Domain – –Pattern recognition –Logic How much is mechanisable

6. 09-09-05Prof. Pushpak Bhattacharyya, IIT Bombay 6 Principlia Mathematica Bertrand Russell & subsequent Mathematicians & Logicians – PRINCIPIA MATHEMATICA - Formal Systems

7. 09-09-05Prof. Pushpak Bhattacharyya, IIT Bombay 7 Mechanical Procedure Axioms Theorems Inference rules A B D C O  AOB =  COD EUCLID’s AXIOMS

8. 09-09-05Prof. Pushpak Bhattacharyya, IIT Bombay 8 Can whole of Mathematics be mechanised? Is Mathematical/logical intelligence formalisable. In 1930 came the GÖDEL THEOREM –“In any formal system of sufficient power, soundness & completeness cannot be achieved simultaneously”.

9. 09-09-05Prof. Pushpak Bhattacharyya, IIT Bombay 9 GÖDEL THEOREM In any formal system of “enough power”, there are theorems which cannot be proved by the machinery of the formal system.

10. 09-09-05Prof. Pushpak Bhattacharyya, IIT Bombay 10 Formal Systems Alphabets Well formed formulae Axioms Inference rules Semantics HILBERT’s AXIOMATISATION OF PROPOSITIONAL CALCULUS or BOOLEAN ALGEBRA

11. 09-09-05Prof. Pushpak Bhattacharyya, IIT Bombay 11 Alphabets in Propositional Calculus Propositions denoted by capital letters towards the end of English letters P, Q, R, S …. -propositions Special Alphabets –  Implication ℱ false (Gothic F) (open parenthesis )close parenthesis

12. 09-09-05Prof. Pushpak Bhattacharyya, IIT Bombay 12 Well Formed Formula (WFF) S  ℱ | P | (S  S) syntax Example: (P  (Q  R))

13. 09-09-05Prof. Pushpak Bhattacharyya, IIT Bombay 13 Axioms (Starting Structures) A1:(A  (B  A)) A2:((A  (B  C))  ((A  B)  (A  C))) A3:(((A  ℱ )  ℱ )  A)

14. 09-09-05Prof. Pushpak Bhattacharyya, IIT Bombay 14 Inference Rule Modus Ponens Given (A  B) & A Write B

15. 09-09-05Prof. Pushpak Bhattacharyya, IIT Bombay 15 Semantics Invoking Meta-symbols T : true F : false

16. 09-09-05Prof. Pushpak Bhattacharyya, IIT Bombay 16 Valuation V : W  {T, F} W is WFF in Propositional Calculus By definition, V( ℱ ) = F

17. 09-09-05Prof. Pushpak Bhattacharyya, IIT Bombay 17 V(A  B) By definition, V(A  B) is obtained from a table called the “truth table”. V(A)V(B) V(A  B) T TT T FF F TT F FT

18. 09-09-05Prof. Pushpak Bhattacharyya, IIT Bombay 18 Metaconcept If V(W) = T for all assigned values in the components of W, then W is called a TAUTOLOGY METACONCEPT

19. 09-09-05Prof. Pushpak Bhattacharyya, IIT Bombay 19 Define “PROOF” A PROOF is a sequence of WFF, where each line is a hypothesis or an axiom or result of inference.

20. 09-09-05Prof. Pushpak Bhattacharyya, IIT Bombay 20 Example Proof of “R” from “P”, “P  Q” & “Q  R” -Starting hypothesis H1: P H2:P  Q H3:Q  R

21. 09-09-05Prof. Pushpak Bhattacharyya, IIT Bombay 21 Example (Contd.) L1:P, H1 L2:P  Q, H2 L3:Q  R, H3 L4:Q, MP L1, L2 L5:R, MP L4, L3 Proof

22. 09-09-05Prof. Pushpak Bhattacharyya, IIT Bombay 22 Theorem Proving Proof of structures where no hypothesis is given is called Theorem Proving. Proved structure is Theorem.

23. 09-09-05Prof. Pushpak Bhattacharyya, IIT Bombay 23 Example Show that P  P is a theorem Proof: L1:(P  (P  P)), A1 L2:(P  ((P  P)  P)), A1 L3:[(P  ((P  P)  P))  ((P  (P  P))  (P  P))], A2 L4:[(P  (P  P))  (P  P)], MP L2, L3 L5: (P  P), MP L1, L4 Last line of the “picture”. So, (P  P) is a theorem.