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

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

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

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

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

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

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

Prof. 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”.

Prof. 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.

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

Prof. 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

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

Prof. 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)

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

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

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

Prof. 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

Prof. 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

Prof. 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.

Prof. 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

Prof. 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

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

Prof. 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.