CS344 : Introduction to Artificial Intelligence

1. CS344 : Introduction to Artificial Intelligence Pushpak BhattacharyyaCSE Dept., IIT Bombay Lecture 9,10,11- Logic; Deduction Theorem 23/1/09 to 30/1/09

2. Logic and inferencing Vision NLP • Search • Reasoning • Learning • Knowledge Expert Systems Robotics Planning Obtaining implication of given facts and rules -- Hallmark of intelligence

3. Propositions • Stand for facts/assertions • Declarative statements • As opposed to interrogative statements (questions) or imperative statements (request, order) Operators => and ¬ form a minimal set (can express other operations) - Prove it. Tautologies are formulae whose truth value is always T, whatever the assignment is

4. Model In propositional calculus any formula with n propositions has 2n models (assignments) - Tautologies evaluate to T in all models. Examples: 1) 2) • e Morgan with AND

5. Formal Systems • Rule governed • Strict description of structure and rule application • Constituents • Symbols • Well formed formulae • Inference rules • Assignment of semantics • Notion of proof • Notion of soundness, completeness, consistency, decidability etc.

6. Hilbert's formalization of propositional calculus 1. Elements are propositions : Capital letters 2. Operator is only one :  (called implies) 3. Special symbolF(called 'false') 4. Two other symbols : '(' and ')' 5. Well formed formula is constructed according to the grammar WFF P|F|WFFWFF 6. Inference rule : only one Given AB and A write B known asMODUS PONENS

7. 7. Axioms : Starting structures A1: A2: A3 This formal system defines the propositional calculus

8. Notion of proof 1. Sequence of well formed formulae 2. Start with a set of hypotheses 3. The expression to be proved should be the last line in the sequence 4. Each intermediate expression is either one of the hypotheses or one of the axioms or the result of modus ponens 5. An expression which is proved only from the axioms and inference rules is called a THEOREM within the system

9. Example of proof From P and and prove R H1: P H2: H3: i) P H1 ii) H2 iii) Q MP, (i), (ii) iv) H3 v) R MP, (iii), (iv)

10. Prove that is a THEOREM i) A1 : P for A and B ii) A1: P for A and for B iii) A2: with P for A, for B and P for C iv) MP, (ii), (iii) v) MP, (i), (iv)

11. Formalization of propositional logic (review) Axioms : A1 A2 A3 Inference rule: Given and A, write B A Proof is: A sequence of i) Hypotheses ii) Axioms iii) Results of MP A Theorem is an Expression proved from axioms and inference rules

12. Example: To prove i) A1 : P for A and B ii) A1: P for A and for B iii) A2: with P for A, for B and P for C iv) MP, (ii), (iii) v) MP, (i), (iv)

13. Shorthand 1. is written as and called 'NOTP' 2. is written as and called 'PORQ’ 3. is written as and called 'P AND Q' Exercise: (Challenge) - Prove that

14. A very useful theorem (Actually a meta theorem, called deduction theorem) Statement If A1, A2, A3 ............. An├ B then A1, A2, A3, ...............An-1├ ├ is read as 'derives' Given A1 A2 A3 . . . . An B A1 A2 A3 . . . . An-1 Picture 1 Picture 2

15. Use of Deduction Theorem Prove i.e., ├ F(M.P) A├(D.T) ├(D.T) Very difficult to prove from first principles, i.e., using axioms and inference rules only

16. Prove i.e. ├ F ├ (D.T) ├ Q (M.P with A3) P├ ├

17. Formalization of propositional logic (review) Axioms : A1 A2 A3 Inference rule: Given and A, write B A Proof is: A sequence of i) Hypotheses ii) Axioms iii) Results of MP A Theorem is an Expression proved from axioms and inference rules

18. Example: To prove i) A1 : P for A and B ii) A1: P for A and for B iii) A2: with P for A, for B and P for C iv) MP, (ii), (iii) v) MP, (i), (iv)

19. Shorthand 1. is written as and called 'NOTP' 2. is written as and called 'PORQ’ 3. is written as and called 'P AND Q' Exercise: (Challenge) - Prove that

20. A very useful theorem (Actually a meta theorem, called deduction theorem) Statement If A1, A2, A3 ............. An├ B then A1, A2, A3, ...............An-1├ ├ is read as 'derives' Given A1 A2 A3 . . . . An B A1 A2 A3 . . . . An-1 Picture 1 Picture 2

21. Use of Deduction Theorem Prove i.e., ├ F(M.P) A├(D.T) ├(D.T) Very difficult to prove from first principles, i.e., using axioms and inference rules only

22. Prove i.e. ├ F ├ (D.T) ├ Q (M.P with A3) P├ ├

23. More proofs

24. Proof Sketch of the Deduction Theorem To show that If A1, A2, A3,… An |- B Then A1, A2, A3,… An-1 |- AnB

25. Case-1: B is an axiom One is allowed to write A1, A2, A3,… An-1 |- B |- B(AnB) |- (AnB); mp-rule

26. Case-2: B is An AnAn is a theorem (already proved) One is allowed to write A1, A2, A3,… An-1 |- (AnAn) i.e. |- (AnB)

27. Case-3: B is Ai where (i <>n) Since Aiis one of the hypotheses One is allowed to write A1, A2, A3,… An-1 |- B |- B(AnB) |- (AnB); mp-rule

28. Case-4: B is result of MP Suppose B comes from applying MP on Ei and Ej Where, Ei and Ej come before B in A1, A2, A3,… An |- B

29. B is result of MP (contd) If it can be shown that A1, A2, A3,… An-1 |- An Ei and A1, A2, A3,… An-1 |- (An (EiB)) Then by applying MP twice A1, A2, A3,… An-1 |- An B

30. B is result of MP (contd) This involves showing that If A1, A2, A3,… An |- Ei Then A1, A2, A3,… An-1 |- AnEi (similarly for AnEj)

31. B is result of MP (contd) Adopting a case by case analysis as before, We come to shorter and shorter length proof segments eating into the body of A1, A2, A3,… An |- B Which is finite. This process has to terminate. QED

32. Important to note • Deduction Theorem is a meta-theorem (statement about the system) • PP is a theorem (statement belonging to the system) • The distinction is crucial in AI • Self reference, diagonalization • Foundation of Halting Theorem, Godel Theorem etc.

33. Example of ‘of-about’ confusion • “This statement is false” • Truth of falsity cannot be decided