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

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

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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

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

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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.

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Hilbert's formalization of propositional calculus 1. Elements are propositions : Capital letters 2. Operator is only one : (called implies) 3. Special symbol F (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

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

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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

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Example of proof From P and and prove R H1: P H2: H3: i) PH1 ii) H2 iii) QMP, (i), (ii) iv) H3 v) RMP, (iii), (iv)

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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)

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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

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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)

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Shorthand 1. is written as and called 'NOT P' 2. is written as and called 'P OR Q’ 3. is written as and called 'P AND Q' Exercise: (Challenge) - Prove that

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A very useful theorem (Actually a meta theorem, called deduction theorem) Statement If A 1, A 2, A A n ├ B then A 1, A 2, A 3, A n-1 ├ ├ is read as 'derives' Given A1A2A3....AnBA1A2A3....AnB Picture 1 A 1 A 2 A 3. A n-1 Picture 2

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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

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

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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

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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)

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Shorthand 1. is written as and called 'NOT P' 2. is written as and called 'P OR Q’ 3. is written as and called 'P AND Q' Exercise: (Challenge) - Prove that

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A very useful theorem (Actually a meta theorem, called deduction theorem) Statement If A 1, A 2, A A n ├ B then A 1, A 2, A 3, A n-1 ├ ├ is read as 'derives' Given A1A2A3....AnBA1A2A3....AnB Picture 1 A 1 A 2 A 3. A n-1 Picture 2

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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

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

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More proofs

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Proof Sketch of the Deduction Theorem To show that If A 1, A 2, A 3,… A n |- B Then A 1, A 2, A 3,… A n-1 |- A n B

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Case-1: B is an axiom One is allowed to write A 1, A 2, A 3,… A n-1 |- B |- B (A n B) |- (A n B); mp-rule

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Case-2: B is A n A n A n is a theorem (already proved) One is allowed to write A 1, A 2, A 3,… A n-1 |- (A n A n ) i.e.|- (A n B)

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Case-3: B is A i where (i <>n) Since A i is one of the hypotheses One is allowed to write A 1, A 2, A 3,… A n-1 |- B |- B (A n B) |- (A n B); mp-rule

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Case-4: B is result of MP Suppose B comes from applying MP on E i and E j Where, E i and E j come before B in A 1, A 2, A 3,… A n |- B

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B is result of MP (contd) If it can be shown that A 1, A 2, A 3,… A n-1 |- A n E i and A 1, A 2, A 3,… A n-1 |- (A n (E i B)) Then by applying MP twice A 1, A 2, A 3,… A n-1 |- A n B

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B is result of MP (contd) This involves showing that If A 1, A 2, A 3,… A n |- E i Then A 1, A 2, A 3,… A n-1 |- A n E i (similarly for A n E j )

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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 A 1, A 2, A 3,… A n |- B Which is finite. This process has to terminate. QED

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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.

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

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