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CS 344 Artificial Intelligence By Prof: Pushpak Bhattacharya Class on 05/Feb/2007

CS 344 Artificial Intelligence By Prof: Pushpak Bhattacharya Class on 05/Feb/2007. Deduction Theorem: Application and Proof. Shortens the proof procedure. Statement: If A 1 , A 2 , A 3 , …, A n |-- B then A 1 , A 2 , A 3 , …, A n-1 |-- A n → B ( |-- means derives)

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CS 344 Artificial Intelligence By Prof: Pushpak Bhattacharya Class on 05/Feb/2007

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  1. CS 344 Artificial Intelligence By Prof: Pushpak Bhattacharya Class on 05/Feb/2007

  2. Deduction Theorem: Application and Proof • Shortens the proof procedure. • Statement: If A1, A2, A3, …, An |-- B then A1, A2, A3, …, An-1 |-- An → B (|-- means derives) • We will try to prove the following: • p → p • p → p v q • A) (p → q) → (~q → ~p) B) (~q → ~p) → (p → q) 4. (p → q) → ((~p → q) → q) 3 is called the law of composition.

  3. Example proofs • Proofs for axioms are tough to obtain from 1st principles. • Proof for example 3A: (p → q) → (~q → ~p) (p → q), ~q, p |-- ℱ (p → q), ~q |-- (p → ℱ) (p → q) |-- (~q → ~p) |-- (p → q) → (~q → ~p)

  4. Example proofs (contd.) • Proof of example 3B: (~q → ~p) → (p → q) (~q → ~p), p, (q → ℱ)|--ℱ (~q → ~p), p |-- ((q → ℱ) → ℱ) (~q → ~p), p |-- q (~q → ~p) |-- (p → q) |-- (~q → ~p) → (p → q)

  5. Example proofs: uses a previous theorem • Proof of example 4: (p → q) → ((~p → q) → q) (p → q), (~p → q), (q → ℱ) |-- (p → q) → (~q → ~p) (prev theorem) ℱ Now repeated application of DT will get the result

  6. Exercise • Prove the laws of Propositional Calculus using Deduction Theorem • Associativity • Commutativity • Distributivity • De Morgan’s Laws

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