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Logic Charles L Dodgson 1832 - 1898 It s so easy even - PowerPoint PPT Presentation

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IIT Bombay :: Autumn 2020 :: CS 207 :: Discrete Structures :: Manoj Prabhakaran Logic Charles L Dodgson 1832 - 1898 It s so easy even Propositions, computers can do Predicates, it! Operators, Formulas c i Expert Systems g o L

  1. IIT Bombay :: Autumn 2020 :: CS 207 :: Discrete Structures :: Manoj Prabhakaran Logic Charles L Dodgson 
 1832 - 1898 It’ s so easy even Propositions, computers can do Predicates, it! Operators, Formulas

  2. c i Expert Systems g o L n o i t c a n i From a repository consisting of “facts” derive answers to questions posed on the fly To automate decision making mother_child(trude, sally). father_child(tom, sally). father_child(tom, erica). father_child(mike, tom). e.g., Prolog: a programming sibling(X, Y) ← parent_child(Z, X) 
 language that can be used to ∧ parent_child(Z, Y). parent_child(X, Y) ← father_child(X, Y). implement such a system parent_child(X, Y) ← mother_child(X, Y). ?- sibling(sally, erica). Yes

  3. The Pointless Game Alice and Bob sit down to play a new board game, where they take turns to make “moves” that they can choose (no dice/randomness) The rules of the game guarantee that The game can’ t go on for ever There are no ties — Alice or Bob will win when the game terminates Alice and Bob (smart as they are) decide that there is no point in playing the game, because they already know who is going to win it! But how?

  4. Propositions Goal: reasoning about whether statements are true or false These statements are called propositions Propositions refer to things in a “domain of discourse” (e.g., characters in Alice in Wonderland) A proposition could simply refer to a property of an element in the domain (e.g., Alice doesn’ t have wings) These properties are formalised as predicates Alice Jabberwock Flamingo

  5. Predicates Predicate is a function that assigns a value of TRUE or FALSE to each element in the domain of discourse If you apply a predicate to an element you get a proposition A proposition will have truth value True or False More complex propositions can be built from such simple propositions e.g.: Pink(Flamingo) Winged? Flies? Pink? Alice FALSE FALSE FALSE Jabberwock TRUE TRUE FALSE Flamingo TRUE TRUE TRUE

  6. Propositional Calculus Binary operators Binary operators Unary operator “Operate on” propositions using operators p or q p and q if p then q not p Symbol: p ∨ q Symbol: p ∧ q Symbol: p → q Symbol: ¬p True if and only if True if and only if (¬p) ∨ (p ∧ q) Negates the truth at least one of p both of p and q Same as ¬p ∨ q value. and q is true are true George Boole 
 1815 - 1864 e.g.: ¬Flies(Alice) ∧ e.g.: Flies(Alice) → e.g.: ¬Flies(Alice) e.g.: ¬Flies(Alice) ∨ Pink(Jab’wock) Pink(Jab’wock) Pink(Jab’wock) has value True has value True has value True has value False → T F T F ∧ ∨ T F T T F T T F T T T F F F F T T F T F

  7. Operator Gallery (T) T F (F) T F T T T T F F F T T F F F ( 1) T F ( 2) T F ¬(1) T F ¬(2) T F T T T T T F T F F T F T F F F F T F F T T F F T ↔ ⊕ IFF XOR T F T F = ≠ T F T T T F F T F F F T ↓ ↑ OR ∨ NOR AND NAND T F ∧ T F T F T F T F F T T T T F T T T F F F T F T F F T T F F F ↛ IMPLIES T F IMPLIED-BY ← T F → T F ↚ T F T T F T T T T F T T F F F T T F F T F F F F T F

  8. p implies q. Try an example: whenever p holds, q holds IMPLIES T F → p: you’re in the kitchen if p then q. T T F q: you’re in the house q if p. F T T either not p or (p and q). Important: p only if q. Not a causal if not q then not p. Contrapositive relation! not p if not q. p is implied by q. ← IMPLIED-BY T F q implies p. T T T p if q. F F T p if and only if q. ↔ IFF T F p iff q. = T T F p if q and not p if not q. F F T

  9. Formulas A recipe for creating a new proposition from given propositions, using operators p q f e.g. f(p,q) ≜ (p ∧ q) ∨ ¬(p ∨ q) F F T f T F F T F ∧ T T F p ∨ T F F q ∨ F F T ¬ T T T Can also use “logic circuits” instead of formulas Different formulas can be equivalent to each other e.g., g(p,q) ≜ ¬(p ⊕ q). Then f ≡ g. A formula on two variables is equivalent to a binary operator

  10. Another Example g(p,q,r) ≜ (p ∧ q ∧ ¬r) ∨ (¬p ∧ q ∧ r) ∨ (p ∧ ¬q ∧ r) ∨ (p ∧ q ∧ r) “Majority operator” p q r g h h(p,q,r) ≜ (p ∧ (q ∨ r)) ∨ (q ∧ r) F F F F F F F T F F g ≡ h F T F F F T F F F F F T T T T T F T T T T T F T T ((¬p) ∧ q) ∧ r T T T T T

  11. More Equivalences [Exercise] T ∧ q ≡ q F ∨ q ≡ q Conjunction and disjunction with T and F F ∧ q ≡ F T ∨ q ≡ T T → q ≡ q F → q ≡ T Implication involving T and F q → F ≡ ¬q q → T ≡ T q → ¬q ≡ ¬q ¬q → q ≡ q Implication involving negation Contrapositive p → q ≡ (¬q) → (¬p) p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) Distributive Property p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r) De Morgan’ s Law ¬(p ⋀ q) ≡ ¬p ⋁ ¬q ¬(p ⋁ q) ≡ ¬p ⋀ ¬q

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