CSE not every positive integer is prime 311 some positive integer - PowerPoint PPT Presentation

Negations of quantifiers CSE • not every positive integer is prime 311 • some positive integer is not prime • prime numbers do not exist Foundations of • every positive integer is not prime Computing I Fall 2014 Negations of Quantifiers De Morgan’s Laws for Quantifiers • ∀ x PurpleFruit(x) Domain: ¬∀ x P(x) ≡ ∃ x ¬ P(x) Fruit • “All fruits are purple” • What is ¬∀ x PurpleFruit(x)? ¬∃ x P(x) ≡ ∀ x ¬ P(x) PurpleFruit(x) • “Not all fruits are purple” • How about ∃ x PurpleFruit(x) ? • “There is a purple fruit” • If it’s the negation, all situations should be covered by a statement and its negation • Consider the domain {Orange} : Neither statement is true! • No! • How about ∃ x ¬ PurpleFruit(x) ? • “There is a fruit that isn’t purple” • Yes!

De Morgan’s laws for Quantifiers Scope of Quantifiers Example: NotLargest(x) ≡ ∃ y Greater (y, x) ¬∀ x P(x) ≡ ∃ x ¬ P(x) ≡ ∃ z Greater (z, x) ¬ ∃ x P(x) ≡ ∀ x ¬ P(x) truth value: “ There is no largest integer ” doesn’t depend on y or z “bound variables” ¬ ∃ x ∀ y ( x ≥ y) does depend on x “free variable” ≡∀ x ¬ ∀ y ( x ≥ y) ≡∀ x ∃ y ¬ ( x ≥ y) quantifiers only act on free variables of the formula ≡∀ x ∃ y (y > x) they quantify ∀ x ( ∃ y (P(x,y) → ∀ x Q(y, x))) “ For every integer there is a larger integer ” scope of quantifiers CSE 311: Foundations of Computing Fall 2014 vs. ∃ x (P(x) ∧ ∧ Q(x)) ∃ x P(x) ∧ ∧ ∃ x Q(x) ∧ ∧ ∧ ∧ Lecture 6: Predicate Logic, Logical Inference This one asserts P This one asserts P and Q and Q of the same x. of potentially different x’s.

Turtles All The Way Down Nested Quantifiers If the tortoise walks at a rate of one node per step, and the • Bound variable names don’t matter hare walks at a rate of two nodes per step, then the ∀ x ∃ y P(x, y) ≡ ∀ a ∃ b P(a, b) distance between them increases by one node per step. • Positions of quantifiers can sometimes change If the tortoise is on node x, and the hare is on node 2x, then ∀ x (Q(x) ∧ ∃ y P(x, y)) ≡ ∀ x ∃ y (Q(x) ∧ P(x, y)) the distance between them increases by one node per step • But: order is important... OnNode(x) Domain: Non-negative Integers Predicate with Two Variables Quantification with Two Variables expression when true when false y ∀ x ∀ y P(x, y) ∃ x ∃ y P(x, y) x P(x,y) ∀ x ∃ y P(x, y) ∃ y ∀ x P(x, y)

Logical Inference Applications of Logical Inference • So far we’ve considered: • Software Engineering – Express desired properties of program as set of logical – How to understand and express things using constraints propositional and predicate logic – Use inference rules to show that program implies that – How to compute using Boolean (propositional) logic those constraints are satisfied – How to show that different ways of expressing or • Artificial Intelligence computing them are equivalent to each other – Automated reasoning • Algorithm design and analysis • Logic also has methods that let us infer implied – e.g., Correctness, Loop invariants. properties from ones that we know • Logic Programming, e.g. Prolog – Equivalence is a small part of this – Express desired outcome as set of constraints – Automatically apply logic inference to derive solution Proofs An inference rule: Modus Ponens • Start with hypotheses and facts • If p and p → q are both true then q must be true • Use rules of inference to extend set of facts • Result is proved when it is included in the set p, p → q • Write this rule as ∴ q • Given: – If it is Monday then you have a 311 class today. – It is Monday. • Therefore, by modus ponens: – You have a 311 class today.

Proofs Proofs can use equivalences too Show that r follows from p, p → q, and q → r Show that ¬ p follows from p → q and ¬ q 1. p given 1. p → q given 2. p → q given 2. ¬ q given 3. q → r given 3. ¬ q → ¬ p contrapositive of 1 4. q modus ponens from 1 and 2 4. ¬ p modus ponens from 2 and 3 5. r modus ponens from 3 and 4 Inference Rules Simple Propositional Inference Rules • Each inference rule is written as: Excluded middle plus two inference rules per binary A, B ...which means that if both A and B connective, one to eliminate it and one to introduce it ∴ C,D are true then you can infer C and you can infer D. p ∧ q p, q – For rule to be correct (A ∧ B) → C and ∴ p, q ∴ p ∧ q (A ∧ B) → D must be a tautologies p x p ∨ q , ¬ p • Sometimes rules don’t need anything to start with. ∴ p ∨ q, q ∨ p ∴ q These rules are called axioms: p ⇒ q – e.g. Excluded Middle Axiom p, p → q Direct Proof Rule ∴ p ∨¬ p ∴ q ∴ p → q Not like other rules

Important: Applications of inference rules Direct Proof of an Implication • p ⇒ q denotes a proof of q given p as an • You can use equivalences to make substitutions assumption of any sub-formula. • The direct proof rule: • Inference rules only can be applied to whole formulas (not correct otherwise). If you have such a proof then you can conclude that p → q is true e.g. 1. p → q given 2. ( p ∨ r) → q intro ∨ from 1. proof subroutine Example: 1. p assumption Does not follow! e.g . p=F F F, q=F F F, r=T F F T T T 2. p ∨ q intro for ∨ from 1 3. p → (p ∨ q) direct proof rule