Negations of quantifiers CSE • not every positive integer is prime 311 • some positive integer is not prime • prime numbers do not exist Foundations of • Computing I every positive integer is not prime Fall 2014 Negations of Quantifiers De Morgan’s Laws for Quantifiers • ∀x PurpleFruit(x) Domain: • “All fruits are purple” Fruit ¬∀x P(x) ≡ ∃x ¬P(x) • ¬∀ What is x PurpleFruit(x)? PurpleFruit(x) ¬∃ x P(x) ≡ ∀x ¬P(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 ¬∀ x P(x) ≡ ∃x ¬P(x) Example: NotLargest(x) ≡ ∃ y Greater (y, 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 ∀ ∃ → ∀ “For every integer there is a larger integer ” x ( y (P( x,y) x Q( y, x))) scope of quantifiers CSE 311: Foundations of Computing ∃x (P( x) ∧∧∧ Q( x)) vs. ∃x P( x) ∧∧∧ ∃x Q( x) Fall 2014 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 ∀ ∃ ≡ ∀ ∃ distance between them increases by one node per step. x y P(x, y) a b P(a, b) • 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 y expression when true when false ∀x ∀ y P(x, y) ∃ ∃ x P(x,y) x y 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 – How to understand and express things using – Express desired properties of program as set of logical propositional and predicate logic constraints – How to compute using Boolean (propositional) logic – Use inference rules to show that program implies that – those constraints are satisfied How to show that different ways of expressing or • computing them are equivalent to each other Artificial Intelligence – Automated reasoning • • Logic also has methods that let us infer implied Algorithm design and analysis – properties from ones that we know e.g., Correctness, Loop invariants. • – Equivalence is a small part of this Logic Programming, e.g. Prolog – 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 • Write this rule as p, p → q ∴ 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: A, B Excluded middle plus two inference rules per binary ...which means that if both A and B connective, one to eliminate it and one to introduce it are true then you can infer C and ∴ C,D 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 ∨ q , ¬p p x • Sometimes rules don’t need anything to start with. ∴ q ∴ p ∨ q, q ∨ p These rules are called axioms : – e.g. Excluded Middle Axiom → p ⇒ q 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 • You can use equivalences to make substitutions • p ⇒ q denotes a proof of q given p as an of any sub-formula. assumption • Inference rules only can be applied to whole • The direct proof rule: formulas (not correct otherwise). If you have such a proof then you can conclude e.g. 1. p → q given that p → q is true 2. ( p ∨ r) → q intro ∨ from 1. Example: proof subroutine 1. p assumption Does not follow! e.g . p=FFFF, q=FFFF, r=TTTT 2. p ∨ q intro for ∨ from 1 3. p → (p ∨ q) direct proof rule.