About the course. CSE 321 Discrete Structures. Why this material is important. Topic List. Propositional Logic. Administration. From the CSE catalog:

Transcription

1 About the course CSE 321 Discrete Structures Winter 2008 Lecture 1 Propositional Logic From the CSE catalog: CSE 321 Discrete Structures (4) Fundamentals of set theory, graph theory, enumeration, and algebraic structures, with applications in computing. Prerequisite: CSE 143; either MATH 126, MATH 129, or MATH 136. What I think the course is about: Foundational structures for the practice of computer science and engineering Why this material is important Language and formalism for expressing ideas in computing Fundamental tasks in computing Translating imprecise specification into a working system Getting the details right Topic List Logic/boolean algebra: hardware design, testing, artificial intelligence, software engineering Mathematical reasoning/induction: algorithm design, programming languages Number theory/probability: cryptography, security, algorithm design, machine learning Relations/relational algebra: databases Graph theory: networking, social networks, optimization Administration Propositional Logic Instructor Richard Anderson Teaching Assistant Natalie Linnell Quiz section Thursday, 12:30 1:20, or 1:30 2:20 CSE 305 Recorded Lectures Available on line Text: Rosen, Discrete Mathematics 6 th Edition preferred 5 th Edition okay Homework Due Wednesdays (starting Jan 16) Exams Midterms, Feb 8 Final, March 17, 2:30-4:20 pm All course information posted on the web Sign up for the course mailing list

2 Propositions Compound Propositions A statement that has a truth value Which of the following are propositions? The Washington State flag is red It snowed in Whistler, BC on January 4, Hillary Clinton won the democratic caucus in Iowa Space aliens landed in Roswell, New Mexico Ron Paul would be a great president Turn your homework in on Wednesday Why are we taking this class? If n is an integer greater than two, then the equation a n + b n = c n has no solutions in non-zero integers a, b, and c. Every even integer greater than two can be written as the sum of two primes This statement is false Propositional variables: p, q, r, s,... Truth values: T for true, F for false Negation (not) Conjunction (and) Disjunction (or) Exclusive or Implication Biconditional p p q p q p q p q p q Truth Tables p p p q p q Understanding complex propositions Either Harry finds the locket and Ron breaks his wand or Fred will not open a joke shop p q p q p q p q Understanding complex propositions with a truth table h r f h r f (h r) f Aside: Number of binary operators How many different binary operators are there on atomic propositions?

3 p q Implication p implies q whenever p is true q must be true if p then q q if p p is sufficient for q p only if q p q p q If pigs can whistle then horses can fly Converse, Contrapositive, Inverse Implication: p q Converse: q p Contrapositive: q p Inverse: p q Biconditional p q p iff q p is equivalent to q p implies q and q implies p p q p q Are these the same? English and Logic You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old q: you can ride the roller coaster r: you are under 4 feet tall s: you are older than 16 Logical equivalence Terminology: A compound proposition is a Tautology if it is always true Contradiction if it is always false Contingency if it can be either true or false p p (p p) p p p q q (p q) p (p q) (p q) ( p q) ( p q)

4 Logical Equivalence p and q are Logically Equivalent if p q is a tautology. The notation p q denotes p and q are logically equivalent Example: (p q) ( p q) Computing equivalence Describe an algorithm for computing if two logical expressions are equivalent What is the run time of the algorithm? p q p q p p q (p q) ( p q) Understanding connectives Reflect basic rules of reasoning and logic Allow manipulation of logical formulas Simplification Testing for equivalence Applications Query optimization Search optimization and caching Artificial Intelligence Program verification Properties of logical connectives Identity Domination Idempotent Commutative Associative Distributive Absorption Negation De Morgan s Laws (p q) p q (p q) p q What are the negations of: Casey has a laptop and Jena has an ipod Clinton will win Iowa or New Hampshire Equivalences relating to implication p q p q p q q p p q p q p q (p q) p q (p q) (q p) p q p q p q (p q) ( p q) (p q) p q

5 Logical Proofs To show P is equivalent to Q Apply a series of logical equivalences to subexpressions to convert P to Q To show P is a tautology Apply a series of logical equivalences to subexpressions to convert P to T Why bother with logical proofs when we have truth tables? Show (p q) (p q) is a tautology Show (p q) r and p (q r) are not equivalent Predicate Calculus Predicate or Propositional Function A function that returns a truth value x is a cat x is prime student x has taken course y x > y x + y = z Quantifiers x P(x) : P(x) is true for every x in the domain x P(x) : There is an x in the domain for which P(x) is true

6 Statements with quantifiers x Even(x) x Odd(x) x (Even(x) Odd(x)) x (Even(x) Odd(x)) x Greater(x+1, x) x (Even(x) Prime(x)) Even(x) Odd(x) Prime(x) Greater(x,y) Equal(x,y) Statements with quantifiers x y Greater (y, x) For every number there is some number that is greater than it y x Greater (y, x) x y (Greater(y, x) Prime(y)) x (Prime(x) (Equal(x, 2) Odd(x)) x y(equal(x, y + 2) Prime(x) Prime(y)) Greater(a, b) a > b Statements with quantifiers There is an odd prime If x is greater than two, x is not an even prime Even(x) Odd(x) Prime(x) Greater(x,y) Equal(x,y) Goldbach s Conjecture Every even integer greater than two can be expressed as the sum of two primes x y z ((Equal(z, x+y) Odd(x) Odd(y)) Even(z)) There exists an odd integer that is the sum of two primes Even(x) Odd(x) Prime(x) Greater(x,y) Equal(x,y) Systems vulnerability Reasoning about machine status Specify systems state and policy with logic Formal domain reasoning about security automatic implementation of policies Domains Machines in the organization Operating Systems Versions Vulnerabilities Security warnings Predicates RunsOS(M, O) Vulnerable(M) OSVersion(M, Ve) LaterVersion(Ve, Ve) Unpatched(M) System vulnerability statements Unpatched machines are vulnerable There is an unpatched Linux machine All Windows machines have versions later than SP1

7 Prolog Logic programming language Facts and Rules RunsOS(SlipperPC, Windows) RunsOS(SlipperTablet, Windows) RunsOS(CarmelLaptop, Linux) OSVersion(SlipperPC, SP2) OSVersion(SlipperTablet, SP1) OSVersion(CarmelLaptop, Ver3) LaterVersion(SP2, SP1) LaterVersion(Ver3, Ver2) LaterVersion(Ver2, Ver1) Later(x, y) :- Later(x, z), Later(z, y) NotLater(x, y) :- Later(y, x) NotLater(x, y) :- SameVersion(x, y) MachineVulnerable(m) :- OSVersion(m, v), VersionVulnerable(v) VersionVulnerable(v) :- CriticalVulnerability(x), Version(x, n), NotLater(v, n) Nested Quantifiers Iteration over multiple variables Nested loops Details Use distinct variables x( y(p(x,y) x Q(y, x))) Variable name doesn t matter x y P(x, y) a b P(a, b) Positions of quantifiers can change (but order is important) x (Q(x) y P(x, y)) x y (Q(x) P(x, y)) Quantification with two variables Expression When true When false x y P(x,y) x y P(x,y) x y P(x, y) y x P(x, y)