CSE 321 Discrete Structures Winter 2008 Lecture 1 Propositional Logic
About the course • 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 • 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 • 6th Edition preferred • 5th 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
Propositional Logic
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, 2008. • 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 an + bn = cn 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
Compound Propositions • Negation (not) p • Conjunction (and) p q • Disjunction (or) p q • Exclusive or p q • Implication p q • Biconditional p q
Truth Tables x-or example: “you may have soup or salad with your entre”
Understanding complex propositions • Either Harry finds the locket and Ron breaks his wand or Fred will not open a joke shop Atomic propositions h: Harry finds the locket r: Ron breaks his wand f: Fred opens a joke shop (h r) f
Understanding complex propositions with a truth table
Aside: Number of binary operators • How many different binary operators are there on atomic propositions?
pq • 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
If pigs can whistle then horses can fly
Converse, Contrapositive, Inverse • Implication: pq • Converse: qp • Contrapositive: qp • Inverse: pq • Are these the same? Example p: “x is divisible by 2” q: “x is divisible by 4”
Biconditional pq • p iff q • p is equivalent to q • p implies q and q implies p
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 ( rs) q s (rq)
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 pp (p p) p ppqq (pq) p (pq) (pq) (pq) (pq)
Logical Equivalence • p and q are Logically Equivalent if pq is a tautology. • The notation pq denotes p and q are logically equivalent • Example: (pq) (pq)
Computing equivalence • Describe an algorithm for computing if two logical expressions are equivalent • What is the run time of the algorithm?
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 pTp pFF ppp pqqp (pq) r p (qr) p (qr) (pq) (pr) p (pq) p p p F
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
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? Truth tables only practical for small problems Understanding of logic Understanding of a wide class of algorithms based on rewrite algorithms Basis for formal reasoning in other domains. Most formal domains extend propositional calculus Algorithms for important practical problems: Query optimization Search equivalence Program verification
Show (pq) (pq) is a tautology
Show (pq) r and p (qr) are not equivalent All false works
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” Prime(65353)
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 Relate and to and
Statements with quantifiers Domain: Positive Integers • 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 • xy Greater (y, x) • y x Greater (y, x) • xy (Greater(y, x) Prime(y)) • x (Prime(x) (Equal(x, 2) Odd(x)) • xy(Equal(x, y + 2) Prime(x) Prime(y)) For every number there is some number that is greater than it Domain: Positive Integers Greater(a, b) “a > b”
Statements with quantifiers Domain: Positive Integers Even(x) Odd(x) Prime(x) Greater(x,y) Equal(x,y) • “There is an odd prime” • “If x is greater than two, x is not an even prime” • xyz ((Equal(z, x+y) Odd(x) Odd(y)) Even(z)) • “There exists an odd integer that is the sum of two primes”
Goldbach’s Conjecture • Every even integer greater than two can be expressed as the sum of two primes Even(x) Odd(x) Prime(x) Greater(x,y) Equal(x,y) • xyz ((Greater(x, 2) Even(x)) (Equal(x, y+z) Prime(y) Prime(z)) Domain: Positive Integers
Systems vulnerabilityReasoning 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
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
