Transcript CSE 321, Discrete Structures - Computer Science & Engineering

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?
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–
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
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•
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•
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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
pq
p
q
q
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
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?
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
• Are these the same?
Biconditional p  q
• p iff q
• p is equivalent to q
• p implies q and q implies p
p
q
pq
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)
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)
p
q
pq
p
pq
(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?
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
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•
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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
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•
•
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•
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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?
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
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“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
Statements with quantifiers
•
 x Even(x)
•
 x Odd(x)
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 x (Even(x)  Odd(x))
•
 x (Even(x)  Odd(x))
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 x Greater(x+1, x)
•
 x (Even(x)  Prime(x))
Domain:
Positive Integers
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))
Domain:
Positive Integers
Greater(a, b)  “a > b”
Statements with quantifiers
• “There is an odd prime”
Domain:
Positive Integers
• “If x is greater than two, x is not an even prime”
Even(x)
Odd(x)
Prime(x)
Greater(x,y)
Equal(x,y)
• 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”
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)
Domain:
Positive Integers
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
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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
Expression
 x  y P(x,y)
 x  y P(x,y)
 x  y P(x, y)
 y  x P(x, y)
When true
When false