introduction chapter 0. three central areas 1.automata 2.computability 3.complexity
TRANSCRIPT
Introduction
Chapter 0
Three Central Areas
1. Automata2. Computability3. Complexity
Complexity Theory
• Central Question: What makes some problems computationally hard and other easy?– Sorting (easy) – Computers can easily sort a billion items
in seconds.– Scheduling (hard) – With real constraints scheduling a
thousand courses to avoid room and instructor conflicts can take years of computation.
• Cryptography (secret encryption) depends on the existence of sufficiently hard problems.
Computability
• Based on the work of Kurt Gödel, Alan Turing, and Alonzo Church.
• Some very basic problems cannot be solved by computers.– Example: Determining if a mathematical
statement is true or false.• Instead of hard vs. easy, Computability centers
on solvable vs. unsolvable problems.
Automata Theory
• We construct a simple, formal and precise model of a computer. – The precision and formality allow us to prove the
computability of problems• Three types of computer machines
1. Finite Automata2. Pushdown Automata3. Turing Machine
Sets
{7, 57, 7, 7, 21, 7} = {21, 7, 57}
Order does not matterRepetition is not a factor
Infinite Sets
Subsets
Set Descriptions
Set Operations
Venn Diagrams
Venn Diagrams
Venn Diagrams
Venn Diagrams
• What would A-B look like?
A
B
Sequences (Tuples)
Order and repetition matter
(7, 21, 57) different than (57, 21, 7)
(4,3) ordered pair,whereas {4,3} set of size 2
(11, 21, 3, 24, 57) is a 5-tuple
Functions (Relations)
Domain & Range
Example
Binary Function (two inputs)
g(x,y) = zwherex, y and z are in {0,1,2,3}
Predicate
• function whose range is {True, False}
• Example: f(a,b) = a Beats b
f(scissors,paper) = scissors Beats paper = TRUE
Binary Relation
• Predicate whose domain is a set of ordered pairs
P: A x A {True False}
Example:A = {paper, stone, scissors}(scissors, stone) False(stone, scissors) True
Predicates can be described as setsS= { (scissors,stone), (paper, stone), (stone, scissors) }
Equivalence Relation
Binary Relation that is
1. Reflexive: f(x,x) is always true
2. Symmetric: if f(x,y) is true then f(y,x) is true
3. Transitive: if f(x,y) and f(y,z) are both true then f(x,z) is true.
Example
f(a,b) = a Beat b;• Not reflexive: f(paper,paper) = false• Not symmetric: f(paper, stone) = true,
but f(stone, paper) = false• Not transitive:
f(paper, stone) and f(stone, scissors) is true, but f(paper, scissors) is false
Example
f(a, b) true if a <= b, otherwise falsef(a, b) a <= bR = {(a,b) | a <= b}• Always reflexive: (a,a) is always true for all a’s• Not always symmetric:
If (a,b) is true then (b,a) might be false.• Always transitive: (a,b) and (b,c) imply (a,c)
Question
• Is this an equivalence relation (i.e., reflexive, symmetric, and transitive)?
f(a,b) a != 1 AND b != 1R = {(a,b) | a != 1 AND b != 1}
Equivalence Relations
• f: S x S {True, False}• S = {s1, s2, s3, s4, s5}
f s1 s2 s3 s4 s5
s1 True True
s2 True True
s3 True True True
s4 True True True
s5 True True True
Equivalence Relations & Automata
• We are interested in functions that can separate a set of items into disjoint classes.
• {{s1, s2}, {s3, s4, s5}}
• In automata theory, the items will be languages.• The functions will be automata and Turing machines• The classes will be
– Regular languages – Context-free languages– Context-sensitive languages
Equivalence & Automata Theory
The theoretical “heart” of computation• Inputs, outputs and algorithms can all be encoded
as strings of a language• Solving problems can be reduced to defining
languages (set of strings) that correspond to correct solutions to a problem.
• Then, solving a problem reduces to creating an automata that can recognize if a string is in a particular language– i.e., is a string a solution to the encoded problem?
Graph Theory
Terminology:• Nodes/vertices• Edges• Degree• Self-loop• Directed vs.
Undirected Graphs• Labeled Graph• Subgraph
Graph Theory
• Paths, cycles, trees• Outdegree, Indegree• Directed path• Strongly connected
Meaningful graphs
• Directed graphs can show relations
Equivalence Relations as Graphs
• Example to be drawn
Strings & Languages
• Defining alphabets (symbol domain)
String terminology
• Length w = abc, |w| = 3
• Empty string ɛ• Reverse w = abc, wR = cba• Substrings of w = {ɛ, a, b, c, ab, bc, abc} – ba, cb, ca, cba, and ac are not substrings of w,
but they are subsets if w were considered a set– ac is not a substring, but it is subsequence (confusing)
• Lexicographic order {ɛ , 0, 1, 00, 01, 10, 11, 000, … }
Language
• A language is a set of strings• Languages can be explicitly described
Animals = {cat, dog, cow, …}{w | an, n > 1} = {a, aa, aaa, aaaa, … }
• Languages can also be described using– grammar rules and – automata that can recognize strings in the
language.
Boolean Logic
• Review on your own, but
Boolean Logic
Proof by Construction
• State your claim. Be very specific and describe an object you claim exists
• Construct the object. Show details and describe parameters
• Conclusion. Restate your goal in constructing the object.
Example
Proof By Contradiction
• Assume that the theorem is false– Even though you know it to be true
• Show that this assumption leads to a contradiction with something that has already been proven true
Example
• Prove that• First, assume that it is rational, i.e.,
• Where m and n are integers • Also, m/n is reduced, so one of them has to be even.
Example
• Prove that
• Do some algebra
• We know that m must be even. Why?
Example
• Prove that
• We know that n must be even. Why?• Thus, m and n are even. Why is this a
contradiction?