Transcript Document

CS 454 Theory of Computation
Sonoma State University, Fall 2011
Instructor: B. (Ravi) Ravikumar
Office: 116 I Darwin Hall
Original slides by Vahid and Givargis, Mani Srivastava and others
Extensive editing by Ravikumar
Lecture 1
Goals:
•
•
•
•
overview & logistics
Quiz 0
Quiz 0 discussion
chapter 0
(45 minutes)
(15 minutes)
(30 minutes)
(if any time left)
What is theory of computation?
Theory of computation deals with computational models
– what they can and can’t do.
Computational models: instead of defining it, we will give
some examples.
• Cellular automaton
• L-system
• Boolean circuit with restrictions
Course overview
• finite automata
• DFA, NFA, regular expressions
• proof of equivalence, algorithms for conversions
• what can’t be done by FA?
• DFA minimization
• context-free languages
• grammar, pushdown automaton
• equivalence
• what can’t be done by cfg’s?
• Turing machines
• computability
• variations of Turing machines
• reductions, unsolvability
• other computational models
• Complexity Theory
• time complexity
• NP-completeness
• Other topics
• applications (e.g. cellular automata, compilers etc.)
• cryptography, interactive proof systems
• biologically inspired models of computation
Chapter 0
• Sets
• Set operations and set relations
• union, intersection, complement
• member, subset, equality
• Venn diagrams
Problem:
Show that (A U B) C = (A C)U (B C)
Proof:
Let x be in LHS set. Then, x is in both A U B and is in
C. i.e., x is A and C, or x is in B and C. I.e., x is in A C
or in B  C. This means,
x is in (A  C)U (B  C). Converse is similar.
Sequences and tuples
A sequence is a list of objects where order is important.
Thus, <1, 2, 4, 10> is a sequence that is different than <2,
1, 4, 10>.
A finite sequence of length k is called a k-tuple. Thus, the
above sequence is a 4-tuple.
• Power-set of a set A is the set of all its subsets.
Ex: A = {1, 2} P(A) = {{1}, {2}, {1,2}, f}
• Cartesian product
A X B = {<i,j> | i is in A and j is in B }
Example: A = {a, b, c}, B = { 1, 2}
A X B = {(a,1), (a,2), (b,1), (b,2), (c,1), (c,2)}
A X A X … X A is often denoted by Ak.
Functions and relations
• A relation from set A to set B is a subset of A X B.
• A function (from A to B) is a relation R in which for
every i, there is a unique j such that <i,j> is in R.
• onto
• one-one
• bijective function
Some special relations:
• equivalence relation
• partial-order relation
Graphs:
• Definition
• Example
Strings and languages
• Boolean logic
• quantifiers
Summary of mathematical terms
Definitions, theorems and proofs
• Definitions: A definition is the way to describe an object
in a way that its characteristics are completely captured in
the description.
• Assertions: Mathematical statement expresses some
property of a set of defined objects. Assertions may or may
not be true.
• Proof: is a convincing logical argument that a statement
is true. The proofs are required to follow rigid rules and
are not allowed any room for uncertainties or ambiguities.
• Theorem: is a proven assertion of some importance.
Rules for carrying out proof
Usually assertions are compound statements that are
connected using Boolean connectives and quantifiers. You
can use theorems of Boolean logic in the proof.
• For example, if the assertion is P or Q,
you can show it as follows: Suppose P is not true. Then, Q
must be true.
• Similarly, to show P  Q, you assume P is true. From
this, show Q is true. To show that P <-> Q, you should
show P  Q and Q  P.
Example:
Every positive integer is either a prime, or is a product
of two integers both of which are strictly smaller than
itself.
Is this a definition? Is it an assertion?
Example:
Every positive integer is either a prime, or is a product
of two integers both of which are strictly smaller than
itself.
It is an assertion. Is it a true assertion?
Example:
Every positive integer is either a prime, or is a product
of two integers both of which are strictly smaller than
itself.
Yes, it is a true assertion.
How to prove this assertion?
Example:
Every positive integer is either a prime, or is a product
of two integers both of which are strictly smaller than
itself.
Yes, it is a true assertion.
How to prove this assertion?
This requires knowing the definition of a prime number.
Prime number: A number x is prime if it has exactly two
divisors, namely 1 and x.
Most of the theorems assert properties of a collection of
objects. If the collection is finite, usually it is easy: you
show it for every member one by one. Need for proof
really arises when the assertion is of the form : “Every
object in a set X has some property Y.” where X is an
infinite set.
Now you can’t prove it one by one!
• Proof techniques: (not an exhaustive list!)
• proof by construction
• proof by contradiction
• proof by induction
Proof by construction:
Definition: A k-regular graph is a graph in which every
node has degree k.
Theorem: For each even number n greater than 2, there
exists a 3-regular graph with n nodes.
The construction below needs no further explanation.
Proof by contradiction:
Theorem: 2 is not rational, i.e., it can’t be written as a
ratio of two integers.
Proof is presented in page 22.
Exercise: Use the same idea to show that
rational.
3 is not
Proof by induction.
Example 1: Formula for mortgage calculation.
Page 24 of text.
Example 2: Show that the set of all binary strings of
length n can be arranged in a way that every adjacent
string differs in exactly one bit position, and further the
first and the last string also differ in exactly one position.
For n = 2, one such is 00, 01, 11, 10.
Assertion: Every integer is a sum of squares of two integers.
This is not true. To disprove it, it is enough to find one integer
(counter-example) that can’t be written as sum of two squares.
Consider 3. Suppose x2 + y2 = 3 for some integers x and y. This
means, x2 is either 0, 1 or 2. (Why not any other number?)
case 1: x2 is 0. Thus y2 is 3. But from previous slide, we know that
there is no integer y such that y2 = 3.
Case 2: x2 is 1. Thus y2 is 2. But from previous slide, we know that
there is no integer y such that y2 = 2.
Case 3: x2 is 2. But from previous slide, we know that there is no
integer x such that x2 = 2. End of proof.