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EOTTOC.lyx
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#LyX 2.0 created this file. For more info see http://www.lyx.org/
\lyxformat 413
\begin_document
\begin_header
\textclass scrbook
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\language english
\language_package default
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\index Index
\shortcut idx
\color #008000
\end_index
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\paragraph_indentation default
\quotes_language english
\papercolumns 1
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\end_header
\begin_body
\begin_layout Title
Reading Notes of
\begin_inset Newline newline
\end_inset
Elements of the Theory of Computation
\end_layout
\begin_layout Author
Tianyi Cui
\end_layout
\begin_layout Date
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
today
\end_layout
\end_inset
\end_layout
\begin_layout Chapter*
Introduction
\end_layout
\begin_layout Standard
Fundamental questions in computer science answered by theory of computation:
\end_layout
\begin_layout Itemize
What is an algorithm?
\end_layout
\begin_layout Itemize
What can and what cannot be computed?
\end_layout
\begin_layout Itemize
When should an algorithm be considered practically feasible.
\end_layout
\begin_layout Standard
The theory of computation is the mathematical abstractions of computers,
but its origin is even before the advent of the electronic computer.
\end_layout
\begin_layout Quote
It is based on very few and elementary concepts, and draws its power and
depth from the careful, patient, extensive, layer-by-layer manipulation
of these concepts -- just like the computer.
\end_layout
\begin_layout Chapter
Sets, Relations, and Languages
\end_layout
\begin_layout Section
Sets
\end_layout
\begin_layout Standard
Power set:
\begin_inset Formula $2^{A}$
\end_inset
, the collection of all subsets of set
\begin_inset Formula $A$
\end_inset
.
\end_layout
\begin_layout Standard
Partition of set
\begin_inset Formula $A$
\end_inset
, subset of
\begin_inset Formula $2^{A}$
\end_inset
whose elements are nonempty and disjoint when contain all elements of
\begin_inset Formula $A$
\end_inset
.
\end_layout
\begin_layout Section
Relations and functions
\end_layout
\begin_layout Standard
\begin_inset Formula $a$
\end_inset
and
\begin_inset Formula $b$
\end_inset
are called the
\emph on
components
\emph default
of the ordered pair
\begin_inset Formula $(a,b)$
\end_inset
.
\end_layout
\begin_layout Standard
The
\emph on
Cartesian product
\emph default
of two sets.
\end_layout
\begin_layout Standard
Ordered tuples: ordered triples, quadruples, quintuples, sextuples...
\end_layout
\begin_layout Standard
\begin_inset Formula $n$
\end_inset
-ary relation: unary, binary, ternary...
\end_layout
\begin_layout Standard
The domain, image,
\emph on
range
\emph default
, of function; one-to-one + onto = bijection; inverse.
\end_layout
\begin_layout Section
Special types of binary relations
\end_layout
\begin_layout Standard
The relation
\begin_inset Formula $R\in A\times A$
\end_inset
is called a
\emph on
directed graph
\emph default
.
\end_layout
\begin_layout Standard
Properties of relations: reflexive, symmetric, antisymmetric, transitive.
\end_layout
\begin_layout Standard
Equivalence relation: r, s, t.
Partial order: r, a, t.
Total order.
\end_layout
\begin_layout Section
Finite and infinite sets
\end_layout
\begin_layout Standard
Call two sets
\emph on
equinumerous
\emph default
if there is a bijection between them.
\end_layout
\begin_layout Standard
Finite (equinumerous with
\begin_inset Formula $\{{1,2,\ldots n}\}$
\end_inset
, infinite, countably infinite (equinumerous with
\begin_inset Formula $\mathbf{{N}}$
\end_inset
), countable, uncountable.
\end_layout
\begin_layout Section
Three fundamental proof techniques
\end_layout
\begin_layout Standard
The Principle of Mathematical Induction: Let
\begin_inset Formula $A$
\end_inset
be a set of natural numbers such that (1)
\begin_inset Formula $0\in A$
\end_inset
, and (2) for each natural number
\begin_inset Formula $n$
\end_inset
`$n$`, if
\begin_inset Formula $\{0,1,\ldots,n\}\subseteq A$
\end_inset
, then
\begin_inset Formula $n+1\in A$
\end_inset
.
Then
\begin_inset Formula $A=\mathbf{N}$
\end_inset
.
\end_layout
\begin_layout Standard
The Pigeonhole Principle: if
\begin_inset Formula $A$
\end_inset
and
\begin_inset Formula $B$
\end_inset
are finite sets and
\begin_inset Formula $|A|>|B|$
\end_inset
, then there is no one-to-one function from
\begin_inset Formula $A$
\end_inset
to
\begin_inset Formula $B$
\end_inset
.
\end_layout
\begin_layout Standard
The Diagonalization Principle: Let
\begin_inset Formula $R$
\end_inset
be a binary relation on a set
\begin_inset Formula $A$
\end_inset
, and let
\begin_inset Formula $D$
\end_inset
, the diagonal set for
\begin_inset Formula $R$
\end_inset
, be
\begin_inset Formula $\{a:a\in A\text{ and }(a,a)\notin R\}$
\end_inset
.
For each
\begin_inset Formula $a\in A$
\end_inset
, let
\begin_inset Formula $R_{a}=\{b:b\in A\text{ and }(a,b)\in R\}$
\end_inset
.
Then
\begin_inset Formula $D$
\end_inset
is distinct from each
\begin_inset Formula $R_{a}$
\end_inset
.
Lemma: the set
\begin_inset Formula $2^{\mathbf{N}}$
\end_inset
is uncountable.
\end_layout
\begin_layout Section
Closures and algorithms
\end_layout
\begin_layout Standard
The
\emph on
reflexive transitive closure
\emph default
of a directed graph.
\end_layout
\begin_layout Standard
The
\emph on
rate of growth
\emph default
of a function
\begin_inset Formula $f$
\end_inset
on
\begin_inset Formula $\mathbf{{N}}$
\end_inset
.
\end_layout
\begin_layout Standard
The proof of correctness of the Floyd algorithm: define
\emph on
rank of a path
\emph default
as the biggest index among its intermediate nodes, and prove that after
the
\begin_inset Formula $j$
\end_inset
th iteration, all path with rank less than or equal to
\begin_inset Formula $j$
\end_inset
will be found.
\end_layout
\begin_layout Standard
Closure property: Let
\begin_inset Formula $D$
\end_inset
be a set, let
\begin_inset Formula $n\geq0$
\end_inset
, and let
\begin_inset Formula $R\subseteq D^{n+1}$
\end_inset
be a
\begin_inset Formula $(n+1)$
\end_inset
-ary relation on
\begin_inset Formula $D$
\end_inset
.
Then a subset
\begin_inset Formula $B$
\end_inset
of
\begin_inset Formula $D$
\end_inset
is said to be
\emph on
closed under
\emph default
\begin_inset Formula $R$
\end_inset
if
\begin_inset Formula $b_{n+1}\in B$
\end_inset
whenever
\begin_inset Formula $b_{1},\ldots,b_{n}\in B$
\end_inset
and
\begin_inset Formula $(b_{1},\ldots,b_{n},b_{n+1}\}\in R$
\end_inset
.
Any property of the form "the set
\begin_inset Formula $B$
\end_inset
is closed under relation
\begin_inset Formula $R_{1},R_{2},\ldots,R_{m}$
\end_inset
" is called a
\emph on
closure property
\emph default
of
\begin_inset Formula $B$
\end_inset
.
\end_layout
\begin_layout Standard
The minimal set
\begin_inset Formula $B$
\end_inset
that contains
\begin_inset Formula $A$
\end_inset
and has property
\begin_inset Formula $P$
\end_inset
is unique if
\begin_inset Formula $P$
\end_inset
is a closure property defined by relations on a set
\begin_inset Formula $D$
\end_inset
while
\begin_inset Formula $A\subseteq D$
\end_inset
.
Then we call
\begin_inset Formula $B$
\end_inset
the
\emph on
closure
\emph default
of
\begin_inset Formula $A$
\end_inset
under the relation
\begin_inset Formula $R_{1},\ldots,R_{m}$
\end_inset
.
\end_layout
\begin_layout Standard
Inclusion property: unary closure (take
\begin_inset Formula $n=0$
\end_inset
in definition).
\end_layout
\begin_layout Standard
Any closure property over a finite set can be computed in polynomial time
(see ex1.6.9).
\end_layout
\begin_layout Section
Alphabets and languages
\end_layout
\begin_layout Standard
Here is the
\emph on
mathematics of strings of symbols
\emph default
.
\end_layout
\begin_layout Description
symbol: any object, but often only common characters are used.
\end_layout
\begin_layout Description
alphabet: a finite set of symbols.
\end_layout
\begin_layout Description
string: finite sequence of symbols from the alphabet, which has
\emph on
length
\emph default
, operation of
\emph on
concatenation
\emph default
(
\begin_inset Formula $\circ$
\end_inset
),
\emph on
substring
\emph default
,
\emph on
prefix
\emph default
,
\emph on
suffix
\emph default
,
\begin_inset Formula $s^{n}$
\end_inset
, operation of
\emph on
reversal
\emph default
(
\begin_inset Formula $s^{R}$
\end_inset
) defined
\end_layout
\begin_layout Description
language: any set of strings over an alphabet
\begin_inset Formula $\Sigma$
\end_inset
, that is, any subset of
\begin_inset Formula $\Sigma^{*}$
\end_inset
.
It might be able to be enumerated
\emph on
lexicographically
\emph default
.
It has
\emph on
complement
\emph default
(
\begin_inset Formula $\overline{L}$
\end_inset
),
\emph on
concatenation of languages
\emph default
,
\emph on
Kleene star
\emph default
(the set of all strings obtained by concatenating zero or more strings
from it).
We write
\begin_inset Formula $L^{+}$
\end_inset
for
\begin_inset Formula $LL^{*}$
\end_inset
, which is the
\emph on
closure
\emph default
of
\begin_inset Formula $L$
\end_inset
under the function of concatenation.
\end_layout
\begin_layout Section
Finite representations of languages
\end_layout
\begin_layout Standard
This section discusses how to use
\emph on
regular expressions
\emph default
to represent languages.
\end_layout
\begin_layout Standard
A
\emph on
regular expression
\emph default
is the representation of language using empty set, characters in alphabet,
concatentaion (symbol usually omitted), function of union (the
\emph on
or
\emph default
operator in regex), star, and parentheses.
We can define the function
\begin_inset Formula $\mathcal{L}$
\end_inset
from regular expressions to lanuages, whose range is called the class of
\emph on
regular languages
\emph default
.
\end_layout
\begin_layout Standard
A
\emph on
language recognition device
\emph default
is an algorithm that is specifically designed to answer questions of the
form "is string
\begin_inset Formula $w$
\end_inset
a member of
\begin_inset Formula $L$
\end_inset
?".
\end_layout
\begin_layout Standard
A
\emph on
language generator
\emph default
is the description of the way of generating members of a language.
\end_layout
\begin_layout Standard
The relation between the above two types of finite language specifications
is another major subject of this book.
\end_layout
\begin_layout Chapter
Finite Automata
\end_layout
\begin_layout Section
Deterministic Finite Automata
\end_layout
\begin_layout Standard
DFA is computer with no memory; input a string, output indicate whether
it's acceptable.
\end_layout
\begin_layout Standard
DFA definition: quintuple
\begin_inset Formula $M=(K,\Sigma,\delta,s,F)$
\end_inset
, where
\begin_inset Formula $K$
\end_inset
is a finite set of
\emph on
states
\emph default
,
\begin_inset Formula $\Sigma$
\end_inset
is an alphabet,
\begin_inset Formula $s\in K$
\end_inset
is the
\emph on
initial state
\emph default
,
\begin_inset Formula $F\subseteq K$
\end_inset
is the set of
\emph on
final states
\emph default
, and
\begin_inset Formula $\delta$
\end_inset
the
\emph on
transition function
\emph default
, is a function from
\begin_inset Formula $K\times\Sigma$
\end_inset
to
\begin_inset Formula $K$
\end_inset
.
\end_layout
\begin_layout Standard
A
\emph on
configuration of a DFA
\emph default
is any elements of
\begin_inset Formula $K\times\Sigma^{*}$
\end_inset
.
For two configuration
\begin_inset Formula $(q,w)$
\end_inset
and
\begin_inset Formula $(q',w')$
\end_inset
, then
\begin_inset Formula $(q,w)\vdash_{M}(q',w')$
\end_inset
if and only if
\begin_inset Formula $w=aw'$
\end_inset
for some symbol
\begin_inset Formula $a\in\Sigma$
\end_inset
, and
\begin_inset Formula $\delta(q,a)=q'$
\end_inset
.
We say that
\begin_inset Formula $(q,w)$
\end_inset
\series bold
yields
\series default
\begin_inset Formula $(q',w')$
\end_inset
\series bold
in one step
\series default
.
Denote the reflexive transitive closure of
\begin_inset Formula $\vdash_{M}$
\end_inset
by
\begin_inset Formula $\vdash_{M}^{*}$
\end_inset
,
\begin_inset Formula $(q,w)$
\end_inset
\begin_inset Formula $\vdash_{M}^{*}$
\end_inset
\begin_inset Formula $(q',w')$
\end_inset
is read,
\begin_inset Formula $(q,w)$
\end_inset
\series bold
yields
\series default
\begin_inset Formula $(q',w')$
\end_inset
.
\end_layout
\begin_layout Standard
A string
\begin_inset Formula $w\in\Sigma^{*}$
\end_inset
is said to be
\emph on
accepted
\emph default
by
\begin_inset Formula $M$
\end_inset
if and only if there is a state
\begin_inset Formula $q\in F$
\end_inset
such that
\begin_inset Formula $(s,w)\vdash_{M}^{*}(q,e)$
\end_inset
.
\emph on
The language accepted
\emph default
by
\begin_inset Formula $M$
\end_inset
,
\begin_inset Formula $L(M)$
\end_inset
is the set of all strings accepted by
\begin_inset Formula $M$
\end_inset
.
\end_layout
\begin_layout Section
Nondeterministic Finite Automata
\end_layout
\begin_layout Standard
NFAs are simply a useful
\emph on
notational generalization
\emph default
of finite automata, as they can greatly simplify the description of these
automata.
Moreover, every NFA is equivalent to a DFA.
\end_layout
\begin_layout Standard
NFA definition: quintuple
\begin_inset Formula $M=(K,\Sigma,\Delta,s,F)$
\end_inset
where
\begin_inset Formula $\Delta$
\end_inset
the
\emph on
transition relation
\emph default
, is a subset of
\begin_inset Formula $K\times(\Sigma\cup\{e\})\times K$
\end_inset
.
Each triple
\begin_inset Formula $(q,u,p)\in\Delta$
\end_inset
is called a
\emph on
transition
\emph default
of
\begin_inset Formula $M$
\end_inset
.
\end_layout
\begin_layout Standard
Two finite automata
\begin_inset Formula $M_{1}$
\end_inset
and
\begin_inset Formula $M_{2}$
\end_inset
are
\emph on
equivalent
\emph default
if and only if
\begin_inset Formula $L(M_{1})=L(M_{2})$
\end_inset
.
For each NFA, there is an equivalent DFA.
\end_layout
\begin_layout Section
Finite Automata and Regular Expressions
\end_layout
\begin_layout Standard
The class of languages accepted by DFA or NFA, is the same as the class
of
\emph on
regular languages
\emph default
-- those that can be described by regular expressions.
\end_layout
\begin_layout Standard