Theory of Computation
Mathematical Preliminaries
Vladimir KulyukinDepartment of Computer Science
Utah State University
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Outline
● What is Theory of Computation?● Alphabets, Strings, Languages● Numbers, Sets, Set Formers● Proofs
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What is Theory of Computation?
● Theoretical Computer Science can be broadly divided into algorithms and computability (aka theory of computation)
● The field of algorithms answers the question – how something can be computed?
● The field of computability answers the question – can something be computed?
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Basic Methodology
● Abstraction of hardware details● Focus on what can be solved, not on how it can
be solved● Problem analysis in terms of devices (aka
automata) and inputs (aka strings, languages)
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Basic Methodology
● Abstraction of hardware details● Focus on what can be solved, not on how it can
be solved● Problem analysis in terms of devices (aka
automata) and inputs (aka strings, languages)
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Abstraction of Hardware Details
Junun Mark III Robot
SmartphonePersonal Computer
While these devices have very different hardware, they have the same computational model
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Focus on What, Not How
● Problem: sorting a sequence of numbers from smallest to highest
● Algorithmic answer: merge sort, heap sort, quick sort, etc
● Computability answer: sorting is primitive recursive
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Automata & Languages
● Finite State Automata/Regular Expressions – Regular Languages
● Push Down Automata/Stack Machines – Context-Free Languages
● Linear Bounded Automata – Context Sensitive Languages
● Turing Machines/Universal Programs – Recursively Enumerable Languages
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Alphabets
● An alphabet is a finite set of symbols● The Greek letter Σ is typically used to denote an
alphabet● Examples: Σ1 = {a, b}, Σ2 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.● The symbols in the alphabet do not have any
meaning in and of themselves
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Strings
● A string is a finite sequence of symbols● ε is the empty string; ε is not a symbol in any
alphabet; it denotes the string with zero symbols● In the book (Davis et al.), the empty string is
denoted as 0● In formal language theory, strings are typically
written without quotation marks: aab and 010001 instead of “aab” and “010001”
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Strings
● The length of a string is the number of symbols/characters in it
● The length of a string is denoted with a pair of matching vertical lines around it
● Examples: – if x = aab, then |x| = 3; – if x = ε, then |x| = 0
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String Concatenation
● The concatenation of two strings x and y is the strings containing the symbols of x followed by the symbols of y
● Examples: if x = ab and y = 100, then xy = ab100; if x = ε and y = abc, then xy = abc
● For any string, xε = εx = x
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Power Notation in String Concatenation
● When a natural number n is used as an exponent on a string, it denotes the concatenation of that string with itself n times
● Examples: – x0 = ε – x1 = x – x2 = xx – (ab)3 = ababab
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Languages
● A language is a set strings over a alphabet● Note that we can define multiple languages over
the same alphabet● Example: Σ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. L1 = the set
of all strings over Σ that end in 0 (e.g., 0, 110, 213450, etc.); L2 = the set of all strings over Σ that end in 1 (e.g., 1, 01, 0001, 91, etc.)
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Kleene Closures of Alphabets
● The Kleene closure of an alphabet is the set of all strings over it
● If Σ is an alphabet, its Kleene closure is written as Σ*
● Example: Σ = {a, b}. Σ* = the set of all strings consisting of a's and b's, including the empty string
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Natural Numbers
● N is a set of natural numbers.● N includes 0, 1, 2, 3, …● Some texts exclude 0 from the set of natural
numbers, but we will keep it in● In the texts, the word number refers to natural
number
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Set Former Notation
{ }{ } { }
{ } { }
{ } { }{ }.or by followed is or
wherestrings ofset theis ,,,|
s.' ofnumber the
toequal is s' ofnumber theand s' precede s'that
such ,over stringsempty -non ofset theis 1|
3.or 2, 1, 0, islength whose
,over strings all ofset theis 3|, *
ccaaba
ccaaybaxxy
b
aba
banba
baxbax
nn
∈∈
≥
≤∈
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Subsets
.
:setevery ofsubset a isset empty The
. and if)only and (if iff
R
RSSRSR
⊆∅
⊆⊆=
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Set-Theoretic Equalities
s.complement theofunion theis
onintersecti theof complement thei.e. ,
s.complement theofon intersecti the
isunion theof complement thei.e. ,
).(
. and ofon intersecti -
SRSR
SRSR
SRRSR
SRSR
∪=∩
∩=∪
∩−=−∩
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Sets and N-Tuples
( ) ( ) ( ).,,,,,,
:matter does sequences ain elements oforder The
}.,,{},,{},,{
:matternot doesset ain elements oforder The
set. a is },...,,{ 21
bacacbcba
cabbcacba
aaa n
≠≠
==
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Sets and N-tuples
}.,...,,|),...,,{(
...
:follows as defined isset theseofproduct
Cartesian Then the sets. are ,...,,Let
221121
21
21
nnn
n
n
SaSaSaaaa
SSS
SSS
∈∈∈=×××
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Predicates
.0)(or 1)(or )(or )(either
each for such that on function
valued-Boolean totala is predicate A
====∈
aPaPFaPTaP
SaS
P
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Predicates
R. offunction sticcharacteri a is )(
}1)(|{
Then
if 0
if 1)(
set. a be Let
xP
xPxR
Rx
RxxP
R
==
∉∈
=
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Proof Methods
● In CS, there are, broadly speaking, two methods of proving things: formal and empirical
● Formal methods are used in theory of computation, algorithms, operations research, etc.
● Empirical methods are used in many applied branches of CS
● Many R&D projects combine formal and empirical methods
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Mathematical Proofs
● The corner stone of the formal method is the mathematical proof
● Many online and printed CS materials contain proofs● It is of vital importance for a CS practitioner to read at
least some proofs● The good news is that reading proofs is significantly
easier than doing them
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Proof Techniques
● Proof techniques are independent of their subject matter: valid proofs in calculus use the same proof techniques as valid proofs in algorithms or theory of computation
● Common proof techniques can be identified● Learning to identify common proof techniques will enable you to
study many areas of CS independently● The ability to identify proof techniques is based on your ability to
understand how the technique works and when it is likely to be applicable
● In CS, a prominent proof technique is induction
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Learning to Love the P-Word
● General advice: Do not be afraid of proofs; one can be a mediocre theorem prover but a very good proof reader
● The first step in mastering the art of mathematical proof is to read and do proofs of known facts; do not think of it as a waste of time
● When you read some CS material, do not shy away from it, if it contains proofs
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