theory of computation: math preliminaries: alphabets, strings, languages, automata, numbers, sets,...

Theory of Computation

Mathematical Preliminaries

Vladimir KulyukinDepartment of Computer Science

Utah State University

www.vkedco.blogspot.com

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Outline

● What is Theory of Computation?● Alphabets, Strings, Languages● Numbers, Sets, Set Formers● Proofs



What is Theory of Computation?



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?



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)



Abstraction of Hardware Details

Junun Mark III Robot

SmartphonePersonal Computer

While these devices have very different hardware, they have the same computational model



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



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



Automata & Languages



Alphabets, Languages, Strings



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



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”



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



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



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



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.)



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



Numbers, Sets, Set Formers



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



Sets

{ }{ } { }ε≠∅ sets.empty are or



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

∈∈

≥

≤∈



Subsets

.

:setevery ofsubset a isset empty The

. and if)only and (if iff

R

RSSRSR

⊆∅

⊆⊆=



Proper Subsets

. and iff SRSRSR ≠⊆⊂



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

∪=∩

∩=∪

∩−=−∩



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

≠≠

==



Sets and N-tuples

}.,...,,|),...,,{(

...

:follows as defined isset theseofproduct

Cartesian Then the sets. are ,...,,Let

221121

21

21

nnn

n

n

SaSaSaaaa

SSS

SSS

∈∈∈=×××



Predicates

.0)(or 1)(or )(or )(either

each for such that on function

valued-Boolean totala is predicate A

====∈

aPaPFaPTaP

SaS

P



Predicates

R. offunction sticcharacteri a is )(

}1)(|{

Then

if 0

if 1)(

set. a be Let

xP

xPxR

Rx

RxxP

R

==

∉∈

=



Proofs



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



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



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



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



theory of computation: math preliminaries: alphabets, strings, languages, automata, numbers, sets,...

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strings x

set strings

languages numbers

symbols of x

string concatenation

set of natural numbers

finite set of symbols

enumerable languages