Introduction to the computing theory in automata

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Introduction to Automata
Theory
Reading: Chapter 1

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What is Automata Theory?
 Study of abstract computing devices, or
“machines”
 Automaton = an abstract computing device
 Note: A “device” need not even be a physical
hardware!
 A fundamental question in computer science:
 Find out what different models of machines can do
and cannot do
 The theory of computation
 Computability vs. Complexity

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Alan Turing (1912-1954)
 Father of Modern Computer
Science
 English mathematician
 Studied abstract machines called
Turing machines even before
computers existed
 Heard of the Turing test?
(A pioneer of automata theory)

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Theory of Computation: A
Historical Perspective
1930s • Alan Turing studies Turing machines
• Decidability
• Halting problem
1940-1950s • “Finite automata” machines studied
• Noam Chomsky proposes the
“Chomsky Hierarchy” for formal
languages
1969 Cook introduces “intractable” problems
or “NP-Hard” problems
1970- Modern computer science: compilers,
computational & complexity theory evolve

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Languages & Grammars
Or “words”
Image source: Nowak et al. Nature, vol 417, 2002
 Languages: “A language is a
collection of sentences of
finite length all constructed
from a finite alphabet of
symbols”
 Grammars: “A grammar can
be regarded as a device that
enumerates the sentences of
a language” - nothing more,
nothing less
 N. Chomsky, Information
and Control, Vol 2, 1959

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The Chomsky Hierachy
Regular
(DFA)
Context-
free
(PDA)
Context-
sensitive
(LBA)
Recursively-
enumerable
(TM)
• A containment hierarchy of classes of formal languages

7
The Central Concepts of
Automata Theory

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Alphabet
An alphabet is a finite, non-empty set of
symbols
 We use the symbol ∑ (sigma) to denote an
alphabet
 Examples:
 Binary: ∑ = {0,1}
 All lower case letters: ∑ = {a,b,c,..z}
 Alphanumeric: ∑ = {a-z, A-Z, 0-9}
 DNA molecule letters: ∑ = {a,c,g,t}
 …

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Strings
A string or word is a finite sequence of symbols
chosen from ∑
 Empty string is  (or “epsilon”)
 Length of a string w, denoted by “|w|”, is
equal to the number of (non- ) characters in the
string
 E.g., x = 010100 |x| = 6
 x = 01  0  1  00  |x| = ?
 xy = concatentation of two strings x and y

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Powers of an alphabet
Let ∑ be an alphabet.
 ∑k = the set of all strings of length k
 ∑* = ∑0 U ∑1 U ∑2 U …
 ∑+ = ∑1 U ∑2 U ∑3 U …

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Languages
L is a said to be a language over alphabet ∑, only if L  ∑*
 this is because ∑* is the set of all strings (of all possible
length including 0) over the given alphabet ∑
Examples:
1. Let L be the language of all strings consisting of n 0’s
followed by n 1’s:
L = {,01,0011,000111,…}
2. Let L be the language of all strings of with equal number of
0’s and 1’s:
L = {,01,10,0011,1100,0101,1010,1001,…}
Definition: Ø denotes the Empty language
 Let L = {}; Is L=Ø? NO

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The Membership Problem
Given a string w ∑*and a language L
over ∑, decide whether or not w L.
Example:
Let w = 100011
Q) Is w  the language of strings with
equal number of 0s and 1s?

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Finite Automata
 Some Applications
 Software for designing and checking the behavior
of digital circuits
 Lexical analyzer of a typical compiler
 Software for scanning large bodies of text (e.g.,
web pages) for pattern finding
 Software for verifying systems of all types that
have a finite number of states (e.g., stock market
transaction, communication/network protocol)

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Finite Automata : Examples
 On/Off switch
 Modeling recognition of the word “then”
Start state Final state
Transition Intermediate
state
action
state

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Structural expressions
 Grammars
 Regular expressions
 E.g., unix style to capture city names such
as “Palo Alto CA”:
 [A-Z][a-z]*([ ][A-Z][a-z]*)*[ ][A-Z][A-Z]
Start with a letter
A string of other
letters (possibly
empty)
Other space delimited words
(part of city name)
Should end w/ 2-letter state code

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Deductive Proofs
From the given statement(s) to a conclusion
statement (what we want to prove)
 Logical progression by direct implications
Example for parsing a statement:
 “If y≥4, then 2y≥y2.”
(there are other ways of writing this).
given conclusion

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Example: Deductive proof
Let Claim 1: If y≥4, then 2y≥y2.
Let x be any number which is obtained by adding the squares
of 4 positive integers.
Given x and assuming that Claim 1 is true, prove that 2x≥x2
 Proof:
1) Given: x = a2 + b2 + c2 + d2
2) Given: a≥1, b≥1, c≥1, d≥1
3)  a2≥1, b2≥1, c2≥1, d2≥1 (by 2)
4)  x ≥ 4 (by 1 & 3)
5)  2x ≥ x2 (by 4 and Claim 1)
“implies” or “follows”

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Quantifiers
“For all” or “For every”
 Universal proofs
 Notation*=?
“There exists”
 Used in existential proofs
 Notation*=?
Implication is denoted by =>
 E.g., “IF A THEN B” can also be written as “A=>B”
*I wasn’t able to locate the symbol for these notation in powerpoint. Sorry!
Please follow the standard notation for these quantifiers.

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Proving techniques
 By contradiction
 Start with the statement contradictory to the given
statement
 E.g., To prove (A => B), we start with:
 (A and ~B)
 … and then show that could never happen
What if you want to prove that “(A and B => C or D)”?
 By induction
 (3 steps) Basis, inductive hypothesis, inductive step
 By contrapositive statement
 If A then B ≡ If ~B then ~A

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Proving techniques…
 By counter-example
 Show an example that disproves the claim
 Note: There is no such thing called a
“proof by example”!
 So when asked to prove a claim, an example that
satisfied that claim is not a proof

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Different ways of saying the same
thing
 “If H then C”:
i. H implies C
ii. H => C
iii. C if H
iv. H only if C
v. Whenever H holds, C follows

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“If-and-Only-If” statements
 “A if and only if B” (A <==> B)
 (if part) if B then A ( <= )
 (only if part) A only if B ( => )
(same as “if A then B”)
 “If and only if” is abbreviated as “iff”
 i.e., “A iff B”
 Example:
 Theorem: Let x be a real number. Then floor of x =
ceiling of x if and only if x is an integer.
 Proofs for iff have two parts
 One for the “if part” & another for the “only if part”

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Summary
 Automata theory & a historical perspective
 Chomsky hierarchy
 Finite automata
 Alphabets, strings/words/sentences, languages
 Membership problem
 Proofs:
 Deductive, induction, contrapositive, contradiction,
counterexample
 If and only if
 Read chapter 1 for more examples and exercises
 Gradiance homework 1

Introduction to the computing theory in automata

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