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Introduction to Theory of Computation
COMP2600 — Formal Methods for Software Engineering
Katya Lebedeva
Australian National University
Semester 2, 2016
Slides created by Katya Lebedeva
COMP 2600 — Introduction to Theory of Computation 1
Brief Historical Overview
Already in 1930’s Alan Turing studied an abstract ma-
chine, Turing machine!
In 1940’s and 1950’s simpler kinds of machines, fi-
nite automata, were studied. In the late 1950’s Noam
Chomsky began his study of formal grammar.
In 1969 Stephen Cook separated problems that are
“intractable” (also called “NP-hard”).
COMP 2600 — Introduction to Theory of Computation 2
http://www.nytimes.com/1994/11/24/business/company-news-flaw-undermines-accuracy-of-pentium-chips.html
COMP 2600 — Introduction to Theory of Computation 3
Initially all Intel chips did all arithmetic using integers. Software pro-
grams were instructing the chip how to divide floating-point numbers
using integer arithmetic. The 1993 chips had these instructions inte-
grated into them. This makes calculations much faster.
In 1994 Thomas R. Nicely discovered that the processor could return
incorrect decimal results when dividing large prime numbers.http://www.trnicely.net/pentbug/bugmail1.html.
The replacement of flawed chips costed Intel US$475 million!
Intel CEO Andy Grove:“Intel survived the crisis and was made stronger by it. We dramatically improved our
validation methodology to quickly capture and fix errata, and investigated innovative
ways to design products that are error-free right from the beginning.”http://www.techradar.com/au/news/computing-components/processors/pentium-fdiv-the-processor-bug-that-shook-the-world-1270773
COMP 2600 — Introduction to Theory of Computation 4
Three Major branches of Theory of Computation
1. Automata Theory deals with definitions and properties of computation
models. Examples:
• Deterministic Finite Automata (July 18, July 20)
• Nondeterministic Finite Automata (July 22)
• Context-Free Grammars (July 27)
• Turing Machines (August 1, August 3, August 5)
2. Computability Theory - the study of decidability - classifies problems as
being solvable or unsolvable (August 5)
3. Complexity Theory - the study of intractability - classifies problems ac-
cording to their degree of difficulty
COMP 2600 — Introduction to Theory of Computation 5
Basic Definitions of Automata Theory
Alphabet is a finite, nonempty set of symbols
Examples:
Σ1 = {0,1}
Σ2 = {a,b, . . . ,z}
Σ3 = {A,B, . . . ,Z}
COMP 2600 — Introduction to Theory of Computation 6
String (word) is a finite sequence of symbols chosen from some alphabet
01011 is a string from Σ1
gobbledygook is a string from Σ2
• Empty string is the string with no symbols
ε can be chosen from any alphabet!
• Length of a string the number of positions for symbols in the string
(“number of symbols” is often said though)
|01011| is 5 and not 2
|gobbledygook| is 12 and not 8
|ε| is 0
COMP 2600 — Introduction to Theory of Computation 7
Powers of Alphabet
Σk denotes the set of all strings of length k over Σ
Σk = Σ×Σ×·· ·×Σ︸ ︷︷ ︸k times
Σ0 = {ε} for all Σ
For Σ1 = {0,1}, Σ11 = {0,1}
Σ21 = {00,01,10,11}
COMP 2600 — Introduction to Theory of Computation 8
Closure of an alphabet Σ:
Σ∗ is the set of all strings over Σ
Σ∗ = Σ0∪Σ1∪Σ2∪ . . .
The set of non-empty strings from alphabet Σ:
Σ+ = Σ
1∪Σ2∪Σ
3∪ . . .
Hence Σ∗ = Σ+∪{ε}
Note:
All strings in Σ∗ and Σ+ are finite.
Σ∗ and Σ+ are infinite sets.
COMP 2600 — Introduction to Theory of Computation 9
Concatenation of Strings
If
x = a1a2 . . .am ∈ Σ∗
y = b1b2 . . .bn ∈ Σ∗
then
x · y = a1a2 . . .amb1b2 . . .bn
ε is the identity for concatenation:
ε · x = x · ε = x
Length:
|x · y|= |x|+ |y|
COMP 2600 — Introduction to Theory of Computation 10
Language
L over Σ is any subset of Σ∗ (i.e. L⊆ Σ∗)
• set of all English words is a language over {A,B, . . . ,Z,a,d, . . . ,z}
• {ε,01,0011,000111, . . .}, i.e. all strings with equal number of 0 and 1and with all 0’s preceeding 1’s, is a language over {0,1}
• /0, the empty language, is a language over any alphabet
• {ε}, the language consisting of only the empty string, is a language overany alphabet
Note that a language may be infinite.
Note that L is also a language over any alphabet that is a superset of Σ.
COMP 2600 — Introduction to Theory of Computation 11
Finite State Automata
Neurophysiologists Warren S. McCullough and Walter Pitts published in 1943
the paper “A Logical Calculus of the Ideas Immanent in Nervous Activity” that
is considered as a seminal contribution to the theory of automata.
Nowadays FSA are a useful model for many important kinds of hardware and
software, such as:
• software for digital circuits
• software for validating protocols
• lexical analizers
COMP 2600 — Introduction to Theory of Computation 12
State is a snapshot of a system’s history
The advantage of having finite number of states is that we can implement the
system with fixed set of resources.
An automaton works as follows:
• it is always in one of finitely many states at a time
• starts in some state
• changes state in response to an external input (i.e. makes a transition)
• accepts input by ending in an accepting (also called final) state
The accepted strings constitute the language defined by the automaton!
COMP 2600 — Introduction to Theory of Computation 13
A finite automaton modelling recognition of COMP2600
start C CO COM
COMP
COMP2COMP26COMP260COMP2600
C O M
P
2
600
COMP 2600 — Introduction to Theory of Computation 14
Start state:
start
Accepting
state:
COMP2600
Transition:
COMP26COMP2600 When the FA is in state
COMP26 and it sees a “0” in
the input, it moves to state
COMP260 and advances on
the input.
COMP 2600 — Introduction to Theory of Computation 15
Vending Machine
• accepts $1 and $2 coins
• refunds all money if more than $4 is added
• is ready to deliver if exactly $4 has been added
����- $0 -2
@@R1
����$1 -
2
���1����
$2 -2
@@R1
����$3���1
��6
2
����� ��$4
��? 1, 2
Slide created by Ranald Clouston. Adapted by Katya Lebedeva.
COMP 2600 — Introduction to Theory of Computation 16
• Σ = {1,2}
• The start state is “$0”. This is indicated by the→ at the left of it.
• At the accepting state “$4” (double circled) you have the exact credit foryour purchase
Which of the following strings does this automaton accept?
• the empty string εו 22 X• 1222ו 1222221111 X
Slide created by Ranald Clouston. Adapted by Katya Lebedeva.
COMP 2600 — Introduction to Theory of Computation 17
Finite Automaton and Language
• each FA defines a language: the set of all strings that result in a sequence
of state transitions from the start state to an accepting state
• each string is a sequence of input labels along the path from the start
state to an accepting state
• languages that can be described by FA are regular languages
The study of FA and the study of formal languages are therefore linked:
• given a FA, find the language that is accepted by this automaton
• given a language, find the FA that accepts this language
COMP 2600 — Introduction to Theory of Computation 18