THEORY OF COMPUTATION

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THEORY OF COMPUTATION

• CSC 422

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INTRODUCTION

• Are all problems programmable?
• What statement of a problem constitutes an
  implementable program?
• Do the specifications of a program always lead to
  a program?
• Is it always possible to find specifications of a
  problem that lead to a program?

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Purpose of Theory of Computation

• Design a language for the mathematical
  specification of computer languages in general
  for all computers
• Describe the workings of a general computer in
  the simplest and most basic terms possible
• Find a mathematical language acceptable to the
  above general, basic computer

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SETS FUNCTIONS

• A collection
• a,b,c A A set
• CxS (c1,s1),,(cn,sn) Cartesian
  product
• 0,1 Binary alphabet
• e e0 Empty set its length
• wr Reverse of w (if wwr gt
  palindrome)
• A Complement of A (A even nos gt
  A odd nos)

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RELATIONS

• Reflexive if (a,a) ? R ? a ? S
• Symmetric if (a,b) ? R gt (b,a) ? R
• Transitive if (a,b) ? R , (b,c) ? R gt (a,c) ?
  R
• Equivalence relation is reflexive, symmetric
  transitive

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Examples of reflexivity

• Let relation R be a lt b then a lt a R is
  reflexive
• Let relation R be a lt b then a is not lt a R
  is not reflexive
• Let set S be oaks let R be oaks ? trees then
  a ? S gt a ? trees, so (a,a) ? R

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Examples of symmetry

• If R is (big,large) then a ? big gt a ?
  large a ? large gt a ? big
• If R is relation of synonyms then a synonym of
  b gt b synonym of a so R is symmetric also
  reflexive

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Examples of transitivity

• If R is on top of (a,b) ? R (b,c) ? R then
  (a,c) ? R R is transitive
• If R is lt (a,b) ? R (b,c) ? R then (a,c) ?
  R R is transitive
• If R is child of (Bob, Mary) ? R (Mary,
  John) then R is not transitive
• If R is successor of (Bob, Mary) ? R (Mary,
  John) then R is transitive

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Union, Intersection Complement
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Laws of Set Operations

• Idempotency A?A A, A?A A
• Commutativity A?BB?A, A?BB?A
• Associativity (A?B)?C A?(B?C)
• (A?B)?CA?(B?C)
• Distributivity (A?B)?C(A?C)?(B?C)
• (A?B) ? C(A?C) ?(B?C)
• Absorption (A?B)?AA, (A?B)?AA
• DeMorgans A-(B?C)(A-B)?(A-C)
• A-(B?C)(A-B)?(A-C)

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Examples

• Apples ? apples apples
• Apples ? oranges oranges ? apples
• Dad ? Mom Mom ? Dad

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FUNCTIONS

• R is a function if ? x ? X ?! y ? Y
  (x,y) ? R
• There are two notations
• f X --gt Y
• f ? X x Y
• Alternate definition of a function ? x ? X
  ?! y ? Y ? f(x) y

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x
y
y1
For each x ? X there is a unique y ? Y
In other words, the dotted line going from x to
y1 cannot exist
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BIJECTIVE FUNCTION1-1 function

• If ?y ? Y ?! x ?X then we have a Bijection or
  1-1 function
• If f is bijective gt ? f--1(y) x for f(x) y

x
y
x1
y1
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FINITE AUTOMATA

• Are all problems as easily programmable?
• Consider a program to compute the integral of xn
  vs a sort program
• The first has constant memory requirements , the
  second can be of arbitrary length
• Can the implementation of a program be designed
  as a program?

input
output
program
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Deterministic Finite Automata

• A DFA is a program to design programs that use a
  constant amount of memory.
• A DFA can be thought of as a tape which reads one
  character at the time until the end of the tape.
  Each position on the tape puts DFA in a different
  state.
• It can elucidate errors that may show up in a
  programming problem.

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Example 1
When a customer pays for goods with electronic
money, that is with an ATM card, all
possibilities must be accounted for

• The customer may decide to pay, i.e. sends the
  money to the store
• The customer may cancel money is sent to bank
  to be deposited in customers account
• The store may ship the goods to the customer
• The store may redeem the money, i.e. the money
  is sent to the bank to be given to the store
• The bank may send the money to the store

Can the store ship the goods without ever getting
paid?
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DFA for previous problem
For the store
For the bank
For the customer
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Example 2Memory machine
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Example 3Parity machine
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Example 4Adding machine
1 1 1 101101 111001 1100110
States are ( i1 i2 carry) (000), (001), (010),
(011), (100), (101), (110), (111)
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Definition of a DFA
DFA A (Q, ?, ?, s, F) Q - finite set of
states ? - finite input alphabet ? -
transition function from Q x ? --gt Q s -
initial state s ? Q F - favorable, or
accepting, states F ? Q
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Applications of a DFA

• Search engines on Web can use them
• News analysts searching on-line for special
  topics
• Stock analysts searching for stock names
• Shopping robots searching for best price
  on-line
• Searching for all books on Amazon containing a
  certain phrase or word
• grep, egrep, fgrep in Unix

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Functioning of a DFA

• Automaton A in state q ? Q
• reads a ? ? (letter in the alphabet)
• enters state q1 ? (q, a) which is determined
  completely by current state which is the
  content of the current cell.
• Set of all accepted input words by A is called
  the language L(A) of the DFA.

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Finite State Diagram
It is a directed graph representing a DFA.
a
State q changing to state q1 after reading input a
q
q1
A favorable state is doubly circled.
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Programs Finite State Diagrams
If-else
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Example 1
a
a
b
s
r
q
b
a
b
Accepts language, L(anbm)
Ex statement, statement,, if-then, if-then, ,
end
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Example 2
b
a
a
s
r
q
a
b
b
Accepts any language with 2 consecutive
as L(, a, a, )
Ex Any program with a nested pair of while
loops
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Example 3
b
a
a
s
r
q
a
b
b
It will not accept two consecutive as L(, a,
a, ) (Ex no nested for loops)
It is the complement of the previous example
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Example 4
b

• Initial a must be followed by a to be
  accepted

• Initial b can be followed by any number of as
  bs

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Example 5
b
b
a
a
s
f
q
a
b
L(a, bn, a, bm)
a
b
p
Ex one case followed by multiple statements
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Configurations

• In chess there are individual moves standard
  patterns for the opening or end game.
• Imagine that the players have reached the end
  game the pieces left are BK, BQ, 1 BR, 1 BKt, 3
  P WK,WQ, 1 WB, 1 WKt, 2 P.
• Imagine they are in a certain configuration, that
  is a certain pattern on the board.
• Then the next moves can be figured out from that
  point on.

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Definition of a configuration
A configuration is a composite of state (pieces
left), position (pattern on the board), input
(next moves). Ex In reading input aaaabba,
after state aaa has been reached, position q
can be reached by reading input abba (q,abba)
is a configuration towards acceptance. w is
accepted if it yields a favorable state. If
(q,w) --gt(q1,w1) then ? s ? ? w s w1 d
(q, s ) q1 then (q,w) yields (q1,w1) in one
step (q,w) yields (q1,w1) if ? a sequence of
configurations (q1,w1) (qk,wk) (q1,w1)
(q,w) , (qk,wk) (q1,w1) (qi,wi) yields
(qi1,wi1) in one step. All configurations
yield unique configurations as it is
deterministic.
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Example 1
A vending machine for newspapers. The cover is
released when 0.25 is reached. It does not
return money if gt 0.25 is put in.
? ( 5c, 10c, 25c )
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Examples of applications in Computer Science

• Programming sequences, branching, loops
• Pattern matching (for WWW AI)
• Lexical analysis in compilers
• Finite state machines in software specs design
• Word processors
• Design of telecommunication protocols
• Design of circuits for VLSI
• Hardware design
• Control mechanisms

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Non-deterministic Finite Automata

• Union of 2 DFAs gives a NDFA as more than 1 arrow
  from any state with the same input.
• For example, in pattern matching of A ? B, we may
  match A or we may match B.
• NDFA A ( Q,?,?,s,F )
• where ? ? Q x (? ? e) x Q is a transition
  relation (q, a, p) ? ?
• The empty string as input permits the
  automaton to jump from one state to the other. So
  after A is matched, whatever state the other
  branch is at, can jump

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Example

• The ability to be in several different states at
  once, can be expressed as the ability to guess
  what will come next.
• When a system searches for a certain sequence of
  characters, such as a keyword, we can use a
  sequence of states to do nothing but jump from
  state to state until we find the keyword.

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Example 1Find first occurrence of keywords
Searching for the keywords Web or
Internet Finding either gets us to a favorable
state.
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Functioning of a NDFA vs DFA

• Automaton A in state q ? Q
• reads a ? ? (letter in the alphabet)
• enters state q1 or q2 ? (q, a) which is
  determined completely by current state which is
  the content of the current cell.

State q changing to state q1 or state q2 after
reading input a ?b
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Interpretation

• One input can send you to 2 different
  sub-programs (matching 2 Jones in a DB)
• This is useful when you build a program from
  multiple sub-programs. In industry, whole teams
  work on complicated programs, each member on
  separate sub-programs.
• In parallel programming, this represents
  different threads.
• NDFAs are easier to design can be changed to
  DFAs

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Examples
a
No jumping from q to p Therefore no need for trap
state (q,w) can yield many different
configurations (q1, e)
Automaton that begins ends with b
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Definition theorem

• Definition A A1 that accept the same language
  are equivalent.
• Theorem For all non-deterministic automaton A
  there exists a deterministic finite automaton A1
  equivalent to A.

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Example
e
q
r
This NDFA is equivalent to the following
deterministic transitions
e
a
b
a
b
t
s
b
Accepting states are t r,t q,r,t Trap
state is 0
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Equivalent Deterministic Automaton
b
a
s
b
a
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Regular Expressions

• Regular expressions are used to
• Design a language for mathematical specification
  of languages acceptable by a finite automata
  (computers of all types)
• Do that by a general algorithmic procedure
• To convert FA back to its specs through an
  algorithmic procedure.
• These are used in turn to do simulations.

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Reg. Exp. vs FA

• DFA NDFA are machine-like descriptions
• Regular Expressions are algebraic-like
  descriptions.

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Applications of Regular Expressions

• Lexical analyzers such as Lex Flex which take
  source code and convert it into tokens
• Grep in Unix
• Alternative to FA notation for describing
  software components
• A declarative way to express which strings are
  acceptable

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Language for regular expressions
Some of the complex programming languages can be
obtained from simpler languages using ?, ?, ?,
concatenation Kleene stars. Concatenation of
strings u, v is uv of languages L1, L2 is
L1L2 uv u ?L1, v ? L2 Kleene Star L of
L is the infinite union e ? L ? L2 ?
L3 L w1w2 wk when wi ? L
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Examples
L ? M 001, 10, 111 ? e, 001 e, 001, 10,
111 L ? M 001 LM 001, 10, 111, 001001,
10001, 111001 L for 0, 1 all strings of
0s 1s for 0, 11 all strings of 0s
1s such that 1s come in pairs L0 e L1
0, 11 L2 00, 011, 110, 1111 L3
000, 0011, 0110, 01111, 1100, 11011, 11110,
111111
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Languages accepted by FA
Theorem If languages L M are acceptable by a
finite automaton, so are L ? M, L ? M, ? - L
(complement), L M, LM (conc), L (Kleene Star)
Suppose L accepted by
Suppose M accepted by
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• Then L ? M accepted by

_ 2. Then L is accepted by a
FA which is the DFA equivalent of the NDFA and by
flipping favorable unfavorable states
_______ _ __ 3. L ? M (L
? M) and is accepted
__ 4. L M L ? M and is accepted
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5. LM is accepted by
6. L is accepted by
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Definition of a Regular Expression

• It is a string over an alphabet
• ? ? (,), e, ?, ?,
• It is mostly used in lexical analyzers.
• ? , e, ?a ? ? ( the ground elements) are regular
  expressions.
• If ?, ? are regular expressions then
  ? ? ?, ??, ? are regular expressions by
  induction
• No other string is a regular expression.

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Mapping of regular expression ? to language L(?)

• L(?) ?, L(e) e, L(a) a ?a ? ?
• If ?, ? are regular expressions then
• L(? ? ? ) L(?) ? L(?)
• L(??) L(?)L( ?)
• L(?) (L(?))

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Examples

• L((ab)a) w w of form abna
• Identifier in C begins with a letter may be
  followed by a string of letters digits. The
  identifiers can thus be expressed by the regular
  expression
• (a-z ? A-Z) (a-z ? A-Z ? 0-9)
• Identifiers of languages with underscores can be
  expressed by
• (a-z ? A-Z) ((-(a-z ? A-Z ? 0-9))
  (a-z ? A-Z ? 0-9))

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Examples

• Strings with alternating 0s 1s
• (01) (10) 0(10) 1(01)
• Same as starting with (01) and adding
  optional 1 at beginning and optional 0 at end
• (e1)(01)(e0)
• Precedence of operators in Reg. Exp. is
• , . , or by grouping with parentheses

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Regular Expression --gt FA

• For reg. exp. ( (a ? ab) ba )
• ground singletons become
• doubletons become
• (a ? ab) becomes

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(a ? ab)
gt
Add ba
Add ( ... )
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FA --gt Reg. Exp.
Start with FA

• Replace 2 favorable states
• with 1.
• Label nodes 1 to n.

• Replace arrows from i to j from j to k with an
  arrow labeled li,jlj,k

Label from 1 to 3 will be ab Label from 1 to 4
will be abb

• If there is an arrow from j to j, add label
  li,jlji

Labels b,a from 2 to 3 will be inserted as
ab(ab)b
Labels from 1 to 4 and 1 to 7 will be replaced
by ab(ab)b ? ba

• Different arrows from i to j will be replaced by
  l1 ? l2

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Example

• Single favorable state
• Number nodes
• 3 ? 2 ? 4 becomes ab from 3 to 4 1? 2 ? 4
  becomes ab from 1 to 4 4 ? 2 ? 4 becomes bb
• Multiple arrows from one node are replaced by a
  union

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After 1, 2, 3
After 4
ab(bab ? bb)a
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Non-regular languages

• In the design of circuits on a chip, it is
  important to know whether two automata define the
  same language. This might permit to minimize the
  area cost of the chip.
• FA regular languages can describe programs
  using a fixed amount of memory regardless of
  input. This is not true for loops.

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Example

• L(anbn) ab, a2b2, a3b3, is not a regular
  language because after accepting an the
  automaton is in a state q and that state cannot
  know what came before. So it does not know how
  many bs it should accept.
• A loop in a program can be represented by a
  regular language or a FA if we can insert or
  remove iterations without changing the nature of
  the program.

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Pumping Lemma for Reg. Exp.

• For every Reg.lang. L there exists a constant n
  such that every string w in L, where w gt n, we
  can break w into 3 strings, w xyz
• y ? e
• xy lt n
• For all k gt0 the string xykz is also in L

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Steps in proof

• Assume it is regular.
• Find the defining property of language
• Define a y such that xy lt n and wxyz
• Find a k such that xykz destroys the property of
  the language
• You have now found one example that is not in the
  language
• Therefore the language cannot be regular

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Example 1

• L(anbn) w is not a reg.lang.
• Proof
• w 2n
• Let w xyz, then if
• xy lt n, then y am for some m
• Therefore xykz anamkbn and is not in the
  language

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Example 2

• L(1n) w w is a prime is not a reg.lang.
• Proof
• If it were a reg.lang., there would exist a prime
  p gt n, such that w 1p, and w p.
• Then w xyz, xy lt n
• Let y m. Then xz p-m
• Consider string xyp-mz which must then be in L.
• xyp-mz xz (p-m)y (p m) (p m)m
  (m 1)(p m)
• Therefore xyp-mz is not a prime.

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Example 3

• Language of palindromes is not regular.
• Proof
• If it were a regular language, then there would
  exist an n every w ? L, where w gt n can be
  represented as w xyz.
• Consider anban. Since xy lt n, y is a substring
  of an on the left. Let y am
• Then w an-myban and by the Pumping Lemma
  an-mykban ? L. But an-mykban an-mamkban and it
  is not a palindrome.

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The closure properties of RL

• The union of 2 RLs is regular
• The intersection of 2 RLs is regular
• The complement of 2 RLs is regular
• The difference of 2 RLs is regular
• The reversal of a RL is regular
• The Kleene star of a RL is regular
• The concatenation of 2 RLs is regular
• A homomorphism (substitution of strings for
  symbols) of a RL is regular
• The inverse homomorphism of a RL is regular

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Equivalence Minimization of Automata

• When are 2 descriptors of a RL equivalent?
• Consequently, when can we minimize a DFA? This
  would be a unique minimal DFA.
• We start by asking when 2 states can be replaced
  by a single state behaving like both.

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Equivalent distinguishable states

• p q are equivalent if
• ?w ?(p, w) is an accepting state iff ?(q,w)
  is an accepting state
• If 2 states are not equivalent, then they are
  distinguishable, and one is accepting while the
  other is not.

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Writing efficient FAs REs

• There many various FAs and REs that will
  satisfy a language, just as there are many
  different programs that will solve a particular
  problem.
• FAs REs are used to design VLSI chips.
• In all cases it is more efficient cost
  effective to find the minimal FA or RE to
  represent the problem at hand.

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State Minimization Problem

• The idea is to find the FA (or RE) with the
  minimal number of states equivalent to the one
  being used.
• Equivalent automata are indistinguishable by the
  time they get to an accepting state. That is, if
  they both get accepted with the same inputs, they
  are indistinguishable.

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Example 1

• State n is unreachable
• States t p are both reachable from input a and
  are indistinguishable
• n will be eliminated and t p will be merged
• q r are both accepted through a and reach a
  dead state through b. They are indistinguishable.

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Equivalent automaton
76
Example 2

• A E are both non-accepting, so input e does not
  apply.
• On input 1 they both go to F.
• On input 0, A goes to B E goes to H. Then on
  input 1, B goes to C H goes to C. So on input
  01, both A E go to C, a favorable state.
• Therefore A E are equivalent.
• D F both go to C on 0

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Equivalent minimal FA
0
B,H
G
0
1
A,E
1
1
F
1
0
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Definition of equivalent states
States p q are equivalent if for all input
strings w, ?(p, w) is an accepting if and only
if ?(q, w) is an accepting state. Any pair of
states that are not distinguishable as they
proceed to be accepted are equivalent.
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Finding equivalent states

• Any state that is not accepting cannot be
  equivalent to any accepting state.
• States that reach an accepting state with the
  same single input are equivalent.
• States that reach an accepting state with the
  same multiple input are equivalent.

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Proof of equivalency

• It is easier to show the states that are not
  equivalent than the ones that are equivalent on a
  very large FA. To prove they are equivalent all
  inputs (0, 1, 00, 01, 11, 000, ) must be
  considered.
• The accepting state(s) is(are) not equivalent to
  any non-accepting state
• Going back one step from the accepting state(s),
  the states getting there on different inputs are
  not equivalent.
• Going back two steps from the accepting state(s),
  the states getting there on different inputs are
  not equivalent.
• Etc.

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Table of inequivalencesfor example 2
x
82
Minimization of FA through tables

• Eliminate any state that cannot be reached from
  the start state
• Partition remaining states into equivalent blocks
  so that no pair of states from different blocks
  are equivalent

83
Context-Free Grammars

• Some languages cannot be recognized by FAs
• Can FAs recognize legal English statements?
• Can FAs recognize the syntactical correctness of
  statements in programs?

84
Uses of Context-Free Grammars

• Since the 1960s CFGs have been used to turn out
  parsers automatically.
• They are used to describe document formats with
  Document Type Definitions, DTDs used in XML for
  creating Web pages.
• Grammars can define languages through the use of
  Parse Trees.

85
Parse Tree example 1
sentence ? noun-phrase, verb-phrase noun-phrase
?(proper-noun determiner), common-noun verb-
phrase ? verb (verb, adverb)
A Parse Tree is a recursive definition of a
language.
86
Palindromes example 2

• The language of palindromes cannot be represented
  by REs, but it can be defined recursively as
• e, 0, 1 are palindromes
• If w is a palindrome, so are 0w0 and 1w1

• As a Parse Tree
• P ? e
• P ? 0
• P ? 1
• P ? 0P0
• P ? 1P1

P
0 P 0
e
87
Definition of a CFG

• Set of symbols that form the strings of the
  grammar are called terminals, ?
• Set of variables, or strings, are called
  non-terminals, NT
• A start symbol, S ? NT
• Set of productions or rules, R
• R ? NT x (? ? NT)
• which serve to derive the terminals from the
  non-terminals

88
Derivation mechanism

• Left part of arrow always has a non-terminal
• Right part of arrow has a sentential form
• A sentential form v is one-step derivable from
  sentential form u
• u ? v if u xAy and v xzy and A ? z is in the
  rules R
• v is derivable from u, u ? v, if there exist a
  sequence, called a derivation,
• u0, ,un uu0 and vun and u0?u1,
  u1?u2,,un-1?un
• The language L(G) generated or derivable in a
  grammar is defined as
• L(G) w w ? ?, S ? w

89
Example 3 - a pseudo-language
Example of a pgm
Inst ? Assign Inst-phrase END Assign ? (Id)
( )(Id) (op) (Id) Id ? (a-z A-Z) (0-9
a-z A-Z) op ? - Inst-phrase ?
If-phrase If-else-phrase If-phrase ?(IF) (
condition) condition ? (Id) (Cond-op)(Inst)
Cond-op ? lt gt lt gt If-else-phrase ?
(If-phrase) (ELSE) (Inst)
Id Id Id Id Id Id IF (Id lt gtlt gt
Id) Inst ELSE Inst END
90
Example 4
Let grammar G have non-terminals NT S, Np,
Vp, Ap, N, V, A and terminals ? big, stout,
John, bought, white, car and rules, or
productions S ? NpVp Np ? N ApN e Ap ? ApA
e Vp ? VNp A ? big stout white N ? John
car V ? bought

• Strings generated by G
• John bought car
• John bought big car
• big stout John bought big white car
• big stout car bought big white car

91
Different derivations of a sentence
Given S John bought car we can have
derivation S ? NpVp ? NVp ?John Vp ? John
bought Np ? John bought N ? John bought
car or S ? NpVp ? NpVNp ? NVNp ? NVN ? NVcar
? N bought car ? John bought car Both
derivations give the same result and can best be
described in a parse tree. All derivations with
same parse tree are equivalent. We can always
replace leftmost non-terminal, called leftmost
derivations, or we can always replace the
rightmost non-terminal, called rightmost
derivations. CFG derivations are
non-deterministic.
92
Parse tree for example 4
S
93
Parse trees for abc
They are not equivalent. That is why we need a
precedence rule for algebraic expressions. Factors
are parenthesized expressions that cannot be
broken down. Terms are expressions that cannot be
broken down by the or - operators, such as
ab. A parser will accept only leftmost or only
rightmost derivations.
94
Applications
1. Parsers were the first applications of CFG 2.
The YACC command in Unix is a CFG that creates
either a tree or a piece of object code. It
allows to state the precedence of operators
in expressions. 3. XML (Extensible Mark-up
Language) was the precursor is a superset
of HTML (Hypertext Markup Language), a language
with which Web pages are created, both require
a DTD (Document Type Definition) which is a
CFG describing the tags allowed.
95
Pushdown Automata
FA can easily be expressed as derivations, so
that any FA can be expressed as a CFG ??(a,s)
s1 can be expressed as s ? as1 However, there
are non-regular, context-free languages such as
L(anbn). If we add a stack to a FA such that
every time it reads a b it pops an a, then it
does not need to remember how many as there
were.
96
Stack operation
Finite state control
input
Accept/ reject
stack
97
Example - language of palindromes

• Take the language of palindromes of even length
• Lwwr wwr w is in (0,1)
• q0 is the state that represents a guess that
  we are not yet in the middle. In state q0 we read
  input symbols push them onto the stack.
• At any time we guess that we have seen the
  middle go to state q1. Here the right part of w
  will be on top of the stack and the left part on
  the bottom.
• In state q1 we compare input symbols with the
  symbol at the top of the stack. If they do not
  match, the guess was wrong this branch dies.
• If the input symbol matches the symbol on the
  top of the stack, we start popping until the
  stack is empty enter an accepting state.

98
Example for w1111
(q0,1111,a0) (q0,111,1a0) (q1,1111,a0) ?
(f,1111,a0) (q0,11,11a0) (q1,111,1a0) ?
(q1,11,a0) (q0,1,111a0) (q1,11,11,a0)
(f,11,a0) (q0,e,1111a0) (q1,1,111a0)
(q1,1,1a0) (q1,e,1111a0) (q1,e,11a0)
(q1,e,a0) (f,e,a0)
99
Definition of Pushdown Automata
P (Q, ?, ?, s0, a0, ?, F) where Q is a finite
set of states ? is the input alphabet ? is the
set of stack symbols s0? Q is the initial
state a0 is the start symbol needed at bottom of
stack to get to a
favorable state after stack has been
emptied ?(s,a,X s?Q, a???e, X??) is the
transition relation with output (p,S p?Q, S is
string of symbols replacing X on top of stack)
F?Q is the set of favorable states.
100
PDA for language of palindromes

• P ( Q , ? , ? , s0, a0,
  ?, F )
• P (s0,s1,f, 0,1, 0,1,a0, s0, a0, ?, f)
• Originally in state s0 the stack contains a0 and
  we input 0 or 1
• ?(s0,0,a0) (s0,0) and ?(s0,1,a0)
  (s0,1)
• Reading another 0 or 1 we obtain transitions
  ?(s0,0,0) (s0,00), ?(s0,0,1) (s0,01),
  ?(s0,1,0) (s0,10) and ?(s0,1,1) (s0,11)
• We can go from state s0 to state s1 on input e
  ?(s0,e,a0) (s1,a0) ?(s0,e,0) (s1,0)
  and ?(s0,e,1) (s1,1)
• In state s1 we match input symbols to stack
  symbols ?(s1,0,0) (s1,e) and ?(s1,1,1)
  (s1,e)
• After emptying the stack of input we are left
  with a0 in it, and e takes us to a favorable
  state s2 ?(s1,e,a0) (f,a0)

101
PDA for Lanbn)

• Q s0,s1,f , ? a,b, ? a, F f
• ? (s0,e,e) (s1,a0)
• ? (s1,a,e) (s1,a)
• ? (s1,a,e) (s1,a)
• ? (s1,b,a) (s1,a0)
• ? (s1,e,a0) (f,e)

102
PDA for La,b

• Language with the same number of as and bs
• Q s0,s1,f , ? a,b, ? e, a0 , F f
• ? (s0,e,e) (s1,a0)
• ? (s1,a,a0) (s1,aa0)
• ? (s1,a,a) (s1,aa)
• ? (s1,a,b) (s1,e)
• ? (s1,b,a0) (s1,ba0)
• ? (s1,b,b) (s1,bb)
• ? (s1,b,a) (s1,a0)
• ? (s1,e,a0) (f,e)

103
Transitions accepting abbbaaaabb
104
Acceptance by empty stack
In the last example, we reached the end when the
stack was empty. In the example, we did not reach
a favorable state. This is called a PDA that
accepts input strings by empty stack rather than
by a favorable state, and it is described by P
(Q, ?, ?, s0, a0, ?) Theorem There is an
algorithm that accepts a string by empty stack
rather than by favorable state, and the two
algorithms are equivalent.
105
CFG ? PDA

• Theorem
• Given any CFG, G, there exists an algorithm that
  constructs a PDA, A, such that L(A) L(G)
• Proof
• Let A have 2 states, s0 and f.
• Push s0 onto the stack.
• If topmost symbol on stack is N, a non-terminal,
  A picks rule N?w in G and replaces N on the top
  of the stack by w.
• If topmost symbol on stack is T, a terminal, A
  advances to next symbol if it matches T, it pops
  top of the stack.

• Transitions
• ? (s0,e,e) (f,s0)
• ? (f,e,N) (f,w) for each rule N ? w in
  grammar G
• ? (f,T,T) (f,e) for each terminal T

106
Example 1 for CFG ? PDA
Grammar G with following rules S ? e S ? aSa S ?
bSb For language L wwr w ?a,b

• Transitions of PDA for CFG A is
• ? (s0,e,e) (f,S)
• ? (f,e,S) (f,aSa)
• ? (f,e,S) (f,bSb)
• ? (f,e,S) (f,e)
• ? (f,a,a) (f,e)
• ? (f,b,b) (f,e)

107
PDA ? CFG
Theorem Given any PDA, A, there exists an
algorithm that constructs a CFG, G, such that
L(G) L(A) Proof Consider a simple PDA
accepting by empty stack. A simple PDA is one
where every transition replaces the top of the
stack. For any transition ?(s,a,?) --gt (q,?) Let
? be A1, A2, , An. These represent the popped
symbols. Then the transition ?(s,a,?) --gt (q,?)
can be replaced by ?(s,e,A1) --gt (sA1,e)
?(sA1,e,A2) --gt (sA1A2,e)
?(sA1An-2,e,An-1) --gt (sA1A2An-1,e)
?(sA1An-1,e,An) --gt (q,?)
108

• The grammar can now be defined by the following
  rules
• For every s?Q, S --gt s0, a0, s
• For every s,q?Q, every a?? ? e, A??, if
  ?(s,a,A) --gt (q,e), then s,A,q --gt a
• For every s,q?Q, every a?? ? e, A??, if
  ?(s,a,A) --gt (q,B1B2Bk) then
• s,A,qk --gt aq,B1,q1 q1,B2,q2... qk-1,
  Bk,qk

109
RL ? PDA ? CFG
Theorem Every Regular Language is context
free. Proof Given a FA, A, accepting a
language, L. View A as a PDA that does not use
its stack. Therefore L is accepted by a
PDA. Because every PDA ? CFG, every Regular
Language, or FA, is context free.
110
Languages that are not Context-Free
Take language S --gt uAz --gt uvAyz --gt
uvxyz Then it must be true that A --gt x and A
--gt vAy Then we can derive further, getting S
--gt uAz --gt uvAyz --gt uv2Ay2z --gt uv3Az3z --gt ..
--gt uvnAynz Now we can define a type of Pumping
Lemma for context-free languages
111
Pumping Lemma for Context-Free Languages
Lemma Let G (?, NT, R, S) be a context-free
grammar. Then there exists a number n such that
any string w ? L(G) with length w ? n can be
written as w uvxyz for some strings u, v, x, y,
z ? ?, and such that 1. v gt 0, or y gt 0 2.
vxy ? n 3. For any k ? 0, uvkxykz ? L(G)
112
Example 1
Show that the language L akbkck k 0, 1, 2,
is not context-free. Proof If it is
context-free, let w anbncn with length w
3n Let w uvxyz with length vxy ? n Then vxy
contains only as and bs, or only bs and
cs. If vxy contains only as and bs, then v or
y contains at least one symbol. Then the string
uv2xy2z contains more than n as, or more that n
bs But the number of cs is still the same.
Therefore the string is not in the language.