AUBER F4

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CD5560 FABER Formal Languages, Automata and
Models of Computation Lecture 4 Mälardalen
University 2010
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Regular Expressions and Regular
LanguagesNFA?DFA Subset Construction
(Delmängdskonstruktion)FA ? RE State
Elimination (Tillståndseliminering)Minimizing
DFA by Set Partitioning (Särskiljandealgoritmen)
Grammar Linear grammar. Regular grammar. Regular
Lng. .
Content
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Two Basic Theorems
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1 KleeneTheoremNFA?DFA Let M be NFA accepting
L. Then there exists DFA M that accepts L as
well.
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II Finite Language TheoremAny finite language
is FSA-acceptable.
Example L abba, abb, abab
FSA Finite State Automaton DFA/NFA
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Regular ExpressionsandRegular Languages
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Theorem - Part 1
Languages Generated by Regular Expressions
Regular Languages
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Theorem - Part 2
Languages Generated by Regular Expressions
Regular Languages
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Proof - Part 1
For any regular expression the
language is regular
Proof by induction on the size of
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Induction Basis

• Primitive Regular Expressions

(where )
Regular Languages
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Inductive Hypothesis
that and are regular languages
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Inductive Step

• We will prove

regular languages
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• By definition of regular expressions

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It is known, and we will prove in this lecture
Regular languages are closed under
union
concatenation
star
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• Therefore

regular languages
is a regular language
And trivially
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Proof Part 2
For any regular language there is
a regular expression with
Proof by construction of regular expression
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Since is regular take the NFA that
accepts it
Single final state
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• From construct the equivalent
• Generalized Transition Graph
• transition labels are regular expressions

Example
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Reverse of a Regular Language
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Theorem
The reverse of a regular language is a
regular language
Proof idea
Construct NFA that accepts
invert the transitions of the NFA that accepts
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Proof
Since is regular, there is NFA that accepts
Example
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Invert Transitions
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Make old initial state a final state and vice
versa
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Add a new initial state
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Resulting machine accepts
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Some Properties of Regular Languages, Summary
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Properties
For regular languages and we will
prove that
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Regular languages are closed under
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Example
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Union (Thompsons construction)

• NFA for

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Example
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Concatenation (Thompsons construction)

• NFA for

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Example

• NFA for

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Star Operation (Thompsons construction)

• NFA for

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Example

• NFA for

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Summary Operations on Regular Expressions

• RE Regular language description
• ab a,b
• (ab)(ab) aa, ab, ba, bb
• a ?, a, aa, aaa,
• ab b,ab, aab, aaab,
• (ab) ?, a, b, aa, ab, ba, aaa,
  bbb

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Algebraic Properties
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Operator Precedence

• 1. Kleene star
• 2. Concatenation
• 3. Union
• allows dropping unnecessary parentheses.

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Converting Regular Expression to a DFA
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Example From a Regular Expression to an NFA
Example (ab)abb step by step construction
(ab)
a
?
3
2
?
1
?
?
4
5
b
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Example From a Regular Expression to an NFA
Example (ab)abb
(ab)
?
a
2
3
?
?
?
?
0
1
6
?
?
4
5
b
?
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Example From a Regular Expression to an NFA
Example (ab)abb
(ab)abb
?
a
2
3
?
?
?
0
?
1
a
6
7
8
?
?
4
5
b
b
9
b
?
10
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Converting FA to a Regular ExpressionState
Eliminationhttp//www.math.uu.se/salling/Movies/
FA_to_RegExpression.mov
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Example

• Add a new start (s) and a new final (f) state
• From s to the ex-starting state (1)
• From each ex-final state (1,3) to f

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,
2
1
The original
3
f
Lets remove state 1! Each combination of
input/output to 1 will generate a new path once
state 1 is gone
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When state 1 is gone we must be able to make all
those transitions!
,
2
1
Previous
3
,
f
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,
2
1
3
f
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,
2
1
3
f
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A common mistake having several arrows between
the same pair of states. Join the two arrows (by
union of their regular expressions) before going
on to the next step.
,
2
1
3
f
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,
2
1
3
f
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Union again..
?
(
)
,
2
1
3
?
f
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Without state 1...
?
(
)
s
2
3
?
f
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Now we repeat the same procedure for the state
2...
?
(
)
2
?
(
?
3
?
f
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Following the path s-2-3 we concatenate all
strings on our waydont forget a in 2!
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When 2 is removed, the path 3 - 2 3 has also to
be preserved, so we concatenate 3-2 string, take
a loop and go back 2-3 with string b!
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This is how the FA looks like without state 2
3
f
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Finally we remove state 3
3
f
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...so we concatenate strings s-3, loop in 3 and
3-f
3
f
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Now we can omit state 3!
f
From s we have two choices, empty string or the
long expression OR is represented by union ?, as
usually
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So union the arrows...
...and we are done!
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Converting FA to a Regular Expression-An
Algorithm for State Elimination
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• We expand our notion of NFA- to allow transitions
  on arbitrary regular expressions, not simply
  single symbols or ? .
• Successively eliminate states, replacing
  transitions that enter and leave a state with a
  more complicated regular expression, until
  eventually there are only two states left a
  start state, an accepting state, and a single
  transition connecting them, labeled with a
  regular expression.
• The resulting regular expression is then our
  answer.

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• To begin with, the automaton should have a start
  state that has no transitions into it (including
  self-loops), and which is not accepting.
• If your automaton does not obey this property,
  add a new, non-accepting start state s, and add a
  ?-transition from s to the original start state.
• The automaton should also have a single accepting
  final state with no transitions leaving it, and
  no self-loops.
• If your automaton does not have it, add a new
  final state q, change all other accepting states
  to non-accepting, and add ? -transitions from
  them to q.
• This change clearly doesn't change the language
  accepted by the automaton.

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Repeat the following steps, which eliminate a
state 1. Pick a non-start, non-accepting state
q to eliminate. The state q will have i
transitions in and j transitions out. Each will
be labeled with a regular expression. For each of
the ij combinations of transitions into and out
of q, replace
r
p
q
with
r
p
And delete state q.
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2. If several transitions go between a pair of
states, replace them with a single transition
that is the union of the individual transitions.
E.g. replace
p
r
with
r
p
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Example
3
4
1
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• N.B. common mistake!

means
i.e.
NOT
See example on s 46 and 47 in Sallings book
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Minimizing DFAby Set Partitioninghttp//www.math
.uu.se/salling/Movies/SubsetConstruction.mov
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Minimization of DFA

• The deterministic finite automata are not always
  the smallest possible accepting the source
  language.
• There may be states with the same "acceptance
  behavior". This applies to states p and q, if for
  all input words, the automaton always or never
  moves to a final state from p and q.

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State Reduction by Set Partitioning
(Särskiljandealgoritmen)
The set partitioning technique is similar to one
used for partitioning people into groups based on
their responses to questionnaire. The following
slides show the detailed steps for computing
equivalent state sets of the starting DFA and
constructing the corresponding reduced DFA.
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Starting DFA
Reduced DFA
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State Reduction by Set Partitioning

• Step 0 Partition the states into two groups
• accepting and non-accepting.

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State Reduction by Set Partitioning

• Step 1 Get the response of each state for each
  input symbol. Notice that States 3 and 0 show
  different responses from the ones of the other
  states in the same set.
• P1 P2
• p1 p1 p1 p2 p1 p1
• a? ? ? ? a? ? ? ?
• 3, 4, 5 0, 1, 2
• b? ? ? ? b? ? ? ?
• p2 p1 p1 p2 p1 p1
• 
  

P1
P2
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Step 2 Partition the sets according to the
responses, and go to Step 1 until no partition
occurs.
P11 P12 P21 P22 p11 p11 p12
p12 a? ? ? a? ? ? 4, 5
3 1, 2 0
b? ? ? b? ? ? p11 p11
p11 p11
No further partition is possible for the sets P11
and P21 . So the final partition results are as
follows
4, 5 3 1, 2 0
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Minimized DFA consists of four states of the
final partition, and the transitions are the one
corresponding to the starting DFA.
4, 5 3 1, 2 0
Minimized DFA
Starting DFA
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DFA Minimization Algorithm
Why does this work? Partition P ? 2Q (power
set) Start off with 2 subsets of Q F and
Q-F While loop takes Pi?Pi1 by splitting one
or more sets Pi1 is at least one step closer to
the partition with Q sets Maximum of Q
splits Partitions are never combined Initial
partition ensures that final states are intact

• The algorithm

P ? F, Q-F while ( P is still changing)
T ? for each set s ? P for each ?
? ? partition s by ?
into s1, s2, , sk T ? T ? s1, s2, ,
sk if T ? P then P ? T
This is a fixed-point algorithm!
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Minimization and partitioning (Salling)
Example 2.38 A minimal automaton
Theorem 2.4 A DFA with N states is minimal if its
language distinguishes N strings
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Two strings x,y are distinguished by a language
(DFA) if there is a distinguishing stringsuch
that only one of strings xz, yz ends in accepting
state (belongs to language).
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A set of strings is distinguished by language if
each pair of strings is distinguished.
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Why ?
We can try shorter strings. Will do?
No, as both aba and aa get accept! Check!
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Set Partition Algorithm (Salling book)
Example 2.39 We apply step by step set partition
algorithm on the following DFA
Two strings x,y are distinguished by language
(DFA) if there is a (distinguishing) string z
such as that only one of strings xz, yz ends in
accepting state (belongs to language).
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We search for strings that are distinguished by
DFA!
Non-accepting
Accepting
Distinguishing string or
What happens when we read ?
From 1 and 6 by a we reach 3 which is accepting.
They form a special group.
From 5 we end in 6 by b, leaving its group.
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The minimal automaton has four states
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The Chomsky Hierarchy
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Non-regular languages
Context-Free Languages
Regular Languages
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Some Additional Examplesfor your excersize
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NFA N for (ab)abb
Example of subset construction for NFA ? DFA
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Translation table for DFA
Example of subset construction for NFA ? DFA
STATE INPUT SYMBOL INPUT SYMBOL
STATE a b
A B C D E B B B B B C D C E C
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Result of applying the subset construction of NFA
for (ab)abb.
Example of subset construction for NFA ? DFA
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• Another Example of State Elimination

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• Another Example of State Elimination

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Another Example of State Elimination
Resulting Regular Expression
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In General

• Removing states

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• Obtaining the final regular expression

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Example From an NFA to a DFA
b
states a b
A B C
B B D
C B C
D B E
E B C
C
b
b
a
a
b
b
A
B
D
E
a
a
a
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Example From an NFA to a DFA

states a b
A B A
B B D
D B E
E B A
b
b
b
a
b
A
B
D
E
a
a
a
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Example DFA Minimization
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Example DFA Minimization

• What about a ( b c ) ?
• First, the subset construction

?
b
q4
q5
?
?
?
a
?
?
q0
q1
q3
q2
q9
q8
c
?
?
q6
q7
?
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Example DFA Minimization

• Then, apply the minimization algorithm
• To produce the minimal DFA

final states
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Grammars

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Grammars

• Grammars express languages
• Example the English language

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• A derivation of the bird sings

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• A derivation of a dog barks

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• The language of the grammar

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Notation

Non-terminal (Variable)
Terminal
Production rule
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Example
Grammar

• Derivation of sentence

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• Derivation of sentence

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• Other derivations

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• The language of the grammar

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Formal Definition

• Grammar

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Example

• Grammar

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• Sentential Form
• A sentence that contains
• variables and terminals
• Example

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• We write
• Instead of

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• By default

(
)
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Example

Grammar
Derivations
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Example
Grammar
Derivations
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Another Grammar Example
Grammar
Derivations
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More Derivations

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The Language of a Grammar

• For a grammar with start variable

String of terminals
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Example

• For grammar

Since
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Notation

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Linear Grammars

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Linear Grammars

• Grammars with
• at most one variable (non-terminal)
• at the right side of a production

Examples
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A Non-Linear Grammar

Grammar
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Another Linear Grammar

• Grammar

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Right-Linear Grammars

• All productions have form

or
Example
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Left-Linear Grammars

• All productions have form

or
Example
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Regular Grammars

• A grammar is a regular grammar if and only if it
  is right-linear or left-linear.

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Regular Grammars GenerateRegular Languages
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Theorem
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Theorem - Part 1
Any regular grammar generates a regular language
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Theorem - Part 2
Any regular language is generated by a regular
grammar
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Proof Part 1
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The case of Right-Linear Grammars

• Let be a right-linear grammar
• We will prove is regular
• Proof idea We will construct NFA
• with

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• Grammar is right-linear

Example
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• Construct NFA such that every state
• is a grammar variable

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• Add edges for each production

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NFA
Grammar

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In General

• A right-linear grammar
• has variables
• and productions

or
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• We construct the NFA such that
• each variable corresponds to a node

.
special final state
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• For each production
• we add transitions and intermediate nodes

Examplehttp//www.cs.duke.edu/csed/jflap/tutoria
l/grammar/toFA/index.htmlConvert Right-Linear
Grammar to FA by JFLAP
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Example
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The case of Left-Linear Grammars

• Let be a left-linear grammar
• We will prove is regular
• Proof idea
• We will construct a right-linear
• grammar with

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Since is left-linear grammar the
productions look like

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• Construct right-linear grammar

In
In
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Construct right-linear grammar

In
In
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• It is easy to see that
• Since is right-linear, we have

Regular Language
Regular Language
Regular Language
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Proof - Part 2
Any regular language is generated by some
regular grammar
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Any regular language is generated by some
regular grammar
Proof idea
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Example
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Convert to a right-linear grammar

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In General
For any transition
Add production
variable
terminal
variable
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For any final state
Add production
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• Since is right-linear grammar
• is also a regular grammar with

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Regular Grammars

• A regular grammar is any
• right-linear or left-linear grammar

Examples
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Observation

• Regular grammars generate regular languages

Examples
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Chomskys Language Hierarchy
Non-regular languages
Regular Languages
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Application CompilerLexical Analysis(Lexikalisk
analys i Kompilatorteori)
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What is a compiler?

• A compiler is program that translates a source
  language into an equivalent target language

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What is a compiler?
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What is a compiler?
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Phases of a compiler
Source Program
Scanner
Tokens
Parser
Parse Tree
Abstract Syntax Tree with attributes
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Compilation in a Nutshell
Source code (character stream)
if (b 0) a b
Lexical analysis
if
(
b
)
a

b

0

Token stream
Parsing
if



Abstract syntax tree (AST)
b
0
a
b
Semantic Analysis
if
boolean
int



Decorated AST
int b
int 0
int a lvalue
int b
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Stages of Analysis

• Lexical Analysis breaking the input up into
  individual words/tokens
• Syntax Analysis parsing the phrase structure
  ofthe program
• Semantic Analysis calculating the programs
  meaning

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Lexical Analyzer

• Input is a stream of characters
• Produces a stream of names, keywords
  punctuation marks
• Discards white space and comments

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Lexical Analyzer

• Recognize tokens in a program source code.
• The tokens can be variable names, reserved words,
  operators, numbers, etc.
• Each kind of token can be specified as an RE,
  e.g., a variable name is of the form
  A-Za-zA-Za-z0-9.
• We can then construct an ? -NFA to recognize it
  automatically.

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Lexical Analyzer

• By putting all these ?-NFAs together, we obtain
  one which can recognize all different kinds of
  tokens in the input string.
• We can then convert this ? -NFA to NFA and then
  to DFA, and implement this DFA as a deterministic
  program - the lexical analyzer.

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