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Theory of Automata

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Theory of Automata
(CS402)
Theory of Automata
Lecture N0. 41
Reading Material
Introduction to Computer Theory
Chapter 15
Summary
Recap of PDA in conversion form, example of PDA in conversion form, joints of the machine, new pictorial
representation of PDA in conversion form, summary table, row sequence, row language.
Conversion form of PDA
Definition
A PDA is in conversion form if it fulfills the following conditions:
There is only one ACCEPT state.
There are no REJECT states.
Every READ or HERE is followed immediately by a POP i.e. every edge leading out of any READ or HERE
state goes directly into a POP state.
No two POPs exist in a row on the same path without a READ or HERE between them whether or not there are
any intervening PUSH states (i.e. the POP states must be separated by READs or HEREs).
All branching, deterministic or nondeterministic occurs at READ or HERE states, none at POP states and every
edge has only one label.
Even before we get to START, a "bottom of STACK" symbol $ is placed on the STACK. If this symbol is ever
popped in the processing it must be replaced immediately. The STACK is never popped beneath this symbol.
Right before entering ACCEPT this symbol is popped out and left.
The PDA must begin with the sequence
$
START
POP
HERE
PUSH $
The entire input string must be read before the machine can accept the word.
Example
Consider the following PDA accepting the language {a2nbn : n = 1,2,3, ...}
START
D
b
a
a
READ1
POP2
READ2
POP1
b
a
PUSH a
$
POP3
ACCEPT
Which may be converted to
126
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Theory of Automata
(CS402)
$
POP4
START
PUSH $
a
b
READ1
POP2
POP1
HERE
a
a
a
b
READ2
POP5
POP6
D
a
$
$
POP3
ACCEPT
PUSH a
PUSH $
PUSH a
PUSH a
The above PDA accepts exactly the same language
Note
It may be noted that any PDA which is in conversion form can be considered to be the collection of path
segments, where each path segment is of the following form
FROM
TO
READ
POP
PUSH
START
READ
ONE or
Exactly
Any
or READ
or HERE
no input
one
string
or
STACK
onto the
or HERE
letter
ACCEPT
character
STACK
START, READ, HERE and ACCEPT states are called the joints of the machine. Between two consecutive
joints on a path exactly one character is popped and any number of characters can be pushed.
The PDA which is in the conversion form can be supposed to be the set of joints with path segments in
between, similar to a TG
START
READ1
HERE
READ2
ACCEPT
127
img
Theory of Automata
(CS402)
The above entire machine can be described as a list of all joint-to-joint path segments, called summary table.
The PDA converted to the conversion form has the following summary table
FROM
TO
READ
POP
PUSH
ROW
Where
Where
What
What
What
Number
L
START
READ1
$
$
1
READ1
READ1
a
$
a$
2
READ1
READ1
a
a
aa
3
READ1
HERE
b
a
--
4
L
HERE
READ2
a
--
5
READ2
HERE
b
a
--
6
D
READ2
AT
$
--
7
Consider the word aaaabb. This word is accepted by the above PDA through the following path
START-POP4-PUSH$-READ1-POP6-PUSH $-PUSH a-READ1-POP5-PUSH a-PUSH a- READ1-
POP5-PUSH a-PUSH a-READ1-POP5-PUSH a-PUSH a-READ1-POP1- HERE-POP2-READ2-POP1-
HERE-POP2-READ2-POP3-ACCEPT.
The above path can also be expressed by the following path in terms of sequence of rows
Row1 ­Row2 ­Row3 ­Row3 ­Row3 ­Row4 ­Row5 ­Row6 ­Row5 ­Row7
It can be observed that the above path is not only joint-to-joint consistent but STACK consistent as well.
It may be noted that in FAs, paths correspond to strings of letters, while in PDAs, paths correspond to strings of
rows from the summary table.
Note
It may be noted that since the HERE state reads nothing from the TAPE, therefore L is kept in the READ what
column.
It may also be noted that the summary table contains all the information of the PDA which is in the pictorial
representation. Every path through the PDA is a sequence of rows of the summary table. However, not every
sequence of rows from the summary table represents a viable path, i.e. every sequence of rows may not be
STACK consistent.
It is very important to determine which sequences of rows do correspond to possible paths through the PDA,
because the paths are directly related to the language accepted, e.g. Row4 cannot be immediately followed by
Row6 because Row4 leaves in HERE, while Row6 begins in Read2. Some information must be kept about the
STACK before rows are concatenated.
To represent a path, a sequence of rows must be joint-consistent (the rows meet up end to end) and STACK-
consistent (when a row pops a character it should be there at the top of the STACK).
The next target is to define row language whose alphabet is  = {Row1, Row2, ..., Row7} i.e. the alphabet
consists of the letters which are the names of the rows in the summary table.
Note
It may be noted that the words of the row language trace joint-to-joint and STACK consistent paths, which
shows that all the words of this language begin with Row1 and end in Row7. Consider the following row
sequence Row5 Row5 Row3 Row6
This is string of 4 letters, but not word of the row language because
It does not represent a path starting from START and ending in ACCEPT state.
It is not joint consistent.
It is not STACK consistent.
Before the CFG that generates the language accepted by the given PDA, is determined, the CFG that generates
the row language is to be determined. For this purpose new nonterminals are to be introduced that contain the
information needed to ensure joint and STACK consistency.
It is not needed to maintain any information about what characters are read from the TAPE.
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