Transcript ECE-548 Sequential Machine Theory

Deterministic FA/ PDA
Sequential Machine Theory
Prof. K. J. Hintz
Department of Electrical and Computer
Engineering
Lecture 4
Updated by Marek Perkowski
Numerical Acceptor
A Non-intuitive Concept of a Language Is
One Which Can Accept Binary Arithmetic
Values, e.g. L  xB* : x  5 n, n  0 
4
0
1
0
1
0
1
2
1
1
0
5
1
0
3
Moret, Theory of Computation
Check all strings, from shortest that are accepted.
What is their language?
Languages Accepted by FA
• The Class of Languages Accepted by a
Deterministic or Nondeterministic FA Is
Closed Under
–
–
–
–
–
Union
Concatenation
Kleene Star
Complementation
Intersection
Union of Languages
L = L1L 2
since languages are sets of strings
M1
e
>
F = F1  F2
e
M2
Final States of NDFA
Concatenation of Languages
L = L1L 2
Any element of L 1 can be concatenated
with an element of L 2
>
e
F = F2
S1 M1 F1
e
S1 M2 F2
Final States of NDFA
Kleene Star of Languages
L = L1*
Any element of L 1 can be Kleene Starred
e
>
e
e
S1
q0
F = F1  q0
M1
F1
FinalStatesof NDFA
Difference of Languages
L = L1  L2
L1
F = F1  F2
since languages are sets of strings
L2
Final States of NDFA
Intersection of Languages
L = L1L 2
L1
since languages are sets of strings
L
L2
F = F1  F2   F1  F2   F2  F1 
Final Statesof NDFA
I*
Languages and FA
• Kleene’s Theorem
– A Language Is Regular iff It Is Accepted by a
Deterministic or Nondeterministic Finite
Automata
• A Regular Language Is One Which Can Be
Defined by a Regular Expression
– Not all languages are regular
Give examples of languages
that are not regular
Context Free Languages
• Up Until Now we learned the following:
Regular Expression  Regular Languages  DFA  NDFA
But This Is a Limited Class of Languages
• There Are Other Context-Free Languages
Which Cannot Be Recognized by a DFA
• There Are Other Types of Machines Which
Can Accept More Context-Free Languages
A Non-Regular Language
• Since a Language, L, Is a Subset of the Set
of All Strings, I*, Are There Some Strings
Which Cannot Be Produced by a Regular
Expression or Recognized by a DFA? Yes,
e.g.,
L  a b : n  0 
n
n
n
n
a b Regular?
• Theorem1: Let L be an infinite regular
language. Then there are strings x, y, and z
such that y  e and x yn z  L for each n  0
– Proof based on pigeonhole principle
• If there are n + 1 pigeons and n pigeonholes, then at
least two pigeons must be in the same hole
– If a string has more characters than there are
states in the language recognizer, then some
state must be entered more than once
1
Lewis & Papadimitriou, Elements of the Theory of Computation
n
n
a b Regular?
L  a b : n  0 
n
n
• Language Is Infinite, Therefore Theorem
Applies
• Is This Language Regular? If So, the
Theorem Must Apply
• Three Cases to Study
see next slides
n
n
a b Regular?
Case 1: y consists entirely of as
xap
p0
y  aq
q0
z  a r bs
r 0
thenL must contain
x y n z  a p  nq  r b s  n  0
at most one of thesestringscan havean
equal number of as and bs
n
n
a b Regular?
Case 2: y consists entirely of bs
Similar argument to Case 1
n
n
a b Regular?
Case 3: y contains both as and bs
For n > 1, x yn z has an occurrence of b
preceding an occurrence of a and
therefore cannot be in L
Context Free Grammars: idea
• A More General Way to Describe and/or
Generate Languages
• Context-Free
– Replacement Rules Can Be Applied
Independently of the Preceding or Following
Elements. A  aa B  q
abAaBc  ab aa aBc  ab aa aqc
abAac  ab A aqc  ab aa aqc
Context Free Grammar: definition
A Quadruple, ( I, , R, s) where
I

R
An alphabet
A set of terminals,   I
A finite set of rules, a subset of the
crossproduct of the set of non-terminals
and all strings,
R   I    I *
Context Free Grammar: definition
cont
s
The start symbol, a non-terminal
s  I   
I
The set of non-terminals
AND ...
Context Free Grammar: definition
cont
A I  
For All Non-terminals,
u I *
And Strings
The Grammar, G, Maps the Non-Terminals to
Strings
A 
u
G
for strings
and non-terminals
u, v, x, y, v’  I*
A I  
Context Free Grammar: definition of
G-related
u is G-related to v
u 
v
G
iff
u=xAy
v = x v’ y
and
A 
 v
G
Context Free Grammar: the set of all
possible relations
• The relation,


, the * meaning the set of
G
all possible relations, is
u 
u
– reflexive
G
u  v , v  w, then, u  w
– transitive
– has closure
G
I*
Context Free Language
• The Language Generated by the Context
Free Grammar, G, Is the Set of Strings
Which Map From the Start Symbol, s,
Under the Reflexive, Transitive, and Closed
Relation,



Language G    w  I * : s  w 
G


Context Free Language
• A Context Free Language Is One Which
Can Be Generated by a Context Free
Grammar, and,
• The Context Free Grammar (CFG) Can Be
Used to Generate Strings of the Language
A Derivation Example
• The Steps in Applying the Rules From the
Start Symbol to Any String Is Called a
Derivation in G, e.g.,
I   a, b, c, d , r, p, q, t , v, w  s  w
s w
   a, b, c, d , r 
 p  ra 
q  ab 


R  t  cad 
v  qp 


w  vtv 
 vtv
 qptqp
 qratqra
 abratabra
 abracadabra
Pushdown Automata
• All Context-Free Languages can Be Recognized
by a PDA
• { an bn } Is Context-Free, but Not Regular
– Problem is same number of as and bs
• PDA Assumes
– An unlimited memory in the form of a stack, LIFO
– The machine only has access to the top of the stack.
Pushdown Automata:
formal definition
A PDA is a Sextuple,
PDA   S, I,  ,  , s, F , where
S A finite set of States
I

 inputs , alphabet
 stack symbols 
 A non - deterministic push - down relation
s The unique start state
F
 final states 
Pushdown Relation
 s are written as
   S  I *  *    S   * 
  present state, input string, top of stack ,




 next state, new top of stack 
where
“top of stack”
is replaced by
“new top of stack.”
Pushdown Relation Example
  p, ,  ,  q ,   

Replace the  on top of the
x
stack with 

x
s=p

  p, ,  ,  q , a  
push, i. e., replace the null
on the top of the stack with a
s=q
x
a
x
s=p
s=q
PDA Properties
• One Can Have the Same State Before and
After a Transition, but Have Different Stack
Contents Which Makes It NonDeterministic
 p, i,      p,   
• State Changes May Occur in the Absence of
an Input, but Only If the Stack Is Not
Empty. i.e., non-causal behavior.
PDA Properties: what is the state of
PDA?
• The “State” of the PDA Is Determined Not
Only by S, and Not Only by the Top-ofStack, but Rather by Both the Current State
and the Complete Contents of the Stack, x
• This “State of the PDA” Is Called the
“Configuration of the PDA”
PDA Notation
• The Input May Cause the PDA to Change
From Configuration 1 to Configuration 2
 s1 , x1 
y
  s2 , x 2 
where
y I *
s1 , s2  S
x1 , x 2  *, the set of all possible stack contents
PDA Recognizer
• A string w  I* is accepted by a PDA
iff
 somefinal state,sf  F,
and somestringof stack symbols,   *
where theinput string,w  I *
such that
s0 ,   sf , 
w
may occur
PDA Example*
A Machine to Recognize Strings of the Form
wcwR
i. e.,
L 

* Lewis & Papadimitriou, pg. 114
wcw R : w   a, b 
*

PDA Example
wcwR Can Be Represented by a Context Free
Grammar,
G   I,  , R , S 
where
I

R
 S , a, b, c 
 a, b, c 
 S  aSa,

 S  bSb,
 Sc






Equivalent NPDA
wcwR can be represented by a NPDA where
  p,  ,  ,  q, S    T1 , push non - terminal 


  q,  , S ,  q, aSa    T2 , expand NT on TOS
  q,  , S ,  q, bSb    T , expand NT on TOS 
3


    q,  , S ,  q, c    T4 , replace NT with c 


q
,
a
,
a
,
q
,


T
,
pop
a






5


  q, b, b ,  q,     T6 , pop b



  q, c, c ,  q,     T7 , pop c

Is abccba Accepted?
Present
State
p
Unread
Input
abccba
q
abccba
S
T1
Push NT
q
abccba
aSa
T2
q
bccba
Sa
T5
Pick
T2/T3/T4
pop a
Stack
Transition Comments
(Top on Left)
used
e
Initial Conf
Is abccba Accepted?
Present
State
q
Unread
Input
bccba
q
ccba
q
ccba
cba
T4
q
cba
ba
T7
Stack
Transition Comments
(Top on Left)
used
bSba
T3
Pick
T2/T3/T4
Sba
T6
pop b
replace NT
with c
pop c
Is abccba Accepted?
The machine halts with unread inputs since
there is no
  q, c, b ,  next state, top of stack  
to be executed.
Is bacab Accepted?
Present
State
p
Unread
Input
bacab
q
bacab
S
T1
Push NT
q
bacab
bSb
T3
q
acab
Sb
T6
Pick
T2/T3/T4
pop b
Stack
Transition Comments
(Top on Left)
used
e
Initial Conf
Is bacab Accepted?
Present
State
q
Unread
Input
acab
q
cab
q
cab
cab
T4
q
ab
ab
T7
Stack
Transition Comments
(Top on Left)
used
aSab
T2
Pick
T2/T3/T4
Sab
T5
pop a
replace NT
with c
pop c
Is bacab Accepted?
Present
State
q
Unread
Input
b
q
e
Stack
Transition Comments
(Top on Left)
used
b
T5
pop a
e
T6
pop b
The machine halts with no unread input and
nothing on the stack, therefore,
bacab  language wcwR.