5' AUTOMATA AND LANGUAGES

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5.AUTOMATA AND LANGUAGES
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• This section studies the capabilities and
  limitations of algorithmic computation, using
  effective procedures.
• The effective procedures is defined by a finite
  set of instructions that specifies the operation
  that make up the procedure.
• The computation defined by an effective procedure
  consists of sequentially executing a finite
  number of construction and terminating.
• An effective procedure is a deterministic
  discrete process that halts for all possible
  input values.
• We will learn a series of powerful abstract
  computing machines where computations satisfy our
  term effective process.
• Input to a machine a string over an alphabet
• Result of computation indicate the
  acceptability of the input string.

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• The set of accepted strings is the language
  recognized by the machine.
• 2 focuses
• Languages
• Machine
• The study of abstract machine begin with
  determined finite automata (DFA)
• DFA is a read-one machine in which the
  instruction to be executed is determined by the
  state of the machine and the input symbol being
  processed.
• Kleenes theorem shows that finite automata
  accept precisely he language generated by he
  regular grammars.
• The pushdown Automata (read-once machine) is
  introduced to accept the context-free languages.

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• 6.0 FINITE AUTOMATA
• Language accepter an effective procedure that
  determines whether an input string is in a
  language.
• Objective to define a class of abstract machine
  whose computation, ( like those a person) ,
  determine acceptability of an input string.
• Common properties Processing the input
  generate the output.
• a) Example to physical machine reading machine
  
• - take coin as input
• - returns food or beverages as output.
• b) A combination lock
• - expects a sequence of numbers k opens the
  lock if the input sequence is correct.
• Input to our abstract machine
• - consists of a string over an alphabet
• Result of the computation indicates whether the
  input string is acceptable.

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• 6.1 A FINITE-STATE MACHINE
• A formal definition of a machine is not concerned
  with the hard ware involved, rather the internal
  operation as the machine process input.
• A machine state represented the status of an
  outgoing computation.
• The internal operation of the vending machine can
  be described by the interaction of the following
  7 states.

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• The insertion of a coin cause the machine to
  alter its state. When 30 cents or more is input,
  the latch is opened.
• ? Such state is called accepting, indicates the
  correctness of the input.
• The design of the machine ? state diagram n
• d d
  d d d
• n n n
  n n n
  n,d,q
• q q q q
  q q
• State-diagram of newspaper vending machine
• The arcs labeled n,d,q representing the input of
  a nickel, dime, or quarter.
• Input to the machine consists of string from n,
  d ,q

30
25
20
15
10
0
5
7

• The string dndn is accepted by the vending
  machine.
• The nndn is not accepted, computation terminates
  state 5.
• 6.2 DETERMINISTIC FINITE AUTOMATA.
• A class of abstract machine whose computation
  can be used to determine the acceptability of
  input strings.
• Definition 6.2
• A DFA is quintuple M ( Q, S, d, qo, F)
• where Q a finite set of states
• S a finite set called the alphabet
• qo.? Q - a distinguished state known as
  the start state.
• F a subset of Q called the final or
  accepting states
• d a total function from Q x S to Q known as
  the transition function

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• An automata can be through of as a machine
  consisting of 5 components
• a single register
• a set of values for the register
• a tape
• a tape reader
• an instruction set
• The states of a DFA represent the internal status
  of the machine, daunted as
• qo, q1, q2,.qn.
• The register of the machine , called finite
  control, contain one of the states as its value.
• qo the beginning value of register
• The input is a finite sequence of elements from
  the alphabet S
• 
  tape
• register
  
• containing the state ,qo

qo
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• M Q qo, q1 d (qo, a) q1
• S a, b d(qo, b) qo
• F q1 d(q1, a) q1
• d(q1, b) qo
• 
  Computation in a
  DFA,
• 
  exhibits the
  acceptance of
• 
  the string aba.

qo
q1
qo
q1
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• Definition 6.2.2
• Let M (Q, S, d, qo, F ) be a DFA .
• The language of M, denoted L(M), is the set of
  strings S accepted by M.
• A DFA can be considered to be a language
  accepter, the language recognized by the machine
  is simply the set of strings accepted by its
  computations.At any point during the computation,
  
• The result depends only on the current state and
  the unprocessed input.
• ? represented by the order pair.
• qi , w , where qi is the current state
• ,w ? S is the unprocessed input.
• Rotation
• qi, aw?m qi ,w indicate configure qi,
  aw is obtained from
• qi, aw by execution of instruction cycle.

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• Definition 6.2.3
• The function ?m on Q x S is defined by qi,
  aw ?m d(qi, a), w
• for a ?S and w ?S , where d is the transition
  function of the DFA M.
• The rotation qi,u ? qi, v indicates
  configuration qi,v can be obtained from qi,u
  by zero or more transition.
• Example 6.2.1
• The DFA M, accepts the set of strings over
  a,b that contain the substring bb.
• ? (m) (a ?b) bb (a ?b)
• m qo, q1, q2
• S a,b
• F q2

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• The transition function d is given transition
  table
• input (alphabet)
• State
• The action of the automation is state qi with
  input a can be determined by the intersection of
  row and column corresponding to qi and a.
• The computation of M with input string abba and
  abab are traced using the function ?

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• qo , abba qo, abab
  
• ? qo, bba ? qo, bab
• ? qo, bba ? q1, qb
• ? q2,a ? qo, b
• ? q2, ? ? q1, ?
• accepts rejects
• The string abba is accepted since the
  computation halts in state q2.

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• Example 6.2.2
• The newspaper reading machine ? DFA
• Q 0,5,10,15,20,25,30
• S n, d, q
• F 0
• n nickel
• d dime
• q quarter
• The language of the reading machine consists of
  all string that represent a sun of 30 cents more

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• Definition 6.2.4
• The extended transition function -d of a DFA
  with transition
• function d is a function from Q x S to Q
  defined by recursion the length of the input
  string.
• i) Basis Length (w) 0 Then w ? and -d
  (qi, ?) qi
• length (w) 1 then w a for some a ? S ,
  and -d (qi,a) d (qi,a)
• ii) Recursion step Let w be a string of
  length n gt 1. Then w ua and
• -d (qi, ua) d(-d (qi, u),a ).
• The computations of a machine in a step qi with
  string w halts in state
• -d (qi, w)
• A string w is accepted if -d (qo, w) ? F.
• ? (M) w -d (qo, w) ? F

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• 6.3 State Diagrams
• A labeled directed graph, where nodes represent
  the states of the machines, and the arcs are
  obtained from the transition function.
• Definition 6.3.1
• The state of a DFA M ( Q, S, S, qo, F ) is a
  labeled graph G defined by the following
  conditions
• The nodes of G are the elements of Q
• The labels on the arcs of G are element of S
• qo is the start node, depicted gt 0.
• F is the set of accepting nodes each accepting
  is depicted ?.
• There is an arc from node qi to qj labeled a if
  d(qi,a) qj
• For every node qi and symbol a ? S, there is
  exactly me arc labeled a leaving qi.

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• Let Pw be a path beginning at qo that spell w
  and let qw be the terminal node of Pw.
• Theorem 6.3.2 prove that there is only one such
  path for every string
• w ? S.
• qw is the state of the DFA upon completion of
  the processing of w.
• Theorem 6.3.2
• Let M ( Q, S, S, qo, F ) be a DFA and let w ?
  S. Then w determines a unique path Pw in the
  state diagram of M and -d (qo, w) qw

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• Example 6.3.1
• The state diagram of the DFA in the example
  6.2.1
• L(m) (a ?b) bb (a ?b)
• a
  a,b
• b
  b
• a
• The status are used to record the number of
  consecutive bs processed.
• The state q2 is entered when a substring bb is
  encountered.
• Once of machine enters q2 the reminder of the
  input is processed, leaving the state uncharged.

q2
qo
q1
q2
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• input ababb
• Computation Path
• qo, ababb qo
• ? qo, babb qo
• ? q1, abb q1
• ? qo, bb qo
• ? q1, b q1
• ? q2, ? q2
• The string ababb is accepted since the halting
  state of the computation, which is also the
  terminal state of the path that spells ababb, is
  the accepting state ,q2.

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• Example 6.3.2
• The DFA
• b
  a,b
• a
  a
• b
• accepts ( b ?ab) (a ??), the set of string over
  a,b that do not contain the substring aa.
• A string in this set may contain a prefix of any
  any number of bs. All as must be followed by at
  least one b on terminating the string.
• The regular expression
• b (ab) ? b (a b)a
• generates the desired set by partitioning it
  into 2 disjoint subjects

q2
qo
q1
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• strings that end in b
• strings that end in a
• b(a b) ? b (ab) a
• b(a b) (??a)
• b(a bb) (??a)
• (b?a b) (??a)

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• Example 6.3.3
• Strings are a,b that contain the substring bb
  dc noy contain the substring aa are accepted by
  DFA depicted below.
• 
  a
• 
  
• a
  b
• 
  a
• a a b
  b
• b b
• 
  a, b
• This language is the union of the languages of
  
• Strings that contain the substring bb
• Strings that do not contain substring aa.
• ? (a ?b) bb (a ?b) ? (b?a b) (a ??)

q3
q2
a
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• Example 6.3.4
• The DFA M accepted the language consisting of
  all strings over a,b that contain an even
  number of as and an odd number of bs.
• M
• 
  b
• 
  b
• a a
  a a
• 
  
• 
  b
• 
  b

ea, eb
ea, ob
Oa, eb
Oa, Ob
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• Example 6.3.5
• A DFA M accepted all strings over a, b
  that do not contain an even number of as and an
  odd number of bs.
• ? L (M) a, b - L(M)
• M
• 
  b
• 
  b
• a a
  b b
• 
  b
• 
  b
• 
  
• 
  
  

ea, Ob
ea, eb
Oa, Ob
Oa, eb
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• incomplete determinism
• each configuration has at most one action
  specified.
• defined by a partial function from Q X S to Q.
• Example 6.3.7
• Incompletely DFA that accepts (a b) c. A
  computation terminate unsuccessfully as soon as
  the input varies from the desired pattern.
• a
• c
• The computation with input abcc is rejected
  since the machine is unable process the final c.

qo
q1
b
q2
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• An incompletely DFA can be transformed into an
  equivalent DFA. The transformation requires that
  addition of a non accepting error state.
• This state is entered when ever the incompletely
  specified machines enters a configuration for
  which no action in indicated.
• Example 6.3.8
• DFA
• a, b, c
• qe Error state that ensures the processing of
  the entire string.

a
qo
q1
b
c
a
c
b
qe
q2
a,b,c
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• Example 6.3.9
• DFA accepts language ai bi i n, for a
  fixed integer n.
• The state ak count the number of as, and then
  the bks ensure an equal number of bs.
• a a
  a a
• b
  b b
  b
• b
  b b
• This technique cannot be extended to accept
• ai bi i 0, ? an infinite number of
  states could be needed.

ao
ao
ao
an-1
an
b1
bo
bn-2
bn-1
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• Theorem 6.3.3
• Let M (Q, S, d, qo, F ) be a DFA
• Then M (Q, S, d, qo, Q - F ) is a DFA with L
  (M) S - L (M)
• Example 6.3.6
• Let S (0,1,2,3) .
• A string in S is a sequence of integer from S.
  A DFA determines whether the sum of elements of a
  string is divisible by 4. The string 12302 and
  0130 should be accepted and 0111 rejected by M.
  The states represent the value of the sum of the
  processed input modules 4.

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• 0
  0
• 1
• 1
  1
• 2 2
  2 2
• 1
• 0
  0
• The definition of output can be executed to have
  a value associated with each states, i.e i mod.
• ? moore machine.

1 mod 4
0 mod 4
3
3
3 mod 4
2 mod 4
3
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• NON DETERMINISTIC FINITE AUTOMATON
• There may be several instruction that can be
  executed from a given machine configuration.
• d (qn, a) qi
• d (qn, a) qi, qj, qk d (qn, a) ø
• Definition 6.4.1
• A non deterministic finite automaton is a
  quintuple M (Q, S, d, qo, F )
• where Q is a finite state, S a set called the
  alphabet, qo ? Q a distinguished state known as
  the start state, F a subset of Q called the final
  or accepting states, and d a total function from
  QxS to P(Q) known as the
• transition function.

qi
a
qn
qi
a
qi
a
qn
qn
a
qi
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• The relationship between DFAs and NFAs
• Every DFA is non deterministic
• DFA, the transition function of a DFA specifies
  exactly one state that may be entered from a
  given state and input symbol.
• In NFA, allows zero, one, or more states,
• A string input to an NFA, many generate several
  distinct computation.
• Example 6.4.1
• NFA, M
• M Q qo, q1, q2
• S a, b
• F q2

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qo, ababb qo, ababb qo, ababb ?
qo, babb ? qo, bqbb ? qo, babb? qo,
abb ? q1, abb ? qo, abb ? qo, bb
halt ? qo, bb ? qo, b ? q1,b ?
q1, ? ? q2, ?
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• A string is in the L (NFA) is there is one
  computation that accepts it the existence of
  other computations that do not accept the string
  is irrelevant.
• Definition 6.4.2
• The language of an NFA M, denoted L (M)lt is the
  set of string accepted by the M.
• i.e L(M) w there is a computation .
• qo, w ? qi, ? with qi ? F
• Definition 6.4.3
• The state diagram of an NFA M (Q, S, d, qo, F )
  is a labeled directed graph defined by the
  following conditions
• The nodes of G are elements of Q.
• The labeled on the arcs of G are elements of S
• F is the set of accepting nodes.
• There is an arc from node qi to qj labeled a if
  qj ? d ( qi, a)

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• Example 6.4.2
• The state diagram for the NFA M from example
  6.4.1 is
• a, b
• b
  b
• Language of ( a?b) bb
• Example 6.4.3
• State diagrams M1 and M2 ? ( a?b) bb (a?b)
• a
• b
  b a,b a b
• M1
• a

qo
q1
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• a,b
• b b
  a,b
• M2
• M1 (DFA) q2 is entered when the 1st substring bb
  is encountered
• M2 (NFA), q2 can be entered upon processing any
  occurrence of bb
• Example 6.4.4
• An NFA that accepts string a,b with the
  substring aa or bb by combining a a machine that
  accepts string with bb with a similar machine
  that accept strings with aa.
• 
  a
• a,b a
• b
  a,b
• 
  b
• ? L (M1) ?L (M2)

q1
qo
q3
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• 6.5 LAMBDA TRANSITIONS
• The definition of NFA is relaxed to allow state
  transitions without requiring input to be
  processed.
• ? NFA ?
• Definition 6.5.1
• A NFA with ? transition is quintuple M (Q, S,
  d, qo, F ), where Q, d, qo, and F are the same
  as in a NFA. The transition function is a
  function from Q x (S?? ) to P(Q).
• The input is accepted if there is a computation
  that process the entire string and halts in an
  accepting state.
• Let
• M1 a, b
  a,b
• b
  b

( a?b) bb (a?b)
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• M2
• b
• a
• b
• that accept
• Example 6.5.1
• The language of the NFA ? , L (M) is L (M1)
  ?L (M2)
• 
  a, b
  a,b
• 
  
• b b
• 
  a
• 
  b
• compare the simplicity in 6.3.3

( a?b) bb (a ?d)
q1,0
q1,1
qo
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• Example 6.5.2
• An NFA - ? that accepts L(M1). L(M2)
• ? concatenation
• An input strings is accepted only if it consists
  of a string from L(M1) concatenated with one from
  L(M2).
• M2 is entered when ever a prefix of the input
  string is accepted by M.

a, b
b
a, b
a
q1,1
q1,2
b
b
?
b
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• Example 6.5.3
• NFA- ? that accepts all strings of even length.
• ? allow to repeat the sequence qo ,q1, q2
  i.e q2 ? qo

q1
a, b
a, b
?
a, b
a, b
q1
?