Finite-State Machines with No Output Longin Jan Latecki Temple University

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Finite-State Machines with No Output Longin Jan
Latecki Temple University

• Based on Slides by Elsa L Gunter, NJIT,
• and by Costas Busch

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Kleene closure

• A and B are subsets of V, where V is a
  vocabularyThe concatenation of A and B
  isABxy x string in A and y string in B
• Example A0, 11 and B1, 10,
  110AB01,010,0110,111,1110,11110
• What is BA?
• A0?An1AnA for n0,1,2,

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Let A be any subset of V. Kleene closure of A,
denoted by A, is
Examples If C11, C12n n0,1,2, If
B0,1, BV.
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Regular Expressions

• Regular expressions
• describe regular languages
• Example
• describes the language

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Recursive Definition
Primitive regular expressions
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Examples
A regular expression
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Languages of Regular Expressions

• language of regular expression
• Example

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Definition

• For primitive regular expressions

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Definition (continued)

• For regular expressions and

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Example

• Regular expression

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Example

• Regular expression

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Example

• Regular expression

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Example

• Regular expression

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Example

• Regular expression

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Equivalent Regular Expressions

• Definition
• Regular expressions and
• are equivalent if

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Example
all strings without two consecutive 0

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Example Lexing

• Regular expressions good for describing lexemes
  (words) in a programming language
• Identifier (a ? b ? ? z ? A ? B ? ? Z) (a ?
  b ? ? z ? A ? B ? ? Z ? 0 ? 1 ? ? 9 ? _ ?
  )
• Digit (0 ? 1 ? ? 9)

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Implementing Regular Expressions

• Regular expressions, regular grammars reasonable
  way to generates strings in language
• Not so good for recognizing when a string is in
  language
• Regular expressions which option to choose, how
  many repetitions to make
• Answer finite state automata

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Three Equivalent Representations
Regular expressions
Each can describe the others
Regular languages
Finite automata
Kleenes Theorem For every regular expression,
there is a deterministic finite-state automaton
that defines the same language, and vice versa.
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Regular Expression Regular Grammar
a S ? ? aS
(ab) S ? ? aS bS
a b S ? ? A B A ? a aA B ? b bB
S ? ? A B A ? a aA B ? b bB
S ? ? A B A ? a aA B ? b bB
ab S ? b aS
ba S ? bA A ? ? aA
S ? bA A ? ? aA
(ab) S ? ? abS
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EXAMPLE 1

• Consider the language ambn m, n ? N, which
  is represented by the regular expression ab.
• A regular grammar for this language can be
  written as follows
•  
• S? ? aS B
• B ? b bB.

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Finite (State) Automata

• A FA is similar to a compiler in that
• A compiler recognizes legal programs in some
  (source) language.
• A finite-state machine recognizes legal strings
  in some language.
• Example Pascal Identifiers
• sequences of one or more letters or digits,
  starting with a letter

letter digit
letter
S
A
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Finite Automaton
Input

String
Output
Accept or Reject
Finite Automaton
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Finite State Automata

• A finite state automation over an alphabet is
  illustrated by a state diagram
• a directed graph
• edges are labeled with elements of alphabet,
• some nodes (or states), marked as final
• one node marked as start state

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Transition Graph

initial state
accepting state
transition
state
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Initial Configuration
Input String

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Reading the Input

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29

30

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Input finished
accept
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Rejection

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34

35

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Input finished
reject
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Another Rejection

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reject
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Another Example
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Input finished
accept
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Rejection Example
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Input finished
reject
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Finite State Automata

• A finite state automation M(S,I,f,s0,F) consists
  of
• a finite set S of states,
• a finite input alphabet I,
• a state transition function f S x I ? S,
• an initial state s0,
• a subset F of S that represent the final states.

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Finite Automata

• Transition
• s1 ?a s2
• Is read In state s1 on input a go to state
  s2
• If end of input
• If in accepting state gt accept
• Otherwise gt reject
• If no transition possible (got stuck) gt reject
• FSA Finite State Automata

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Example FSA
Construct the state diagram for M(S,I,f,s0,F),
where Ss0, s1, s2, s3, I0,1, Fs0, s3and
the transition function
state Input 0 Input 1
s0 s0 s1
s1 s0 s2
s2 s0 s0
s3 s2 s1
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Language accepted by FSA

• The language accepted by a FSA is the set of
  strings accepted by the FSA.
• in the language of the FSM shown below x, tmp2,
  XyZzy, position27.
• not in the language of the FSM shown below
• 123, a?, 13apples.

letter digit
letter
S
A
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Example

• FSA that accepts three letter English words that
  begin with p and end with d or t.
• Here we use the convenient notation of making the
  state name match the input that has to be on the
  edge leading to that state.

a
t
p
i
o
d
u
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Languages Accepted by FAs

• FA
• Definition
• The language contains
• all input strings accepted by
• strings that bring
• to an accepting state

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Example

accept
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Example

accept
accept
accept
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Example

trap state
accept
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Formal Definition

• Finite Automaton (FA)

set of states
input alphabet
transition function
initial state
set of accepting states
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Input Alphabet

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Set of States

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Initial State

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Set of Accepting States

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Transition Function

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Transition Function

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Extended Transition Function

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Observation if there is a walk from to
with label then
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Example There is a walk from to
with label
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Recursive Definition
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Language Accepted by FAs

• For a FA
• Language accepted by

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Observation

• Language rejected by

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Example
all strings with prefix

accept
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Example
all strings without substring

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Example

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Deterministic FSAs

• If FSA has for every state exactly one edge for
  each letter in alphabet then FSA is deterministic
• In general FSA in non-deterministic.
• Deterministic FSA special kind of
  non-deterministic FSA

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Example FSA

• Regular expression (0 ? 1) 1
• Deterministic FSA

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Example DFSA

• Regular expression (0 ? 1) 1
• Accepts string 0 1 1 0 1

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Example DFSA

• Regular expression (0 ? 1) 1
• Accepts string 0 1 1 0 1

1
0
1
0
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Example DFSA

• Regular expression (0 ? 1) 1
• Accepts string 0 1 1 0 1

1
0
1
0
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Example DFSA

• Regular expression (0 ? 1) 1
• Accepts string 0 1 1 0 1

1
0
1
0
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Example DFSA

• Regular expression (0 ? 1) 1
• Accepts string 0 1 1 0 1

1
0
1
0
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Example DFSA

• Regular expression (0 ? 1) 1
• Accepts string 0 1 1 0 1

1
0
1
0
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Example DFSA

• Regular expression (0 ? 1) 1
• Accepts string 0 1 1 0 1

1
0
1
0
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Example NFSA

• Regular expression (0 ? 1) 1
• Non-deterministic FSA

0
1
1
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Example NFSA

• Regular expression (0 ? 1) 1
• Accepts string 0 1 1 0 1

• Regular expression (0 1) 1
• Accepts string 0 1 1 0 1

0
1
1
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Example NFSA

• Regular expression (0 ? 1) 1
• Accepts string 0 1 1 0 1

• Regular expression (0 1) 1
• Accepts string 0 1 1 0 1

0
1
1
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Example NFSA

• Regular expression (0 ? 1) 1
• Accepts string 0 1 1 0 1

• Regular expression (0 1) 1
• Accepts string 0 1 1 0 1

0
1
1
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Example NFSA

• Regular expression (0 ? 1) 1
• Accepts string 0 1 1 0 1
• Guess

• Regular expression (0 1) 1
• Accepts string 0 1 1 0 1

0
1
1
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Example NFSA

• Regular expression (0 ? 1) 1
• Accepts string 0 1 1 0 1
• Backtrack

• Regular expression (0 1) 1
• Accepts string 0 1 1 0 1

0
1
1
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Example NFSA

• Regular expression (0 ? 1) 1
• Accepts string 0 1 1 0 1
• Guess again

• Regular expression (0 1) 1
• Accepts string 0 1 1 0 1

0
1
1
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Example NFSA

• Regular expression (0 1) 1
• Accepts string 0 1 1 0 1
• Guess

• Regular expression (0 1) 1
• Accepts string 0 1 1 0 1

0
1
1
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Example NFSA

• Regular expression (0 ? 1) 1
• Accepts string 0 1 1 0 1
• Backtrack

• Regular expression (0 1) 1
• Accepts string 0 1 1 0 1

0
1
1
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Example NFSA

• Regular expression (0 ? 1) 1
• Accepts string 0 1 1 0 1
• Guess again

• Regular expression (0 1) 1
• Accepts string 0 1 1 0 1

0
1
1
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Example NFSA

• Regular expression (0 ? 1) 1
• Accepts string 0 1 1 0 1

• Regular expression (0 1) 1
• Accepts string 0 1 1 0 1

0
1
1
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Example NFSA

• Regular expression (0 ? 1) 1
• Accepts string 0 1 1 0 1
• Guess (Hurray!!)

• Regular expression (0 1) 1
• Accepts string 0 1 1 0 1

0
1
1
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If a language L is recognized by a
nondeterministic FSA, then L is recognized by a
deterministic FSA Example 9, p. 763
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How to Implement an FSA

• A table-driven approach
• table
• one row for each state in the machine, and
• one column for each possible character.
• Tablejk
• which state to go to from state j on character k,
• an empty entry corresponds to the machine getting
  stuck.

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The table-driven program for a Deterministic FSA

• state S // S is the start state
• repeat
• k next character from the input
• if (k EOF) // the end of input
• if state is a final state then accept
• else reject
• state Tstate,k
• if state empty then reject // got stuck