Introduction to Finite Automata

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Introduction to Finite Automata

• Adapted from the slides of Stanford CS154

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Informal Explanation

• Finite automata
• finite collections of states with transition
  rules that take you from one state to another.
• Original application
• sequential switching circuits, where the state
  was the settings of internal bits.
• Today, several kinds of software can be modeled
  by FA.

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Representing FA

• Simplest representation is often a graph.
• Nodes states.
• Arcs indicate state transitions.
• Labels on arcs tell what causes the transition.

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Example Recognizing Strings Ending in ing
nothing
Start
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Automata to Code

• In C/C, make a piece of code for each state.
  This code
• Reads the next input.
• Decides on the next state.
• Jumps to the beginning of the code for that state.

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Example Automata to Code

• 2 / i seen /
• c getNextInput()
• if (c n) goto 3
• else if (c i) goto 2
• else goto 1
• 3 / in seen /
• . . .

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Deterministic Finite Automata

• A formalism for defining languages, consisting
  of
• A finite set of states (Q, typically).
• An input alphabet (S, typically).
• A transition function (d, typically).
• A start state (q0, in Q, typically).
• A set of final states (F ? Q, typically).
• Final and accepting are synonyms.

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The Transition Function

• Takes two arguments
• a state and an input symbol.
• d(q, a) the state that the DFA goes to when it
  is in state q and input a is received.

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Graph Representation of DFAs

• Nodes states.
• Arcs represent transition function.
• Arc from state p to state q labeled by all those
  input symbols that have transitions from p to q.
• Arrow labeled Start to the start state.
• Final states indicated by double circles.

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Example Graph of a DFA
Accepts all strings without two consecutive 1s.
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Alternative Representation Transition Table
0
1
A A B B A C C C C
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Extended Transition Function

• We describe the effect of a string of inputs on a
  DFA
• by extending d to a state and a string.
• Induction on length of string.
• Basis d(q, e) q
• Induction d(q,wa) d(d(q,w),a)
• w is a string a is an input symbol.

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Extended d Intuition

• Convention
• w, x, y, x are strings.
• a, b, c, are single symbols.
• Extended d is computed for state q and inputs
  a1a2an by following a path in the transition
  graph, starting at q and selecting the arcs with
  labels a1, a2,,an in turn.

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Example Extended Delta
d(B,011) d(d(B,01),1) d(d(d(B,0),1),1)
d(d(A,1),1) d(B,1) C
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Language of a DFA

• Automata of all kinds define languages.
• If A is an automaton, L(A) is its language.
• For a DFA A, L(A) is the set of strings labeling
  paths from the start state to a final state.
• Formally L(A) the set of strings w such that
  d(q0, w) is in F.

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Example String in a Language
String 101 is in the language of the DFA
below. Start at A.
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Example String in a Language
String 101 is in the language of the DFA below.
Follow arc labeled 1.
0
0,1
1
1
A
C
B
Start
0
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Example String in a Language
String 101 is in the language of the DFA below.
Then arc labeled 0 from current state B.
0
0,1
1
1
A
C
B
Start
0
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Example String in a Language
String 101 is in the language of the DFA below.
Finally arc labeled 1 from current state A.
Result is an accepting state, so 101 is in the
language.
0
0,1
1
1
A
C
B
Start
0
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Example Concluded

• The language of our example DFA is
• w w is in 0,1 and w does not have
• two consecutive 1s

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Regular Languages

• A language L is regular if it is the language
  accepted by some DFA.
• Note the DFA must accept only the strings in L,
  no others.
• Some languages are not regular.
• Intuitively, regular languages cannot count to
  arbitrarily high integers.

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Example A Nonregular Language

• L1 0n1n n 1
• Note ai is conventional for i as.
• Thus, 04 0000, e.g.
• Read The set of strings consisting of n 0s
  followed by n 1s, such that n is at least 1.
• Thus, L1 01, 0011, 000111,

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Another Example

• L2 w w in (, ) and w is balanced
• Note alphabet consists of the parenthesis
  symbols ( and ).
• Balanced parens are those that can appear in an
  arithmetic expression.
• E.g. (), ()(), (()), (()()),

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But Many Languages are Regular

• Regular Languages can be described in many ways,
  e.g., regular expressions.
• They appear in many contexts and have many useful
  properties.
• Example the strings that represent floating
  point numbers in your favorite language is a
  regular language.

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Nondeterminism

• A nondeterministic finite automaton has the
  ability to be in several states at once.
• Transitions from a state on an input symbol can
  be to any set of states.

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Nondeterminism (2)

• Start in one start state.
• Accept if any sequence of choices leads to a
  final state.
• Intuitively the NFA always guesses right.

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Formal NFA

• A finite set of states, typically Q.
• An input alphabet, typically S.
• A transition function, typically d.
• A start state in Q, typically q0.
• A set of final states F ? Q.

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Transition Function of an NFA

• d(q, a) is a set of states.
• Extend to strings as follows
• Basis d(q, e) q
• Induction d(q, wa) the union over all states p
  in d(q, w) of d(p, a)

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Language of an NFA

• A string w is accepted by an NFA if d(q0, w)
  contains at least one final state.
• The language of the NFA is the set of strings it
  accepts.

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Equivalence of DFAs, NFAs

• A DFA can be turned into an NFA that accepts the
  same language.
• If dD(q, a) p, let the NFA have dN(q, a)
  p.
• Then the NFA is always in a set containing
  exactly one state the state the DFA is in after
  reading the same input.

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Equivalence (2)

• Surprisingly, for any NFA there is a DFA that
  accepts the same language.
• Proof is the subset construction.
• The number of states of the DFA can be
  exponential in the number of states of the NFA.
• Thus, NFAs accept exactly the regular languages.

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Subset Construction

• Given an NFA with states Q, inputs S, transition
  function dN, state state q0, and final states F,
  construct equivalent DFA with
• States 2Q (Set of subsets of Q).
• Inputs S.
• Start state q0.
• Final states all those with a member of F.

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Subset Construction (2)

• The transition function dD is defined by
• dD(q1,,qk, a) is the union over all i 1,,k
  of dN(qi, a).

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Proof of Equivalence Subset Construction

• Show by induction on w that
• dN(q0, w) dD(q0, w)
• Basis w e dN(q0, e) dD(q0, e) q0.

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Induction

• Assume IH (Induction Hypothesis) holds for
  strings shorter than w.
• Let w xa IH holds for x.
• Let dN(q0, x) dD(q0, x) S.
• Let T the union over all states p in S of dN(p,
  a).
• Then dN(q0, w) dD(q0, w) T.
• For NFA the extension of dN.
• For DFA definition of dD plus extension of dD.
• That is, dD(S, a) T then extend dD to w xa.

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NFAs With e-Transitions

• We can allow state-to-state transitions on e
  input.
• These transitions are done spontaneously, without
  looking at the input string.
• A convenience at times, but still only regular
  languages are accepted.

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Example e-NFA
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Subset construction in the Textbook
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Subset construction in the Textbook (cont.)
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