Introduction to Computational Theory

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Introductionto Computational Theory

• Chapter 6

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Automatic Door/FA
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Nondeterministic
When the ultimate path through a machine is not
determined by input alone the machine is
nondeterministic.
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Preamble to Chapter 6

• NFAs
• Non-deterministic finite automata
• Vs. DFAs (deterministic the book calls FAs)
• We allow multiple transitions per letter per
  state
• Including lambda-transitions
• Move on a whim (w/o consuming input)
• Accept if a path exists to a final state
• Transition relation

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Examples
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Why Non-determinism?

• More expressive model
• Easier to find machines for a language
• E.g., unions of two languages/machines

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Examples

• (ab aba)
• Language over a b where last symbol is repeated

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NFA DFA Equivalence

• There is an algorithm to convert a NFA to a DFA
• Just track all the possibilities
• Collapse lambda moves
• States are a subset of 2Q
• Rabin-Scott Algorithm
• Example acb

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Lambda Transitions

• Handy for combining machines
• E.g., union of two languages create a new start
  state with lambda moves to the start states of
  the two machines

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Examples
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Examples
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Examples
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Examples
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Transition Graphs
Abandon the requirement that the edges eat just
one letter at a time.
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Crashes (Formerly, Hell State or Jail)
When an input string that has not been completely
read reaches a state (final or otherwise) that it
cannot leave because there is no outgoing edge
that it may follow, we say that the input (or the
machine) crashes at that state.
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Rejected Input

• Trace a path ending in a non-final state
• Crash while being processed

baa
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Acceptance
A string is accepted by a TG if there is some way
it could be processed as to arrive at a final
state. There may also be ways in which this
string does not get to a final state, but we
ignore all failures.
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Transition Graph
A collection of three things 1. A finite set of
states, at least one of which is designated as
the start state (-) and some (maybe none) of
which are designated as final states () 2. An
alphabet of possible input letters from which
input strings are formed. 3. A finite set of
transitions (edge labels) that show how to go
from some states to some others, based on reading
specified substrings of input letters (possibly
even the null string )
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Successful Path
A successful path through a transition graph is a
series of edges forming a path beginning at some
start state (there may be several) and ending at
a final state.
abbab
abbaa
abb
a
abbababba
1-
4
aa
b
A Lambda transition occurs when you get a free
transition that was not initiated by user or
system action/interaction. Move on a whim
(w/o consuming input).
Slide modified by Seals
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Equivalent Language Acceptors
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Examples
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Examples
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Examples
(a b)b
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Examples
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Examples(EVEN-EVEN cf. p. 69)
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Example(p. 84)
b
a,b
a
b
-

bbb
a
a
a
ab
b
bbb
bb
bb
a
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Examples(p. 85)
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Example(Problem 17, p. 91)

• L a abb bbaab bbbaa
• 1) given a FA that accepts L, construct a TG that
  accepts transpose(L)
• Invert start/final states reverse arrows
• 2) given a TG that accepts L, construct a TG that
  accepts transpose(L)
• Same as 1, but reverse transition strings

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Generalized Transition Graph (GTG)
A collection of three things 1. A finite set of
states, at least one of which is designated as
the start state (-) and some (maybe none) of
which are designated as final states () 2. An
alphabet of possible input letters from which
input strings are formed. 3. Directed edges
connecting some pairs of states, each labeled
with a regular expression.
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Examples