Minimising FSAs Regular Expressions

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Minimising FSAs /Regular Expressions
size of FSAs minimal FSAs Definition of regular
expressions
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How small can a FSA be?

• two strings x and y are distinguishable with
  respect to (w.r.t.) a machine A if there is a
  string z such that (s.t.) exactly one of xz and
  yz are in L(A)
• Let L be a language, L? T
• if there are n elements of T s.t. any two are
  distinguishable w.r.t. a machine A, then any
  machine that recognises L must have at least n
  states
• Thus for any language, there is a minimum size
  for FSAs

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Unique minimal FSAs
Theorem minimal FSAs For any language defined by
a FSA, there exists a unique FSA of minimum size,
and an algorithm to produce it

• Informal Algorithm
• Input A (Q,I,F,T,E)
• 1. reverse A to get A'
• 2. run remove edge choices on A' to get
  (Q',I',F',T,E')
• 3. merge all states in A that always appear
  together in Q'

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Example minimisation
a
2
b
4
a
b
1
a
b
a
b
b
a
5
3
5
Example Minimisation
a
a
2
b
4
2
b
4
a
a
b
b
1
b
1
a
a
b
a
a
b
b
b
b
a
a
5
3
5
3
reversed
the set Q from remove edge choices is 4,5,
2,4,5, 2,3,4,5, 1,2,3,4,5 4 5 always
appear together
2
a,b
a
1
a,b
4,5
a
b
b
3
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Specifying Languages

• Our aim is to be able to specify languages for
  use in the computer
• the FSA sketch is easy for us to understand, but
  difficult to input to a computer
• the FSA formal description is unpleasant, and
  difficult to write
• Can we come up with a simpler method of
  specifying languages, in a style that we can
  write down in text?

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Regular Expressions

• A regular expression over an alphabet T defines a
  language
• ? denotes ?, ? denotes , and t denotes t
  for t ? T
• if r and s a re regular expressions denoting R
  and S, then
• r s denotes R ?S
• rs denotes RS
• r denotes R
• nothing else is a regular expression over T
• interpret st as s(t)
• interpret st as s (t)
• interpret r st as r (st)

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Regular Languages

• A language over T is regular if there is a
  regular expression denoting it.
• If A and B are regular languages, so are A?B, AB
  and A
• If A and B are regular languages, so are A?B and
  A'
• Any finite language is regular.

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Example regular expressions

• All binary numbers with no leading 0s
• All strings over a,b with a substring of aa or
  bb
• All strings over 0,1 not containing two
  adjacent 0s
• All strings over 0,1 with exactly one 00
  substring

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Next lecture
Comparing Regular Expressions and Finite State
Automata...