Specifying Infinite Languages

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Specifying Infinite Languages
the problem of infinite languages finite state
machines
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Ordering Languages

• Let T be an alphabet with a given ordering on its
  symbols. Say T a, b, c, d, .... Strings over
  T can be ordered in two ways
• Dictionary Order
• All strings beginning a are ordered before all
  strings beginning b, and b before c, etc. Within
  groups of strings beginning with the same symbol,
  strings are ordered by their second symbol, and
  so on. ? is always the first string.
• Lexical Order
• Strings are ordered by their length, with the
  shortest first. Within groups of strings of the
  same length, strings are ordered in dictionary
  order. ? is always the first string.

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Specifying Infinite Languages
L1 xn n 1, 2, 3, ... What elements
are in L1?
L2 xn n 1, 4, 9, 16, ... What elements
are in L2?
L3 xn n 1,4, 9, 48, ... What elements are
in L3?
Can we devise a clear and precise method for
defining infinite languages?
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Recognising Languages

• We will tackle the problem of defining languages
  by considering how we would recognise them
• We will define an abstract machine that, given a
  string, produces the answer "yes" or "no".
• The abstract machine will be the specification of
  the language.

Problem Can we recognise infinite
languages? i.e. given a language description and
a string, is there an algorithm that will answer
yes or no (correctly)?
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Finite State Automata

• A finite state automaton is an abstract model of
  a simple machine (or computer).
• The machine can be in a finite number of states.
  It receives symbols as input, and the result of
  receiving a particular input in a particular
  state moves the machine to a specified new state.
  Certain states are finishing states, and if the
  machine is in one of those states when the input
  ends, it has ended successfully (or has accepted
  the input).

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Example FSA
a
2
3
a,b
a
1
b
a
b
4
b
A1
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FSA Formal Definition

• A Finite State Automaton is a 5-tuple (Q, I, F,
  T, E)
• Q is a finite set (called states)
• I is a subset of Q (called initial states)
• F is a subset of Q (called final states)
• T is an alphabet
• E is a subset of Q ?(T ?) ? Q (called edges).
• Essentially, an FSA is a labelled, directed graph
  - that is, a set of nodes with directed arcs
  (arrows) between nodes, and where each arc may
  have a label from an alphabet.

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Example formal definition
Example formal definition of A1 Q 1, 2,
3, 4 I 1 F 4 T a, b E (1,a,2),
(1,b,4), (2,a,3), (2,b,4), (3,a,3),
(3,b,3), (4,a,2), (4,b,4)
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Definitions

• if (x,a,y) is an edge, x is its start state and y
  is its end state
• a path is a sequence of edges such that the end
  state of one is the start state of the next
• a path is successful if the start state of the
  first edge is an initial state, and the end state
  of the last is a final state
• the label of a path is the sequence of edge
  labels
• a string is accepted by a FSA if it is the label
  of a successful path

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Definition of a finite state language

• Let A be a FSA. The language accepted by A is the
  set of strings accepted by A, denoted L(A)
• The language accepted by A1 is the set of strings
  of a's and b's which end in a b, and in which no
  two a's are adjacent.

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Non-determinism

• We wanted to use FSAs as a mechanism to recognise
  languages, simply following the edges as we input
  symbols from a string.
• The way we have defined FSAs gives a problem -
  they are non-deterministic. That is, we may have
  a choice at certain times.
• There could be a number of edges with the same
  label leaving a state some edges could be
  labelled with ? there could be more than one
  start state.

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Deterministic FSA

• A FSA is deterministic (DFSA) if
• there is only one initial state
• there are no ?-labelled edges
• for any pair of state and symbol (q,t), there is
  at most one edge (q,t, p)
• A FSA is non-deterministic (NDFSA) if any one of
  the above does not hold

A DFSA is a specification of an algorithm for
determining whether or not a string is a member
of a given language
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Transition Functions

• The transition function of a FSA A is the
  function
• ? (x, t) ? y ? edge (x, t, y) in A
• The transition matrix of a FSA A is a matrix with
  one row for each state, and one column for each
  symbol in the alphabet, such that the entry in
  row q, column t is ?(q,t)

Example a b 1 2 4 2 3 4 3 3 3
4 2 4
initial state
final state
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Recognition Problem

• Problem Given a DFSA, A (Q,I,F,T,E), and a
  string w, determine whether w ? L(A).
• Note denote the current state by q, and the
  current input symbol t. Since A is deterministic,
  ?(q,t) will always be a singleton set or will be
  undefined. If it is undefined, denote it by ? (?
  Q).

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Recognition Algorithm

• 1. add symbol to end of w
• 2. q initial state
• 3. t first symbol of w
• 4. while (t ? and q ? ?)
• 5. begin
• 6. q ?(q,t)
• 7. t next symbol of w
• 8. end
• 9. return ((t ) (q ? F))

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Next lecture
what is the difference between DFSA and NDFSA?