2. Finite state transducer
• A finite state transducer is a finite state
automaton in which the members of Σ (the
symbols labeling the arcs) are pairs, triples, etc.,
rather than simple symbols.
• Traditionally the members of Σ in a transducer
are just pairs, of which the left-hand member is
the `input symbol' and the right hand member is
the `output symbol'.
3. Finite State Transducers
• Special type of finite state machine
• It has two tapes
• An input string and an output string
• Rather than just traversing (and accepting or
rejecting) an input string
• It translates the contents of its input string to its
output string
4. Finite state transducer
• Fst has a finite set Q of internal states
• Read once-only input file that contains a string from an
input alphabet Σ,
• Output mechanism produces a string from an output
alphabet r, in response to a given input,
5. We put PAIRS of symbols (or symbol strings) on
our transition arrows, viewing one as input and
the other as output
a:b at the arc means that in this transition the
transducer reads a from the first tape and writes
b onto the second.
6. Finite state transducer
• Fst as an implementation of a function.
If M is an fst, we will FM denote the function
represented by M, so
FM : D R
Where D is subset of Σ* and R is a subset of r*
7. Finite state transducer
Interpreting as fst as a function implies
that it is deterministic, that is that output is
uniquely deterministic by the input,
Nondeterministic is no significant role in
the study of finite state transducer.
8. Finite state transducer
• One input symbol result in one output symbol
appears to imply that the mapping FM
length-preserving that is
|FM (w)| = |w|
• We can always include the empty string so that
|FM (w)| < |w|
9. Formal definition of FST
• A Deterministic Finite State Transducer (FST) is a
7-tuple (Q, Σ, Γ, δ, ω, q0, F)
1. Q is a finite set called the states
2. Σ is a finite set called the alphabet
3. Γ is a finite set called the output alphabet
4. δ : Q × Σ → Q is the transition function
5. ω : Q × Σ → Γ is the output function
6. q0 ∈ Q is the start state
7. F ⊆ Q is the set of accept states
10. Frequently the transition function and the
transduction function(output function) are
combined into a single transition-transduction
function, which may also be called δ,
Q × Σ → Q × Γ,
mapping a pair of a state and an input symbol
onto a pair of a state and an output symbol
11. • Each string-to-string finite state transducer
relates the input alphabet Σ to the output
alphabet Γ.
Relations R on Σ*×Γ* that can be implemented
as finite state transducers are called rational
relations.
Rational relations that are partial functions,
i.e. that relate every input string from Σ* to at
most one Γ*, are called rational functions.
12. A finite-state transducer is deterministic if
the output is uniquely determined by the input,
13. A finite-state transducer is non-deterministic if
either the transition mapping or the
transduction mapping fails to be a function, i.e.
if there is more than one possible transition or
more than one possible output symbol for a
given pair of a state and an input symbol.
http://www.coli.unisb.de/~kris/nlp-with-prolog/html/node13.html
http://arxiv.org/pdf/1301.5197.pdf
14. Mealy Machines
Output produced by each transition
depends on the internal state prior to the
transition and the input symbol used in the
transition
Output produced during the
transition.
15. Definition:
Mealy Machine
A Mealy machine is defined by the sextuple
M = ( Q , Σ, r, δ, ,qo )
Where
Q is a finite set of internal states,
Σ is the input alphabet,
R is the output alphabet,
δ : Q × Σ Q is the transition function.
: Q × Σ r is the output function.
qo ∈ Q is the initial state of M.
16. • Starting qo at which time All input is available for
processing.
At the time tn the mealy machine is in state qi, the curent
input symbol is ‘a’ and δ(qi ,a ) = qf , (qi ,a ) =b the
machine enter state qf and produce output ‘b’.
a/b
It is assumed the entire process is process is terminated when
the end of the input is reached.
Note that there are no final states associated with a transducer.
qi qf
