LING124 Finite state transducers, weighted automata - PowerPoint PPT Presentation

Loading...

PPT – LING124 Finite state transducers, weighted automata PowerPoint presentation | free to view - id: a2ea2-MTkxM



Loading


The Adobe Flash plugin is needed to view this content

Get the plugin now

View by Category
About This Presentation
Title:

LING124 Finite state transducers, weighted automata

Description:

Definition of weighted automata. Example applications of FSM. References ... S, so , F, S, d : the same as in the definition of FSA ... – PowerPoint PPT presentation

Number of Views:357
Avg rating:3.0/5.0
Slides: 20
Provided by: hahn7
Category:

less

Write a Comment
User Comments (0)
Transcript and Presenter's Notes

Title: LING124 Finite state transducers, weighted automata


1
LING124 Finite state transducers, weighted
automata
  • September 30, 2008

2
Class outline
  • Finite state transducers
  • Definition
  • Operations on FST
  • Deterministic vs. nondeterministic FST
  • Weighted finite state machines
  • Preliminaries
  • Definition of weighted automata
  • Example applications of FSM
  • References

3
Finite state automata (FSA)
  • An FSA is a quintuple ltS, so, F, S, dgt
  • S Set of states
  • so Initial state
  • F Set of final states
  • S Alphabet (a set of symbols)
  • d Transition relation
  • dS x S -gt S
  • dS x S U e -gt S, where e is a null symbol, for
    (nondeterministic) FSA with epsilon transition

4
Finite state transducer (FST)
  • An FST is a septuple ltS, so, F, S, ?, d, sgt
  • S Set of states
  • so Initial state
  • F Set of final states
  • S Input alphabet (a set of input symbols)
  • ? Output alphabet (a set of output symbols)
  • d Transition relation (dSxS -gtS)
  • s Output relation (sSxS -gt ?)

5
FST as a state diagram
FSA
FST
6
Functions of FST
  • Recognizer
  • Accepts input-output string pair if there is a
    path from the initial state to a final state
  • Generator
  • Generates a string pair the FST encounters along
    its path from the initial state to a final state
  • Translator
  • Reads input string and prints out the
    corresponding output string
  • Set relator
  • Computes relation between sets

7
Inversion
  • Switches the input and output labels
  • T maps from the input alphabet I to the output
    alphabet O
  • T-1 maps from the output alphabet O to the input
    alphabet I

8
Composition
  • T1 maps from I1 to O1
  • T2 maps from O1 to O2
  • T1?T2 maps from I1 to O2

9
Projection
  • Extract one side of the relation (e.g. input) and
    produce an FSA that accepts a string if and only
    if the original FST translates the string (e.g.
    input string) to another string (e.g. output
    string)
  • First projection (p1) Extracts the input (left)
    side
  • Second projection (p2) Extracts the output
    (right) side

10
First projection (p1)
11
Second projection (p2)
12
Deterministic vs. nondeterministic
  • Nondeterministic FST
  • At a given state, each input may be translated to
    one or more output symbols or symbol sequences
  • Deterministic FST
  • At a given state, each input is translated to a
    unique output symbol or output symbol sequence
    (sSxS -gt ? is a function)
  • A state-input pair is mapped to a unique state
    (dSxS -gtS is a function)
  • Unlike FSA, there is not always a deterministic
    FST equivalent to a nondeterministic FST

13
Preliminaries (1)
  • Operand
  • Input to a mathematical operator
  • e.g. 3 6
  • Binary operation
  • Calculation involving two operands
  • e.g. Addition

14
Preliminaries (2)
  • Monoid
  • A pair (M,)
  • M is a set
  • is a binary operation on M, satisfying
  • Closure If a, b are in M, ab is also in M
  • Associativity For a, b, c in M, (ab)c
    a(bc)
  • Identity For a in M, there exists an element e
    in M such that ae ea a
  • e is called the identity element

15
Preliminaries (3)
  • Semiring
  • A set R together with two binary operators and
  • (R,) is a commutative monoid with identity
    element 0
  • a00aa
  • ab ba
  • (R, ) is a monoid with identity element 1
  • a11aa
  • Multiplication () distributes over addition
  • a(bc)abac
  • (ab)cacbc
  • a00a0

16
Weighted automata (1)
  • AltS, so, F, S, d, ?, s, ?gt
  • S, so , F, S, d the same as in the definition
    of FSA
  • ? Initial output function that assigns weight
    for entering FSA (? so -gt K)
  • s Output function that assigns a weight to
    paths between two states (sSxSxS -gt K)
  • ? Final output function that assigns weight for
    leaving FSA (? F -gt K)

17
Weighted automata (2)
  • Function f S-gt (K,, ) associated with A
  • Assuming the operator means addition
  • For example, if K were a set of real numbers
    between 0 and 1, and and meant addition
    (plus) and multiplication (times), f would be
    interpreted as finding the probability of a
    string w assigned by the weighted automaton A

18
Example WFST
Figure from Mohri et al. (2002)
19
Example applications
  • FST implementation of phonological rules
  • JM 11.1 and 11.2
  • Kaplan and Kay (1994)
  • Koskenniemi (1983)
  • Mohri and Sproat (1996)
  • Weighted finite state transducers in speech
    recognition
  • Mohri et al. (2002)
  • And a LOT more
About PowerShow.com