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LING124 Finite state transducers, weighted

automata

- September 30, 2008

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

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

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 ?)

FST as a state diagram

FSA

FST

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

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

Composition

- T1 maps from I1 to O1
- T2 maps from O1 to O2
- T1?T2 maps from I1 to O2

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

First projection (p1)

Second projection (p2)

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

Preliminaries (1)

- Operand
- Input to a mathematical operator
- e.g. 3 6
- Binary operation
- Calculation involving two operands
- e.g. Addition

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

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

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)

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

Example WFST

Figure from Mohri et al. (2002)

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