WFST: Weighted Finite State Transducer. September 12, 2017 Prof. Marie Meteer

Transcription

1 + WFST: Weighted Finite State Transducer September 12, 217 Prof. Marie Meteer

2 + FSAs: A recurring structure in speech 2 Phonetic HMM Viterbi trellis Language model Pronunciation modeling

3 + The language of sheep: /baa+!/ 3 Final state State Transition n We can say the following things about this machine n It has 5 states n b, a, and! are in its alphabet n q is the start state n q 4 is an accept state n It has 5 transitions

4 + Input tape acceptor b a! e , Reject Accept Brandeis CS Meteer

5 + Non-Determinism Both define /baa+/ Input: b Brandeis CS Meteer a a Stay in q2 or go to q3?

6 + Non-Determinism cont. n Yet another technique n Epsilon transitions n Key point: these transitions do not examine or advance the tape during recognition Brandeis CS Meteer

7 + Equivalence n Non-deterministic machines can be converted to deterministic ones with a fairly simple construction n That means that they have the same power: n non-deterministic machines are not more powerful than deterministic ones in terms of the languages they can accept n Two basic approaches to ND recognition (used in all major implementations of regular expressions) 1. Either take a ND machine and convert it to a D machine and then do recognition with that. 2. Or explicitly manage the process of recognition as a statespace search (leaving the machine as is). Brandeis CS Meteer

8 + Non-Deterministic Recognition: Search n In a ND FSA there exists at least one path through the machine for a string that is in the language defined by the machine. n But not all paths directed through the machine for an accept string lead to an accept state. n No paths through the machine lead to an accept state for a string not in the language. n Non-determinism doesn t get us more formal power and it causes headaches so why bother? n More natural (understandable) solutions Brandeis CS Meteer

9 + Compositional Machines n Formal languages are just sets of strings 9 n Therefore, we can talk about various set operations (intersection, union, concatenation) n This turns out to be a useful exercise Union Concatenation Speech and Language Processing - Jurafsky and Martin 9/8/17

10 + Weighted finite state acceptors 1 a/1 1 b/1 2/1 n Like a normal FSA but with costs on the arcs and final-states n Note: cost comes after /. For final-state, 2/1 means final-cost 1 on state 2. n View WFSA as a function from a string to a cost. n In this view, unweighted FSA is f : string {, }. n If multiple paths have the same string, take the one with the lowest cost. n This example maps ab to (3 = 1+1+1), all else to. Thanks for Mirko Hannemann for this slide

11 + Weights vs. costs a/ b/2 1/ 11 2 c/1 3/1 n Use cost to refer to the numeric value, and weight when speaking abstractly, e.g.: n The acceptor above accepts a with unit weight. n It accepts a with zero cost. n It accepts bc with cost 4=2+1+1 n State 1 is final with unit weight. n The acceptor assigns zero weight to xyz. n It assigns infinite cost to xyz. Thanks for Mirko Hannemann for this slide

12 + Combinations red/.5 green/.3 1 blue/ yellow/.6 n Union: parallel 2 /.8 6 eps/ eps/ red/.5 green/.4 blue/1.2 1 / 2 /.3 green/.3 1 blue/ 2 /.8 yellow/.6 3 green/.4 4 / blue/ /.3 n Concatenation: series red/.5 green/.3 1 blue/ yellow/.6 2 eps/.8 3 green/.4 blue/1.2 4 / 5 /.3 n Kleene closure: arbitrary repetition 3 / eps/ green/.4 eps/ blue/1.2 1 / eps/.3 2 /.3

13 + WSFAs in speech 13 Language modeling using/1 1 data/.66 intuition/.33 Pronunciation modeling d/1 1 ey/.5 ae/ (a) t/.3 dx/.7 is/.5 are/.5 is/1 3 4 ax/1 4 better/.7 worse/.3 5

14 + WFS Transducer n Accept an input while producing an output c:a/.3 14 input output weight a:b/.1 1 a:a/.4 a:b/.3 b:a/.2 2 b:b/.5 3/.6 n Input phonemes: output words (a) d:data/1 1 ey:ε/.5 ae:ε/.5 2 t:ε/.3 dx:ε/.7 3 ax: ε /1 4 d:dew/1 5 uw:ε/1 6

15 + FST for morphology: Foxes and Cats 15 9/12/17

16 + Composition c:a/.3 a:b/.1 1 a:a/.4 a:b/.6 16 b:a/.2 2 b:b/.5 3/.6 b:c/.3 1 a:b/.4 2/.7 a c a à b a a à c b b c:b/.9 c:b/.7 (1, 2) a:b/1 a:c/.4 (, ) (1, 1) a:b/.8 (3, 2)/1.3 a c a à c b b

17 + WFSTs in Action with OpenFST 17 n Create text file n Compile n Print n Show info n Union n Concatenate n Compose n Invert