4To prove ( 1 ) â ( 3 ) , express f as an FST and apply the well-known Kleene-Sch Â¨ utzenberger construction ( Berstel and Reutenauer , 1988 ) , taking care to write each regexp in the construction as a constant times a probabilistic regexp .
rational series and their languages
Jean Berstel Christophe Reutenauer
The general form is illustrated by 3Conceptually , the parameters represent the probabilities of reading another a ( A ) ; reading another b ( Î½ ) ; transducing b to p rather than q ( Âµ ) ; starting to transduce p to a rather than x ( p ) . 4To prove ( 1 ) â ( 3 ) , express f as an FST and apply the well-known Kleene-Sch Â¨ utzenberger construction ( Berstel and Reutenauer , 1988 ) , taking care to write each regexp in the construction as a constant times a probabilistic regexp . A full proof is straightforward , as are proofs of ( 3 ) â ( 2 ) , ( 2 ) â ( 1 ) .
4To prove ( 1 ) â ( 3 ) , express f as an FST and apply the well-known Kleene-Sch Â¨ utzenberger construction ( @@CITATION ) , taking care to write each regexp in the construction as a constant times a probabilistic regexp .