Transforming a Single-Valued Transducer Into a Mealy Machine

Abstract

This article deals with the transformation of a single-valued finite transducer into a Mealy machine. The following results are obtained: (1) LetMbe a single-valued real-time (or “letter-to-word”) transducer withnstates, input alphabetΣ, and output alphabetΔwhich is equivalent to some Mealy machineM′. Then,Mcan be effectively transformed into such anM′ having at most 2n+1·min{#Σ, #Δ}n−1states. A similar result holds ifMis not real time. As an important side effect three “Mealy” properties are obtained which characterize the fact that the given transducerMis equivalent to some Mealy machine. (2) The upper bound in result (1) improves to 2n−1 ifMis known to be a letter-to-letter transducer. (3) For every integert⩾2 and every odd integern⩾3 there is a single-valued real-time transducerMwithnstates and input and output alphabets of cardinalitytsuch thatMis equivalent to some Mealy machineM′ and every suchM′ has at leastt(n−1)/2states. (4) Ift=3, then result (3) holds true with letter-to-letter transducers rather than real-time transducers and with a lower bound of 2(n−1)/2. (5) It is a PSPACE-complete problem to decide whether or not a given single-valued transducerMis equivalent to some Mealy machine. The problem remains PSPACE-complete ifMis known to be a letter-to-letter transducer.