Ambiguity, Valuedness and Costs

This survey reports results on the ambiguity of nite tree automata, the valuedness of bottom{up nite state tree transducers and boundedness of cost automata. 0 Introduction Ambiguity, valuedness and maximal cost are diierent measures on the behaviour of nite state devices. The ambiguity of an automaton is the least upper bound on the number of diierent accepting computations for a given input. If we attach outputs to transitions we obtain a transducer. The valuedness of a transducer is the least upper bound on the number of diierent outputs produced for any input. Both the interest in ambiguity of automata and valuedness of transducers is motivated by the equivalence problems of the corresponding devices. Automata are called equivalent ii they accept the same language; transducers are called equivalent ii they deene the same translation. For nite word automata one nds that equivalence is PSPACE{complete in general MS72, SM73], but decidable in polynomial time provided the ambiguities of the automata are bounded by some constant SH85, Ku88]. Accordingly, equivalence of nite word transducers is in general undecidable Gr68]; however if the transducers are nite{valued we obtain a decidable problem CK86, We88]. We prove similar results for tree automata and bottom{up tree transducers. Due to the more complicated tree structure, nontrivial extensions of the proof techniques are necessary. If we attach to the transitions of a nite tree automaton (sequences of) poly-nomials over some semiring R we obtain an R{cost automaton. Producing the 1