Normal Form on Linear Tree-to-Word Transducers

Abstract

We study a subclass of tree-to-word transducers: linear treeto-word transducers, that cannot use several copies of the input. We aim to study the equivalence problem on this class, by using minimization and normalization techniques. We identify a Myhill-Nerode characterization. It provides a minimal normal form on our class, computable in Exptime. This paper extends an already existing result on tree-to-word transducers without copy or reordering (sequential tree-to-word transducers), by accounting for all the possible reorderings in the output.

On the Balancedness of Tree-to-word Transducers

“…Most importantly, both lemmas now state that, if e.g. (u, ε)(s 1 , t 1 ) 2 w = us 2 1 wt 2 1 is a witness of the lcs w.r.t. (u, ε)w = uw, then also (u, ε)(s 1 , t 1 )(s 2 , t 2 )w = us 1 s 2 wt 2 t 1 is a witness w.r.t.…”

“…In [2], a canonical normal form for LTWs without negated output has been provided which allows to reduce equivalence of transducers to syntactic identity. That normal form, however, may increase the sizes of representations of the transducers exponentially.…”

“…|r X | = d X and r X ∈ Prf(ρ(L X )). 2 In order to decide whether G is wf we compute the languages ρ(r X L ≤h X ) modulo ≈ lcs for increasing derivation height h using fixed-point iteration. Assuming inductively that (i) r X L ≤h X is wf and that we have computed (ii)…”

Deciding Equivalence of Linear Tree-to-Word Transducers in Polynomial Time

“…The equivalence for these transducers is decidable in polynomial time [15]. Moreover a normal form for sequential and linear treeto-word transducers, computable in exponential time, is known [7,1]. Two equivalent ltws do not necessarily transform their trees in the same order.…”

“…Suppose there exists co-reachable (and thus equivalent) states q e and q e in M e and M e , respectively, with rules q e , f → v 0 q e 1 (x σ (1) ) . .…”