Copyful Streaming String Transducers

Abstract

Copyless streaming string transducers (copyless SST) have been introduced by R. Alur and P. Cerny in 2010 as a one-way deterministic automata model to define transformations of finite strings. Copyless SST extend deterministic finite state automata with a set of registers in which to store intermediate output strings, and those registers can be combined and updated all along the run, in a linear manner, i.e., no register content can be copied on transitions. It is known that copyless SST capture exactly the class of MSO-definable string-to-string transformations, as defined by B. Courcelle, and are equi-expressive to deterministic two-way transducers. They enjoy good algorithmic properties. Most notably, they have decidable equivalence problem (in PSpace). In this paper, we show that they still have decidable equivalence problem even without the copyless restriction. The proof reduces to the HDT0L sequence equivalence problem, which is known to be decidable. We also show that this latter problem is as difficult as the SST equivalence problem, modulo linear time reduction.

Polynomial automata: Zeroness and applications

“…We show that polynomial automata can be used to simulate streaming string transducers, which were formulated in [6] for the verification of list-processing programs, and shown in [7] to encompass the set of MSOdefinable transductions. Using this, we show that the equivalence problem for general streaming string transducers (which was first proved decidable in [8]) is Ackermannian. We then define a class of copyless polynomial automata, and show that the Zeroness Problem for this class is in PSPACE.…”

“…Line (9) follows by commutativity of the left-hand diagram in (8), while line (10) follows by (repeated application of) commutativity of the right-hand diagram in (8).…”

“…Composing these logarithmic space reductions, we see that: Decidability of the equivalence problem was announced in [8], but to date no complexity bounds have appeared.…”

“…We first recall the definition of copyless SSTs [8]. Let T be an SST, as in Definition 1, with set of variables X and output alphabet Γ.…”

Separability by piecewise testable languages and downward closures beyond subwords

“…Examples of (i) are Eilenberg's algorithm for equivalence of weighted automata over fields [26, p.143-145] and Karr's computation of strongest affine invariants [44] (see also [63] and the stronger result [42]) and applications [38]. Examples for (ii) are the algorithm for polynomial invariants of affine programs [64], several algorithms for equivalence of transducers or register machines [8,12,29,71] (see [13] for a unifying result). An example of (iii) is the classical backward search algorithm to decide coverability in well-structured transition systems [1,30].…”