Reducing Transducer Equivalence to Register Automata Problems Solved by "Hilbert Method"

In the past decades, classical results from algebra, including Hilbert\u27s Basis Theorem, had various applications in formal languages, including a proof of the Ehrenfeucht Conjecture, decidability of HDT0L sequence equivalence, and decidability of the equivalence problem for functional tree-to-string transducers. In this paper, we study the scope of the algebraic methods mentioned above, particularily as applied to the functionality problem for register automata, and equivalence for functional register automata. We provide two results, one positive, one negative. The positive result is that functionality and equivalence are decidable for MSO transformations on unordered forests. The negative result comes from a try to extend this method to decide functionality and equivalence on macro tree transducers. We reduce macro tree transducers equivalence to an equivalence problem for some class of register automata naturally relevant to our method. We then prove this latter problem to be undecidable

Some Remarks on Deciding Equivalence for Graph-To-Graph Transducers

We study the following decision problem: given two mso transductions that input and output graphs of bounded treewidth, decide if they are equivalent, i.e. isomorphic inputs give isomorphic outputs. We do not know how to decide it, but we propose an approach that uses automata manipulating elements of a ring extended with division. The approach works for a variant of the problem, where isomorphism on output graphs is replaced by a relaxation of isomorphism

The many facets of string transducers

Regular word transductions extend the robust notion of regular languages from a qualitative to a quantitative reasoning. They were already considered in early papers of formal language theory, but turned out to be much more challenging. The last decade brought considerable research around various transducer models, aiming to achieve similar robustness as for automata and languages. In this paper we survey some older and more recent results on string transducers. We present classical connections between automata, logic and algebra extended to transducers, some genuine definability questions, and review approaches to the equivalence problem

Equivalence of finite-valued streaming string transducers is decidable

In this paper we provide a positive answer to a question left open by Alur and and Deshmukh in 2011 by showing that equivalence of finite-valued copyless streaming string transducers is decidable

On Rational Recursive Sequences

We study the class of rational recursive sequences (ratrec) over the rational numbers. A ratrec sequence is defined via a system of sequences using mutually recursive equations of depth 1, where the next values are computed as rational functions of the previous values. An alternative class is that of simple ratrec sequences, where one uses a single recursive equation, however of depth k: the next value is defined as a rational function of k previous values. We conjecture that the classes ratrec and simple ratrec coincide. The main contribution of this paper is a proof of a variant of this conjecture where the initial conditions are treated symbolically, using a formal variable per sequence, while the sequences themselves consist of rational functions over those variables. While the initial conjecture does not follow from this variant, we hope that the introduced algebraic techniques may eventually be helpful in resolving the problem. The class ratrec strictly generalises a well-known class of polynomial recursive sequences (polyrec). These are defined like ratrec, but using polynomial functions instead of rational ones. One can observe that if our conjecture is true and effective, then we can improve the complexities of the zeroness and the equivalence problems for polyrec sequences. Currently, the only known upper bound is Ackermanian, which follows from results on polynomial automata. We complement this observation by proving a PSPACE lower bound for both problems for polyrec. Our lower bound construction also implies that the Skolem problem is PSPACE-hard for the polyrec class

Polyregular functions on unordered trees of bounded height

We consider injective first-order interpretations that input and output trees of bounded height. The corresponding functions have polynomial output size, since a first-order interpretation can use a -tuple of input nodes to represent a single output node. We prove that the equivalence problem for such functions is decidable, i.e. given two such interpretations, one can decide whether, for every input tree, the two output trees are isomorphic. We also give a calculus of typed functions and combinators which derives exactly injective first-order interpretations for unordered trees of bounded height. The calculus is based on a type system, where the type constructors are products, coproducts and a monad of multisets. Thanks to our results about tree-to-tree interpretations, the equivalence problem is decidable for this calculus. As an application, we show that the equivalence problem is decidable for first-order interpretations between classes of graphs that have bounded tree-depth. In all cases studied in this paper, first-order logic and mso have the same expressive power, and hence all results apply also to mso interpretations