16 research outputs found
Copyful Streaming String Transducers
International audienceCopyless streaming string transducers (copyless SST) have been introduced by R. Alur and P. Ë CernÂŽyCernÂŽy in 2010 as a one-way determin-istic automata model to define transductions of finite strings. Copyless SST extend deterministic finite state automata with a set of variables in which to store intermediate output strings, and those variables can be combined and updated all along the run, in a linear manner, i.e., no variable content can be copied on transitions. It is known that copyless SST capture exactly the class of MSO-definable string-to-string trans-ductions, and are as expressive as deterministic two-way transducers. They enjoy good algorithmic properties. Most notably, they have decid-able equivalence problem (in PSpace). On the other hand, HDT0L systems have been introduced for a while, the most prominent result being the decidability of the equivalence problem. In this paper, we propose a semantics of HDT0L systems in terms of transductions, and use it to study the class of deterministic copyful SST. Our contributions are as follows: (i) HDT0L systems and total deterministic copyful SST have the same expressive power, (ii) the equivalence problem for deterministic copyful SST and the equivalence problem for HDT0L systems are inter-reducible, in linear time. As a consequence, equivalence of deterministic SST is decid-able, (iii) the functionality of non-deterministic copyful SST is decidable, (iv) determining whether a deterministic copyful SST can be transformed into an equivalent deterministic copyless SST is decidable in polynomial time
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
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
Decision Problems of Tree Transducers with Origin - Emmanuel Filiot
International audienc
Revisiting the growth of polyregular functions: output languages, weighted automata and unary inputs
Polyregular functions are the class of string-to-string functions definable
by pebble transducers (an extension of finite automata) or equivalently by MSO
interpretations (a logical formalism). Their output length is bounded by a
polynomial in the input length: a function computed by a -pebble transducer
or by a -dimensional MSO interpretation has growth rate .
Boja\'nczyk has recently shown that the converse holds for MSO
interpretations, but not for pebble transducers. We give significantly
simplified proofs of those two results, extending the former to first-order
interpretations by reduction to an elementary property of -weighted
automata. For any , we also prove the stronger statement that there is some
quadratic polyregular function whose output language differs from that of any
-fold composition of macro tree transducers (and which therefore cannot be
computed by any -pebble transducer).
In the special case of unary input alphabets, we show that pebbles
suffice to compute polyregular functions of growth . This is obtained
as a corollary of a basis of simple word sequences whose ultimately periodic
combinations generate all polyregular functions with unary input. Finally, we
study polyregular and polyblind functions between unary alphabets (i.e. integer
sequences), as well as their first-order subclasses.Comment: 27 pages, not submitted ye
Equivalence of Symbolic Tree Transducers
International audienceSymbolic tree transducers are programs by which to transform data trees with an infinite signature. In this paper, we show that the equivalence problem of symbolic top-down deterministic tree transducers (DTops) can be reduced to that of classical DTops. As a consequence the equivalence of two symbolic DTops can be decided in NExpTime, when assuming that all operations related to the processing of data values are in PTime. This results can be extended to symbolic DTops with lookahead and thus to symbolic bottom-up deterministic tree transducers
Proactive Synthesis of Recursive Tree-to-String Functions from Examples
Synthesis from examples enables non-expert users to generate programs by specifying examples of their behavior. A domain-specific form of such synthesis has been recently deployed in a widely used spreadsheet software product. In this paper we contribute to foundations of such techniques and present a complete algorithm for synthesis of a class of recursive functions defined by structural recursion over a given algebraic data type definition. The functions we consider map an algebraic data type to a string; they are useful for, e.g., pretty printing and serialization of programs and data. We formalize our problem as learning deterministic sequential
top-down tree-to-string transducers with a single state (1STS).
The first problem we consider is learning a tree-to-string
transducer from any set of input/output examples provided by the user. We show that, given a set of input/output examples, checking whether there exists a 1STS consistent with these examples is NP-complete in general. In contrast, the problem can be solved in polynomial time under a (practically useful) closure condition that each
subtree of a tree in the input/output example set is also
part of the input/output examples.
Because coming up with relevant input/output examples may be
difficult for the user while creating hard constraint problems
for the synthesizer, we also study a more automated
active learning scenario in which the algorithm chooses the
inputs for which the user provides the outputs. Our
algorithm asks a worst-case linear number of queries as a
function of the size of the algebraic data type definition
to determine a unique transducer.
To construct our algorithms we present two new results on
formal languages.
First, we define a class of word equations, called
sequential word equations, for which we prove that
satisfiability can be solved in deterministic polynomial
time. This is in contrast to the general word equations for
which the best known complexity upper bound is in linear space.
Second, we close a long-standing open problem about the
asymptotic size of test sets for context-free languages. A
test set of a language of words L is a subset T of L
such that any two word homomorphisms equivalent on T are
also equivalent on L. We prove that it is possible to
build test sets of cubic size for context-free languages,
matching for the first time the lower bound found 20 years
ago
Determinacy and rewriting of functional topâdown and MSO tree transformations
A query is determined by a view, if the result of the query can be reconstructed from the result of the view. We consider the problem of deciding for two given (functional) tree transformations, whether one is determined by the other. If the view transformation is induced by a tree transducer that may copy, then determinacy is undecidable. For a large class of noncopying views, namely compositions of extended linear topâdown tree transducers, we show that determinacy is decidable, where queries are either deterministic topâdown tree transducers (with regular look-ahead) or deterministic MSO tree transducers. We also show that if a query is determined by a view, then it can be rewritten into a query that works over the view and is in the same class of transducers as the query. The proof relies on the decidability of equivalence for the considered classes of queries, and on their composition closure