186 research outputs found
First-order definable string transformations
The connection between languages defined by computational models and logic
for languages is well-studied. Monadic second-order logic and finite automata
are shown to closely correspond to each-other for the languages of strings,
trees, and partial-orders. Similar connections are shown for first-order logic
and finite automata with certain aperiodicity restriction. Courcelle in 1994
proposed a way to use logic to define functions over structures where the
output structure is defined using logical formulas interpreted over the input
structure. Engelfriet and Hoogeboom discovered the corresponding "automata
connection" by showing that two-way generalised sequential machines capture the
class of monadic-second order definable transformations. Alur and Cerny further
refined the result by proposing a one-way deterministic transducer model with
string variables---called the streaming string transducers---to capture the
same class of transformations. In this paper we establish a transducer-logic
correspondence for Courcelle's first-order definable string transformations. We
propose a new notion of transition monoid for streaming string transducers that
involves structural properties of both underlying input automata and variable
dependencies. By putting an aperiodicity restriction on the transition monoids,
we define a class of streaming string transducers that captures exactly the
class of first-order definable transformations.Comment: 31 page
FO-definable transformations of infinite strings
The theory of regular and aperiodic transformations of finite strings has
recently received a lot of interest. These classes can be equivalently defined
using logic (Monadic second-order logic and first-order logic), two-way
machines (regular two-way and aperiodic two-way transducers), and one-way
register machines (regular streaming string and aperiodic streaming string
transducers). These classes are known to be closed under operations such as
sequential composition and regular (star-free) choice; and problems such as
functional equivalence and type checking, are decidable for these classes. On
the other hand, for infinite strings these results are only known for
-regular transformations: Alur, Filiot, and Trivedi studied
transformations of infinite strings and introduced an extension of streaming
string transducers over -strings and showed that they capture monadic
second-order definable transformations for infinite strings. In this paper we
extend their work to recover connection for infinite strings among first-order
logic definable transformations, aperiodic two-way transducers, and aperiodic
streaming string transducers
The Equivalence Problem for Deterministic MSO Tree Transducers is Decidable
It is decidable for deterministic MSO definable graph-to-string or
graph-to-tree transducers whether they are equivalent on a context-free set of
graphs
Two-Way Visibly Pushdown Automata and Transducers
Automata-logic connections are pillars of the theory of regular languages.
Such connections are harder to obtain for transducers, but important results
have been obtained recently for word-to-word transformations, showing that the
three following models are equivalent: deterministic two-way transducers,
monadic second-order (MSO) transducers, and deterministic one-way automata
equipped with a finite number of registers. Nested words are words with a
nesting structure, allowing to model unranked trees as their depth-first-search
linearisations. In this paper, we consider transformations from nested words to
words, allowing in particular to produce unranked trees if output words have a
nesting structure. The model of visibly pushdown transducers allows to describe
such transformations, and we propose a simple deterministic extension of this
model with two-way moves that has the following properties: i) it is a simple
computational model, that naturally has a good evaluation complexity; ii) it is
expressive: it subsumes nested word-to-word MSO transducers, and the exact
expressiveness of MSO transducers is recovered using a simple syntactic
restriction; iii) it has good algorithmic/closure properties: the model is
closed under composition with a unambiguous one-way letter-to-letter transducer
which gives closure under regular look-around, and has a decidable equivalence
problem
Streaming Tree Transducers
Theory of tree transducers provides a foundation for understanding
expressiveness and complexity of analysis problems for specification languages
for transforming hierarchically structured data such as XML documents. We
introduce streaming tree transducers as an analyzable, executable, and
expressive model for transforming unranked ordered trees in a single pass.
Given a linear encoding of the input tree, the transducer makes a single
left-to-right pass through the input, and computes the output in linear time
using a finite-state control, a visibly pushdown stack, and a finite number of
variables that store output chunks that can be combined using the operations of
string-concatenation and tree-insertion. We prove that the expressiveness of
the model coincides with transductions definable using monadic second-order
logic (MSO). Existing models of tree transducers either cannot implement all
MSO-definable transformations, or require regular look ahead that prohibits
single-pass implementation. We show a variety of analysis problems such as
type-checking and checking functional equivalence are solvable for our model.Comment: 40 page
Aperiodic String Transducers
Regular string-to-string functions enjoy a nice triple characterization
through deterministic two-way transducers (2DFT), streaming string transducers
(SST) and MSO definable functions. This result has recently been lifted to FO
definable functions, with equivalent representations by means of aperiodic 2DFT
and aperiodic 1-bounded SST, extending a well-known result on regular
languages. In this paper, we give three direct transformations: i) from
1-bounded SST to 2DFT, ii) from 2DFT to copyless SST, and iii) from k-bounded
to 1-bounded SST. We give the complexity of each construction and also prove
that they preserve the aperiodicity of transducers. As corollaries, we obtain
that FO definable string-to-string functions are equivalent to SST whose
transition monoid is finite and aperiodic, and to aperiodic copyless SST
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