579 research outputs found
Analyzing Timed Systems Using Tree Automata
Timed systems, such as timed automata, are usually analyzed using their
operational semantics on timed words. The classical region abstraction for
timed automata reduces them to (untimed) finite state automata with the same
time-abstract properties, such as state reachability. We propose a new
technique to analyze such timed systems using finite tree automata instead of
finite word automata. The main idea is to consider timed behaviors as graphs
with matching edges capturing timing constraints. When a family of graphs has
bounded tree-width, they can be interpreted in trees and MSO-definable
properties of such graphs can be checked using tree automata. The technique is
quite general and applies to many timed systems. In this paper, as an example,
we develop the technique on timed pushdown systems, which have recently
received considerable attention. Further, we also demonstrate how we can use it
on timed automata and timed multi-stack pushdown systems (with boundedness
restrictions)
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
Partially Punctual Metric Temporal Logic is Decidable
Metric Temporal Logic \mathsf{MTL}[\until_I,\since_I] is one of the most
studied real time logics. It exhibits considerable diversity in expressiveness
and decidability properties based on the permitted set of modalities and the
nature of time interval constraints . Henzinger et al., in their seminal
paper showed that the non-punctual fragment of called
is decidable. In this paper, we sharpen this decidability
result by showing that the partially punctual fragment of
(denoted ) is decidable over strictly monotonic finite point
wise time. In this fragment, we allow either punctual future modalities, or
punctual past modalities, but never both together. We give two satisfiability
preserving reductions from to the decidable logic
\mathsf{MTL}[\until_I]. The first reduction uses simple projections, while
the second reduction uses a novel technique of temporal projections with
oversampling. We study the trade-off between the two reductions: while the
second reduction allows the introduction of extra action points in the
underlying model, the equisatisfiable \mathsf{MTL}[\until_I] formula obtained
is exponentially succinct than the one obtained via the first reduction, where
no oversampling of the underlying model is needed. We also show that
is strictly more expressive than the fragments
\mathsf{MTL}[\until_I,\since] and \mathsf{MTL}[\until,\since_I]
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