1,875 research outputs found
A Quasi-Linear Time Algorithm Deciding Whether Weak B\"uchi Automata Reading Vectors of Reals Recognize Saturated Languages
This work considers weak deterministic B\"uchi automata reading encodings of
non-negative -vectors of reals in a fixed base. A saturated language is a
language which contains all encoding of elements belonging to a set of
-vectors of reals. A Real Vector Automaton is an automaton which recognizes
a saturated language. It is explained how to decide in quasi-linear time
whether a minimal weak deterministic B\"uchi automaton is a Real Vector
Automaton. The problem is solved both for the two standard encodings of vectors
of numbers: the sequential encoding and the parallel encoding. This algorithm
runs in linear time for minimal weak B\"uchi automata accepting set of reals.
Finally, the same problem is also solved for parallel encoding of automata
reading vectors of relative reals
On the Sets of Real Numbers Recognized by Finite Automata in Multiple Bases
This article studies the expressive power of finite automata recognizing sets
of real numbers encoded in positional notation. We consider Muller automata as
well as the restricted class of weak deterministic automata, used as symbolic
set representations in actual applications. In previous work, it has been
established that the sets of numbers that are recognizable by weak
deterministic automata in two bases that do not share the same set of prime
factors are exactly those that are definable in the first order additive theory
of real and integer numbers. This result extends Cobham's theorem, which
characterizes the sets of integer numbers that are recognizable by finite
automata in multiple bases.
In this article, we first generalize this result to multiplicatively
independent bases, which brings it closer to the original statement of Cobham's
theorem. Then, we study the sets of reals recognizable by Muller automata in
two bases. We show with a counterexample that, in this setting, Cobham's
theorem does not generalize to multiplicatively independent bases. Finally, we
prove that the sets of reals that are recognizable by Muller automata in two
bases that do not share the same set of prime factors are exactly those
definable in the first order additive theory of real and integer numbers. These
sets are thus also recognizable by weak deterministic automata. This result
leads to a precise characterization of the sets of real numbers that are
recognizable in multiple bases, and provides a theoretical justification to the
use of weak automata as symbolic representations of sets.Comment: 17 page
Near-Optimal Complexity Bounds for Fragments of the Skolem Problem
Given a linear recurrence sequence (LRS), specified using the initial conditions and the recurrence relation, the Skolem problem asks if zero ever occurs in the infinite sequence generated by the LRS. Despite active research over last few decades, its decidability is known only for a few restricted subclasses, by either restricting the order of the LRS (upto 4) or by restricting the structure of the LRS (e.g., roots of its characteristic polynomial).
In this paper, we identify a subclass of LRS of arbitrary order for which the Skolem problem is easy, namely LRS all of whose characteristic roots are (possibly complex) roots of real algebraic numbers, i.e., roots satisfying x^d = r for r real algebraic. We show that for this subclass, the Skolem problem can be solved in NP^RP. As a byproduct, we implicitly obtain effective bounds on the zero set of the LRS for this subclass. While prior works in this area often exploit deep results from algebraic and transcendental number theory to get such effective results, our techniques are primarily algorithmic and use linear algebra and Galois theory. We also complement our upper bounds with a NP lower bound for the Skolem problem via a new direct reduction from 3-CNF-SAT, matching the best known lower bounds
Approximated Symbolic Computations over Hybrid Automata
Hybrid automata are a natural framework for modeling and analyzing systems
which exhibit a mixed discrete continuous behaviour. However, the standard
operational semantics defined over such models implicitly assume perfect
knowledge of the real systems and infinite precision measurements. Such
assumptions are not only unrealistic, but often lead to the construction of
misleading models. For these reasons we believe that it is necessary to
introduce more flexible semantics able to manage with noise, partial
information, and finite precision instruments. In particular, in this paper we
integrate in a single framework based on approximated semantics different over
and under-approximation techniques for hybrid automata. Our framework allows to
both compare, mix, and generalize such techniques obtaining different
approximated reachability algorithms.Comment: In Proceedings HAS 2013, arXiv:1308.490
Computing Probabilistic Bisimilarity Distances for Probabilistic Automata
The probabilistic bisimilarity distance of Deng et al. has been proposed as a
robust quantitative generalization of Segala and Lynch's probabilistic
bisimilarity for probabilistic automata. In this paper, we present a
characterization of the bisimilarity distance as the solution of a simple
stochastic game. The characterization gives us an algorithm to compute the
distances by applying Condon's simple policy iteration on these games. The
correctness of Condon's approach, however, relies on the assumption that the
games are stopping. Our games may be non-stopping in general, yet we are able
to prove termination for this extended class of games. Already other algorithms
have been proposed in the literature to compute these distances, with
complexity in and \textbf{PPAD}. Despite the
theoretical relevance, these algorithms are inefficient in practice. To the
best of our knowledge, our algorithm is the first practical solution.
The characterization of the probabilistic bisimilarity distance mentioned
above crucially uses a dual presentation of the Hausdorff distance due to
M\'emoli. As an additional contribution, in this paper we show that M\'emoli's
result can be used also to prove that the bisimilarity distance bounds the
difference in the maximal (or minimal) probability of two states to satisfying
arbitrary -regular properties, expressed, eg., as LTL formulas
Real Hypercomputation and Continuity
By the sometimes so-called 'Main Theorem' of Recursive Analysis, every
computable real function is necessarily continuous. We wonder whether and which
kinds of HYPERcomputation allow for the effective evaluation of also
discontinuous f:R->R. More precisely the present work considers the following
three super-Turing notions of real function computability:
* relativized computation; specifically given oracle access to the Halting
Problem 0' or its jump 0'';
* encoding real input x and/or output y=f(x) in weaker ways also related to
the Arithmetic Hierarchy;
* non-deterministic computation.
It turns out that any f:R->R computable in the first or second sense is still
necessarily continuous whereas the third type of hypercomputation does provide
the required power to evaluate for instance the discontinuous sign function.Comment: previous version (extended abstract) has appeared in pp.562-571 of
"Proc. 1st Conference on Computability in Europe" (CiE'05), Springer LNCS
vol.352
The complexity of linear-time temporal logic over the class of ordinals
We consider the temporal logic with since and until modalities. This temporal
logic is expressively equivalent over the class of ordinals to first-order
logic by Kamp's theorem. We show that it has a PSPACE-complete satisfiability
problem over the class of ordinals. Among the consequences of our proof, we
show that given the code of some countable ordinal alpha and a formula, we can
decide in PSPACE whether the formula has a model over alpha. In order to show
these results, we introduce a class of simple ordinal automata, as expressive
as B\"uchi ordinal automata. The PSPACE upper bound for the satisfiability
problem of the temporal logic is obtained through a reduction to the
nonemptiness problem for the simple ordinal automata.Comment: Accepted for publication in LMC
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