1,917 research outputs found
The prospects for mathematical logic in the twenty-first century
The four authors present their speculations about the future developments of
mathematical logic in the twenty-first century. The areas of recursion theory,
proof theory and logic for computer science, model theory, and set theory are
discussed independently.Comment: Association for Symbolic Logi
A Survey on Continuous Time Computations
We provide an overview of theories of continuous time computation. These
theories allow us to understand both the hardness of questions related to
continuous time dynamical systems and the computational power of continuous
time analog models. We survey the existing models, summarizing results, and
point to relevant references in the literature
Positivity Problems for Low-Order Linear Recurrence Sequences
We consider two decision problems for linear recurrence sequences (LRS) over
the integers, namely the Positivity Problem (are all terms of a given LRS
positive?) and the Ultimate Positivity Problem} (are all but finitely many
terms of a given LRS positive?). We show decidability of both problems for LRS
of order 5 or less, with complexity in the Counting Hierarchy for Positivity,
and in polynomial time for Ultimate Positivity. Moreover, we show by way of
hardness that extending the decidability of either problem to LRS of order 6
would entail major breakthroughs in analytic number theory, more precisely in
the field of Diophantine approximation of transcendental numbers
Synthesis of Data Word Transducers
In reactive synthesis, the goal is to automatically generate an
implementation from a specification of the reactive and non-terminating
input/output behaviours of a system. Specifications are usually modelled as
logical formulae or automata over infinite sequences of signals
(-words), while implementations are represented as transducers. In the
classical setting, the set of signals is assumed to be finite. In this paper,
we consider data -words instead, i.e., words over an infinite alphabet.
In this context, we study specifications and implementations respectively given
as automata and transducers extended with a finite set of registers. We
consider different instances, depending on whether the specification is
nondeterministic, universal or deterministic, and depending on whether the
number of registers of the implementation is given or not.
In the unbounded setting, we show undecidability for both universal and
nondeterministic specifications, while decidability is recovered in the
deterministic case. In the bounded setting, undecidability still holds for
nondeterministic specifications, but can be recovered by disallowing tests over
input data. The generic technique we use to show the latter result allows us to
reprove some known result, namely decidability of bounded synthesis for
universal specifications
History-Register Automata
Programs with dynamic allocation are able to create and use an unbounded
number of fresh resources, such as references, objects, files, etc. We propose
History-Register Automata (HRA), a new automata-theoretic formalism for
modelling such programs. HRAs extend the expressiveness of previous approaches
and bring us to the limits of decidability for reachability checks. The
distinctive feature of our machines is their use of unbounded memory sets
(histories) where input symbols can be selectively stored and compared with
symbols to follow. In addition, stored symbols can be consumed or deleted by
reset. We show that the combination of consumption and reset capabilities
renders the automata powerful enough to imitate counter machines, and yields
closure under all regular operations apart from complementation. We moreover
examine weaker notions of HRAs which strike different balances between
expressiveness and effectiveness.Comment: LMCS (improved version of FoSSaCS
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