76 research outputs found
On the Complexity of Quadratization for Polynomial Differential Equations
Chemical reaction networks (CRNs) are a standard formalism used in chemistry
and biology to reason about the dynamics of molecular interaction networks. In
their interpretation by ordinary differential equations, CRNs provide a
Turing-complete model of analog computattion, in the sense that any computable
function over the reals can be computed by a finite number of molecular species
with a continuous CRN which approximates the result of that function in one of
its components in arbitrary precision. The proof of that result is based on a
previous result of Bournez et al. on the Turing-completeness of polyno-mial
ordinary differential equations with polynomial initial conditions (PIVP). It
uses an encoding of real variables by two non-negative variables for
concentrations, and a transformation to an equivalent quadratic PIVP (i.e. with
degrees at most 2) for restricting ourselves to at most bimolecular reactions.
In this paper, we study the theoretical and practical complexities of the
quadratic transformation. We show that both problems of minimizing either the
number of variables (i.e., molecular species) or the number of monomials (i.e.
elementary reactions) in a quadratic transformation of a PIVP are NP-hard. We
present an encoding of those problems in MAX-SAT and show the practical
complexity of this algorithm on a benchmark of quadratization problems inspired
from CRN design problems
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
Non-polynomial Worst-Case Analysis of Recursive Programs
We study the problem of developing efficient approaches for proving
worst-case bounds of non-deterministic recursive programs. Ranking functions
are sound and complete for proving termination and worst-case bounds of
nonrecursive programs. First, we apply ranking functions to recursion,
resulting in measure functions. We show that measure functions provide a sound
and complete approach to prove worst-case bounds of non-deterministic recursive
programs. Our second contribution is the synthesis of measure functions in
nonpolynomial forms. We show that non-polynomial measure functions with
logarithm and exponentiation can be synthesized through abstraction of
logarithmic or exponentiation terms, Farkas' Lemma, and Handelman's Theorem
using linear programming. While previous methods obtain worst-case polynomial
bounds, our approach can synthesize bounds of the form
as well as where is not an integer. We present
experimental results to demonstrate that our approach can obtain efficiently
worst-case bounds of classical recursive algorithms such as (i) Merge-Sort, the
divide-and-conquer algorithm for the Closest-Pair problem, where we obtain
worst-case bound, and (ii) Karatsuba's algorithm for
polynomial multiplication and Strassen's algorithm for matrix multiplication,
where we obtain bound such that is not an integer and
close to the best-known bounds for the respective algorithms.Comment: 54 Pages, Full Version to CAV 201
Deciding Reachability for Piecewise Constant Derivative Systems on Orientable Manifolds
© 2019 Springer-Verlag. This is a post-peer-review, pre-copyedit version of a paper published in Reachability Problems: 13th International Conference, RP 2019, Brussels, Belgium, September 11–13, 2019, Proceedings. The final authenticated version is available online at: http://dx.doi.org/10.1007/978-3-030-30806-3_14A hybrid automaton is a finite state machine combined with some k real-valued continuous variables, where k determines the number of the automaton dimensions. This formalism is widely used for modelling safety-critical systems, and verification tasks for such systems can often be expressed as the reachability problem for hybrid automata. Asarin, Mysore, Pnueli and Schneider defined classes of hybrid automata lying on the boundary between decidability and undecidability in their seminal paper ‘Low dimensional hybrid systems - decidable, undecidable, don’t know’ [9]. They proved that certain decidable classes become undecidable when given a little additional computational power, and showed that the reachability question remains unsolved for some 2-dimensional systems. Piecewise Constant Derivative Systems on 2-dimensional manifolds (or PCD2m) constitute a class of hybrid automata for which decidability of the reachability problem is unknown. In this paper we show that the reachability problem becomes decidable for PCD2m if we slightly limit their dynamics, and thus we partially answer the open question of Asarin, Mysore, Pnueli and Schneider posed in [9]
Computability and dynamical systems
In this paper we explore results that establish a link between dynamical
systems and computability theory (not numerical analysis). In the last few decades,
computers have increasingly been used as simulation tools for gaining insight into
dynamical behavior. However, due to the presence of errors inherent in such numerical
simulations, with few exceptions, computers have not been used for the
nobler task of proving mathematical results. Nevertheless, there have been some recent
developments in the latter direction. Here we introduce some of the ideas and
techniques used so far, and suggest some lines of research for further work on this
fascinating topic
Robust computations with dynamical systems
In this paper we discuss the computational power of Lipschitz
dynamical systems which are robust to in nitesimal perturbations.
Whereas the study in [1] was done only for not-so-natural systems from
a classical mathematical point of view (discontinuous di erential equation
systems, discontinuous piecewise a ne maps, or perturbed Turing
machines), we prove that the results presented there can be generalized
to Lipschitz and computable dynamical systems.
In other words, we prove that the perturbed reachability problem (i.e. the
reachability problem for systems which are subjected to in nitesimal perturbations)
is co-recursively enumerable for this kind of systems. Using
this result we show that if robustness to in nitesimal perturbations is
also required, the reachability problem becomes decidable. This result
can be interpreted in the following manner: undecidability of veri cation
doesn't hold for Lipschitz, computable and robust systems.
We also show that the perturbed reachability problem is co-r.e. complete
even for C1-systems
Proving Positive Almost-Sure Termination
Rapport interne.In order to extend the modeling capabilities of rewriting systems, it is rather natural to consider that the firing of rules can be subject to some probabilistic laws. Considering rewrite rules subject to probabilities leads to numerous questions about the underlying notions and results. We focus here on the problem of termination of a set of probabilistic rewrite rules. A probabilistic rewrite system is said almost surely terminating if the probability that a derivation leads to a normal form is one. Such a system is said positively almost surely terminating if furthermore the mean length of a derivation is finite. We provide several results and techniques in order to prove positive almost sure termination of a given set of probabilistic rewrite rules. All these techniques subsume classical ones for non-probabilistic systems
Computability of ordinary differential equations
In this paper we provide a brief review of several results about the
computability of initial-value problems (IVPs) defined with ordinary differential
equations (ODEs). We will consider a variety of settings and analyze
how the computability of the IVP will be affected. Computational
complexity results will also be presented, as well as computable versions
of some classical theorems about the asymptotic behavior of ODEs.info:eu-repo/semantics/publishedVersio
The Generic Model of Computation
Over the past two decades, Yuri Gurevich and his colleagues have formulated
axiomatic foundations for the notion of algorithm, be it classical,
interactive, or parallel, and formalized them in the new generic framework of
abstract state machines. This approach has recently been extended to suggest a
formalization of the notion of effective computation over arbitrary countable
domains. The central notions are summarized herein.Comment: In Proceedings DCM 2011, arXiv:1207.682
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