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Abstract-The exact performance analysis of large-scale software systems with discrete-state approaches is difficult because of the well-known problem of state-space explosion. This paper considers this problem with regard to the stochastic process algebra PEPA, presenting a deterministic approximation to the underlying Markov chain model based on ordinary differential equations. The accuracy of the approximation is assessed by means of a substantial case study of a distributed multi-threaded application.

Abstract. We present an algorithm to compute exact aggregations of a class of systems of ordinary differential equations (ODEs). Our approach consists in an extension of Paige and Tarjan's seminal solution to the coarsest refinement problem by encoding an ODE system into a suitable discrete-state representation. In particular, we consider a simple extension of the syntax of elementary chemical reaction networks because i) it can express ODEs with derivatives given by polynomials of degree at most two, which are relevant in many applications in natural sciences and engineering; and ii) we can build on two recently introduced bisimulations, which yield two complementary notions of ODE lumping. Our algorithm computes the largest bisimulations in O(r · s · log s) time, where r is the number of monomials and s is the number of variables in the ODEs. Numerical experiments on real-world models from biochemistry, electrical engineering, and structural mechanics show that our prototype is able to handle ODEs with millions of variables and monomials, providing significant model reductions.

In chemical reaction networks (CRNs) with stochastic semantics based on continuous-time Markov chains (CTMCs), the typically large populations of species cause combinatorially large state spaces. This makes the analysis very difficult in practice and represents the major bottleneck for the applicability of minimization techniques based, for instance, on lumpability. In this paper we present syntactic Markovian bisimulation (SMB), a notion of bisimulation developed in the Larsen-Skou style of probabilistic bisimulation, defined over the structure of a CRN rather than over its underlying CTMC. SMB identifies a lumpable partition of the CTMC state space a priori, in the sense that it is an equivalence relation over species implying that two CTMC states are lumpable when they are invariant with respect to the total population of species within the same equivalence class. We develop an efficient partition-refinement algorithm which computes the largest SMB of a CRN in polynomial time in the number of species and reactions. We also provide an algorithm for obtaining a quotient network from an SMB that induces the lumped CTMC directly, thus avoiding the generation of the state space of the original CRN altogether. In practice, we show that SMB allows significant reductions in a number of models from the literature. Finally, we study SMB with respect to the deterministic semantics of CRNs based on ordinary differential equations (ODEs), where each equation gives the time-course evolution of the concentration of a species. SMB implies forward CRN bisimulation, a recently developed behavioral notion of equivalence for the ODE semantics, in an analogous sense: it yields a smaller ODE system that keeps track of the sums of the solutions for equivalent species.

International audienceWe study the problem of making forecasts about the future availability of bicycles in stations of a bike-sharing system (BSS). This is relevant in order to make recommendations guaranteeing that the probability that a user will be able to make a journey is sufficiently high. To do this we use probabilistic predictions obtained from a queuing theoretical time-inhomogeneous model of a BSS. The model is parametrized and successfully validated using historical data from the Vélib ' BSS of the City of Paris. We develop a critique of the standard root-mean-square-error (RMSE), commonly adopted in the bike-sharing research as an index of the prediction accuracy, because it does not account for the stochasticity inherent in the real system. Instead we introduce a new metric based on scoring rules. We evaluate the average score of our model against classical predictors used in the literature. We show that these are outperformed by our model for prediction horizons of up to a few hours. We also discuss that, in general, measuring the current number of available bikes is only relevant for prediction horizons of up to few hours

Ordinary differential equations (ODEs) with polynomial derivatives are a fundamental tool for understanding the dynamics of systems across many branches of science, but our ability to gain mechanistic insight and effectively conduct numerical evaluations is critically hindered when dealing with large models. Here we propose an aggregation technique that rests on two notions of equivalence relating ODE variables whenever they have the same solution (backward criterion) or if a self-consistent system can be written for describing the evolution of sums of variables in the same equivalence class (forward criterion). A key feature of our proposal is to encode a polynomial ODE system into a finitary structure akin to a formal chemical reaction network. This enables the development of a discrete algorithm to efficiently compute the largest equivalence, building on approaches rooted in computer science to minimize basic models of computation through iterative partition refinements. The physical interpretability of the aggregation is shown on polynomial ODE systems for biochemical reaction networks, gene regulatory networks, and evolutionary game theory.

We present ERODE , a multi-platform tool for the solution and exact reduction of systems of ordinary differential equations (ODEs). ERODE supports two recently introduced, complementary, equivalence relations over ODE variables: forward differential equivalence yields a self-consistent aggregate system where each ODE gives the cumulative dynamics of the sum of the original variables in the respective equivalence class. Backward differential equivalence identifies variables that have identical solutions whenever starting from the same initial conditions. As back-end ERODE uses the well-known Z3 SMT solver to compute the largest equivalence that refines a given initial partition of ODE variables. In the special case of ODEs with polynomial derivatives of degree at most two (covering affine systems and elementary chemical reaction networks), it implements a more efficient partition-refinement algorithm in the style of Paige and Tarjan. ERODE comes with a rich development environment based on the Eclipse plug-in framework offering: (i) seamless project management; (ii) a fully-featured text editor; and (iii) importing-exporting capabilities.

Ordinary differential equations (ODEs) are widespread in many natural sciences including chemistry, ecology, and systems biology, and in disciplines such as control theory and electrical engineering. Building on the celebrated molecules-as-processes paradigm, they have become increasingly popular in computer science, with highlevel languages and formal methods such as Petri nets, process algebra, and rule-based systems that are interpreted as ODEs.We consider the problem of comparing and minimizing ODEs automatically. Influenced by traditional approaches in the theory of programming, we propose differential equivalence relations. We study them for a basic intermediate language, for which we have decidability results, that can be targeted by a class of highlevel specifications. An ODE implicitly represents an uncountable state space, hence reasoning techniques cannot be borrowed from established domains such as probabilistic programs with finite-state Markov chain semantics. We provide novel symbolic procedures to check an equivalence and compute the largest one via partition refinement algorithms that use satisfiability modulo theories.We illustrate the generality of our framework by showing that differential equivalences include (i) well-known notions for the minimization of continuous-time Markov chains (lumpability), (ii) bisimulations for chemical reaction networks recently proposed by Cardelli et al., and (iii) behavioral relations for process algebra with ODE semantics. With a prototype implementation we are able to detect equivalences in biochemical models from the literature that cannot be reduced using competing automatic techniques.

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