45 research outputs found
Syntactic Markovian Bisimulation for Chemical Reaction Networks
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.Comment: Extended version (with proofs), of the corresponding paper published
at KimFest 2017 (http://kimfest.cs.aau.dk/
Lumpability for Uncertain Continuous-Time Markov Chains
The assumption of perfect knowledge of rate parameters in continuous-time Markov chains (CTMCs) is undermined when confronted with reality, where they may be uncertain due to lack of information or because of measurement noise. In this paper we consider uncertain CTMCs, where rates are assumed to vary non-deterministically with time from bounded continuous intervals. This leads to a semantics which associates each state with the reachable set of its probability under all possible choices of the uncertain rates. We develop a notion of lumpability which identifies a partition of states where each block preserves the reachable set of the sum of its probabilities, essentially lifting the well-known CTMC ordinary lumpability to the uncertain setting. We proceed with this analogy with two further contributions: a logical characterization of uncertain CTMC lumping in terms of continuous stochastic logic; and a polynomial time and space algorithm for the minimization of uncertain CTMCs by partition refinement, using the CTMC lumping algorithm as an inner step. As a case study, we show that the minimizations in a substantial number of CTMC models reported in the literature are robust with respect to uncertainties around their original, fixed, rate values
Mean-Field Limits Beyond Ordinary Differential Equations
16th International School on Formal Methods for the Design of Computer, Communication, and Software Systems, SFM 2016, Bertinoro, Italy, June 20-24, 2016, Advanced LecturesInternational audienceWe study the limiting behaviour of stochastic models of populations of interacting agents, as the number of agents goes to infinity. Classical mean-field results have established that this limiting behaviour is described by an ordinary differential equation (ODE) under two conditions: (1) that the dynamics is smooth; and (2) that the population is composed of a finite number of homogeneous sub-populations, each containing a large number of agents. This paper reviews recent work showing what happens if these conditions do not hold. In these cases, it is still possible to exhibit a limiting regime at the price of replacing the ODE by a more complex dynamical system. In the case of non-smooth or uncertain dynamics, the limiting regime is given by a differential inclusion. In the case of multiple population scales, the ODE is replaced by a stochastic hybrid automaton
Efficient Syntax-Driven Lumping of Differential Equations
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⋅logs)O(r⋅s⋅logs) 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
Property-driven State-Space Coarsening for Continuous Time Markov Chains
Dynamical systems with large state-spaces are often expensive to thoroughly
explore experimentally. Coarse-graining methods aim to define simpler systems
which are more amenable to analysis and exploration; most current methods,
however, focus on a priori state aggregation based on similarities in
transition rates, which is not necessarily reflected in similar behaviours at
the level of trajectories. We propose a way to coarsen the state-space of a
system which optimally preserves the satisfaction of a set of logical
specifications about the system's trajectories. Our approach is based on
Gaussian Process emulation and Multi-Dimensional Scaling, a dimensionality
reduction technique which optimally preserves distances in non-Euclidean
spaces. We show how to obtain low-dimensional visualisations of the system's
state-space from the perspective of properties' satisfaction, and how to define
macro-states which behave coherently with respect to the specifications. Our
approach is illustrated on a non-trivial running example, showing promising
performance and high computational efficiency.Comment: 16 pages, 6 figures, 1 tabl
Guaranteed Error Bounds on Approximate Model Abstractions Through Reachability Analysis
It is well known that exact notions of model abstraction and reduction
for dynamical systems may not be robust enough in practice because they are
highly sensitive to the specific choice of parameters. In this paper we consider this
problem for nonlinear ordinary differential equations (ODEs) with polynomial
derivatives. We introduce approximate differential equivalence as a more permissive
variant of a recently developed exact counterpart, allowing ODE variables
to be related even when they are governed by nearby derivatives. We develop
algorithms to (i) compute the largest approximate differential equivalence; (ii)
construct an approximate quotient model from the original one via an appropriate
parameter perturbation; and (iii) provide a formal certificate on the quality of
the approximation as an error bound, computed as an over-approximation of the
reachable set of the perturbed model. Finally, we apply approximate differential
equivalences to study the effect of parametric tolerances in models of symmetric
electric circuits
From electric circuits to chemical networks
Electric circuits manipulate electric charge and magnetic flux via a small set of discrete components to implement useful functionality over continuous time-varying signals represented by currents and voltages. Much of the same functionality is useful to biological organisms, where it is implemented by a completely different set of discrete components (typically proteins) and signal representations (typically via concentrations). We describe how to take a linear electric circuit and systematically convert it to a chemical reaction network of the same functionality, as a dynamical system. Both the structure and the components of the electric circuit are dissolved in the process, but the resulting chemical network is intelligible. This approach provides access to a large library of well-studied devices, from analog electronics, whose chemical network realization can be compared to natural biochemical networks, or used to engineer synthetic biochemical networks
Guaranteed error bounds on approximate model abstractions through reachability analysis
It is well known that exact notions of model abstraction and reduction for dynamical systems may not be robust enough in practice because they are highly sensitive to the specific choice of parameters. In this paper we consider this problem for nonlinear ordinary differential equations (ODEs) with polynomial derivatives. We introduce approximate differential equivalence as a more permissive variant of a recently developed exact counterpart, allowing ODE variables to be related even when they are governed by nearby derivatives. We develop algorithms to (i) compute the largest approximate differential equivalence; (ii) construct an approximate quotient model from the original one via an appropriate parameter perturbation; and (iii) provide a formal certificate on the quality of the approximation as an error bound, computed as an over-approximation of the reachable set of the perturbed model. Finally, we apply approximate differential equivalences to study the effect of parametric tolerances in models of symmetric electric circuits
Comparing chemical reaction networks: A categorical and algorithmic perspective
We study chemical reaction networks (CRNs) as a kernel model of concurrency provided with semantics based on ordinary differential equations. We investigate the problem of comparing two CRNs, i.e., to decide whether the solutions of a source and of a target CRN can be matched for an appropriate choice of initial conditions. Using a categorical framework, we extend and unify model-comparison approaches based on dynamical (semantic) and structural (syntactic) properties of CRNs. Then, we provide an algorithm to compare CRNs, running linearly in time with respect to the cardinality of all possible comparisons. Finally, using a prototype implementation, CAGE, we apply our results to biological models from the literature