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Positive systems naturally arise in situations where the model tracks physical quantities. Although the linear case is well understood, analysis and controller design for nonlinear positive systems remain challenging. Model reduction methods can help tame this problem. Here we propose a notion of model reduction for a class of positive bilinear systems with (bounded) matrix and exogenous controls. Our reduction, called proper positive lumping, aggregates the original system such that states of the corresponding reduced model represent non-negative linear combinations of original state variables. We prove a characterization result showing that the reductions by proper positive lumping are exactly those preserving the optimality of a suitable class of value functions. Moreover, we provide an efficient polynomial-time algorithm for the computation of the minimal lumping. We numerically evaluate our approach by applying it to a number of benchmark case studies.

The ability to control complex networks is of crucial importance across a wide range of applications in natural and engineering sciences. However, issues of both theoretical and numerical nature introduce fundamental limitations to controlling large-scale networks. In this paper, we cope with this problem by introducing a coarse-graining algorithm. It leads to an aggregated network which satisfies control equivalence, i.e., such that the optimal control values for the original network can be exactly recovered from those of the aggregated one. The algorithm is based on a partition refinement method originally devised for systems of ordinary differential equations, here extended and applied to linear dynamics on complex networks. Using a number of benchmarks from the literature we show considerable reductions across a variety of networks from biology, ecology, engineering, and social sciences.

Computing the posterior distribution of a probabilistic program is a hard task for which no one-fit-for-all solution exists. We propose Gaussian Semantics, which approximates the exact probabilistic semantics of a bounded program by means of Gaussian mixtures. It is parametrized by a map that associates each program location with the moment order to be matched in the approximation. We provide two main contributions. The first is a universal approximation theorem stating that, under mild conditions, Gaussian Semantics can approximate the exact semantics arbitrarily closely. The second is an approximation that matches up to second-order moments analytically in face of the generally difficult problem of matching moments of Gaussian mixtures with arbitrary moment order. We test our second-order Gaussian approximation (SOGA) on a number of case studies from the literature. We show that it can provide accurate estimates in models not supported by other approximation methods or when exact symbolic techniques fail because of complex expressions or non-simplified integrals. On two notable classes of problems, namely collaborative filtering and programs involving mixtures of continuous and discrete distributions, we show that SOGA significantly outperforms alternative techniques in terms of accuracy and computational time.

Efficient methods for the simulation of quantum circuits on classic computers are crucial for their analysis due to the exponential growth of the problem size with the number of qubits. Here we study lumping methods based on bisimulation, an established class of techniques that has been proven successful for (classic) stochastic and deterministic systems such as Markov chains and ordinary differential equations. Forward constrained bisimulation yields a lower-dimensional model which exactly preserves quantum measurements projected on a linear subspace of interest. Backward constrained bisimulation gives a reduction that is valid on a subspace containing the circuit input, from which the circuit result can be fully recovered. We provide an algorithm to compute the constraint bisimulations yielding coarsest reductions in both cases, using a duality result relating the two notions. As applications, we provide theoretical bounds on the size of the reduced state space for well-known quantum algorithms for search, optimization, and factorization. Using a prototype implementation, we report significant reductions on a set of benchmarks. In particular, we show that constrained bisimulation can boost decision-diagram-based quantum circuit simulation by several orders of magnitude, allowing thus for substantial synergy effects., Comment: 20 pages, 1 algorithm, 2 tables

In safety-critical applications that rely on the solution of an optimization problem, the certification of the optimization algorithm is of vital importance. Certification and suboptimality results are available for a wide range of optimization algorithms. However, a typical underlying assumption is that the operations performed by the algorithm are exact, i.e., that there is no numerical error during the mathematical operations, which is hardly a valid assumption in a real hardware implementation. This is particularly true in the case of fixed-point hardware, where computational inaccuracies are not uncommon. This article presents a certification procedure for the proximal gradient method for box-constrained QP problems implemented in fixed-point arithmetic. The procedure provides a method to select the minimal fractional precision required to obtain a certain suboptimality bound, indicating the maximum number of iterations of the optimization method required to obtain it. The procedure makes use of formal verification methods to provide arbitrarily tight bounds on the suboptimality guarantee. We apply the proposed certification procedure on the implementation of a non-trivial model predictive controller on 32-bit fixed-point hardware., Comment: 8 pages

Across many disciplines, chemical reaction networks (CRNs) are an established population model defined as a system of coupled nonlinear ordinary differential equations. In many applications, for example, in systems biology and epidemiology, CRN parameters such as the kinetic reaction rates can be used as control inputs to steer the system toward a given target. Unfortunately, the resulting optimal control problem is nonlinear, therefore, computationally very challenging. We address this issue by introducing an optimality-preserving reduction algorithm for CRNs. The algorithm partitions the original state variables into a reduced set of macro-variables for which one can define a reduced optimal control problem from which one can exactly recover the solution of the original control problem. Notably, the reduction algorithm runs with polynomial time complexity in the size of the CRN. We use this result to reduce reachability and control problems of large-scale protein-interaction networks and vaccination models with hundreds of thousands of state variables.

Quantitative notions of bisimulation are well-known tools for the minimization of dynamical models such as Markov chains and ordinary differential equations (ODEs). In \emph{forward bisimulations}, each state in the quotient model represents an equivalence class and the dynamical evolution gives the overall sum of its members in the original model. Here we introduce generalized forward bisimulation (GFB) for dynamical systems over commutative monoids and develop a partition refinement algorithm to compute the coarsest one. When the monoid is $(\mathbb{R}, +)$, we recover probabilistic bisimulation for Markov chains and more recent forward bisimulations for nonlinear ODEs. Using $(\mathbb{R}, \cdot)$ we get nonlinear reductions for discrete-time dynamical systems and ODEs where each variable in the quotient model represents the product of original variables in the equivalence class. When the domain is a finite set such as the Booleans $\mathbb{B}$, we can apply GFB to Boolean networks (BN), a widely used dynamical model in computational biology. Using a prototype implementation of our minimization algorithm for GFB, we find disjunction- and conjunction-preserving reductions on 60 BN from two well-known repositories, and demonstrate the obtained analysis speed-ups. We also provide the biological interpretation of the reduction obtained for two selected BN, and we show how GFB enables the analysis of a large one that could not be analyzed otherwise. Using a randomized version of our algorithm we find product-preserving (therefore non-linear) reductions on 21 dynamical weighted networks from the literature that could not be handled by the exact algorithm., Comment: Accepted at Thirty-Eighth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), 2023

Boolean Networks (BNs) are established models to qualitatively describe biological systems. The analysis of BNs might be infeasible for medium to large BNs due to the state-space explosion problem. We propose a novel reduction technique called \emph{Backward Boolean Equivalence} (BBE), which preserves some properties of interest of BNs. In particular, reduced BNs provide a compact representation by grouping variables that, if initialized equally, are always updated equally. The resulting reduced state space is a subset of the original one, restricted to identical initialization of grouped variables. The corresponding trajectories of the original BN can be exactly restored. We show the effectiveness of BBE by performing a large-scale validation on the whole GINsim BN repository. In selected cases, we show how our method enables analyses that would be otherwise intractable. Our method complements, and can be combined with, other reduction methods found in the literature.

We introduce polynomial couplings, a generalization of probabilistic couplings, to develop an algorithm for the computation of equivalence relations which can be interpreted as a lifting of probabilistic bisimulation to polynomial differential equations, a ubiquitous model of dynamical systems across science and engineering. The algorithm enjoys polynomial time complexity and complements classical partition-refinement approaches because: (a) it implements a local exploration of the system, possibly yielding equivalences that do not necessarily involve the inspection of the whole system of differential equations; (b) it can be enhanced by up-to techniques; and (c) it allows the specification of pairs which ought not to be included in the output. Using a prototype, these advantages are demonstrated on case studies from systems biology for applications to model reduction and comparison. Notably, we report four orders of magnitude smaller runtimes than partition-refinement approaches when disproving equivalences between Markov chains.

Motivation: Stochastic reaction networks are a widespread model to describe biological systems where the presence of noise is relevant, such as in cell regulatory processes. Unfortu-nately, in all but simplest models the resulting discrete state-space representation hinders analytical tractability and makes numerical simulations expensive. Reduction methods can lower complexity by computing model projections that preserve dynamics of interest to the user. Results: We present an exact lumping method for stochastic reaction networks with mass-action kinetics. It hinges on an equivalence relation between the species, resulting in a reduced network where the dynamics of each macro-species is stochastically equivalent to the sum of the original species in each equivalence class, for any choice of the initial state of the system. Furthermore, by an appropriate encoding of kinetic parameters as additional species, the method can establish equivalences that do not depend on specific values of the parameters. The method is supported by an efficient algorithm to compute the largest species equivalence, thus the maximal lumping. The effectiveness and scalability of our lumping technique, as well as the physical interpretability of resulting reductions, is demonstrated in several models of signaling pathways and epidemic processes on complex networks. Availability: The algorithms for species equivalence have been implemented in the software tool ERODE, freely available for download from https://www.erode.eu.

Stochastic reaction networks are a fundamental model to describe interactions between species where random fluctuations are relevant. The master equation provides the evolution of the probability distribution across the discrete state space consisting of vectors of population counts for each species. However, since its exact solution is often elusive, several analytical approximations have been proposed. The deterministic rate equation (DRE) gives a macroscopic approximation as a compact system of differential equations that estimate the average populations for each species, but it may be inaccurate in the case of nonlinear interaction dynamics. Here we propose finite state expansion (FSE), an analytical method mediating between the microscopic and the macroscopic interpretations of a stochastic reaction network by coupling the master equation dynamics of a chosen subset of the discrete state space with the mean population dynamics of the DRE. An algorithm translates a network into an expanded one where each discrete state is represented as a further distinct species. This translation exactly preserves the stochastic dynamics, but the DRE of the expanded network can be interpreted as a correction to the original one. The effectiveness of FSE is demonstrated in models that challenge state-of-the-art techniques due to intrinsic noise, multi-scale populations, and multi-stability., Comment: 33 pages, 9 figures

Motivation: Detailed mechanistic models of biological processes can pose significant challenges for analysis and parameter estimations due to the large number of equations used to track the dynamics of all distinct configurations in which each involved biochemical species can be found. Model reduction can help tame such complexity by providing a lower-dimensional model in which each macro-variable can be directly related to the original variables. Results: We present CLUE, an algorithm for exact model reduction of systems of polynomial differential equations by constrained linear lumping. It computes the smallest dimensional reduction as a linear mapping of the state space such that the reduced model preserves the dynamics of user-specified linear combinations of the original variables. Even though CLUE works with nonlinear differential equations, it is based on linear algebra tools, which makes it applicable to high-dimensional models. Using case studies from the literature, we show how CLUE can substantially lower model dimensionality and help extract biologically intelligible insights from the reduction. Availability: An implementation of the algorithm and relevant resources to replicate the experiments herein reported are freely available for download at https://github.com/pogudingleb/CLUE. Supplementary information: enclosed.

It is well known that building analytical performance models in practice is difficult because it requires a considerable degree of proficiency in the underlying mathematics. In this paper, we propose a machine-learning approach to derive performance models from data. We focus on queuing networks, and crucially exploit a deterministic approximation of their average dynamics in terms of a compact system of ordinary differential equations. We encode these equations into a recurrent neural network whose weights can be directly related to model parameters. This allows for an interpretable structure of the neural network, which can be trained from system measurements to yield a white-box parameterized model that can be used for prediction purposes such as what-if analyses and capacity planning. Using synthetic models as well as a real case study of a load-balancing system, we show the effectiveness of our technique in yielding models with high predictive power.

Principles of feedback control have been shown to naturally arise in biological systems and successfully applied to build synthetic circuits. In this work we consider Biochemical Reaction Networks (CRNs) as a paradigm for modelling biochemical systems and provide the first implementation of a derivative component in CRNs. That is, given an input signal represented by the concentration level of some species, we build a CRN that produces as output the concentration of two species whose difference is the derivative of the input signal. By relying on this component, we present a CRN implementation of a feedback control loop with Proportional-Integral-Derivative (PID) controller and apply the resulting control architecture to regulate the protein expression in a microRNA regulated gene expression model., Comment: 8 Pages, 4 figures, Submitted to CDC 2019

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.

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., Comment: Accepted for publication at QEST 2018

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/)

This volume contains the post-proceedings of the 14th International Workshop on Quantitative Aspects of Programming Languages and Systems (QAPL), held as a satellite workshop of ETAPS 2016 in Eindhoven, The Netherlands, on 2-3 April 2016.

Like with most large-scale systems, the evaluation of quantitative properties of collective adaptive systems is an important issue that crosscuts all its development stages, from design (in the case of engineered systems) to runtime monitoring and control. Unfortunately it is a difficult problem to tackle in general, due to the typically high computational cost involved in the analysis. This calls for the development of appropriate quantitative abstraction techniques that preserve most of the system's dynamical behaviour using a more compact representation. This paper focuses on models based on ordinary differential equations and reviews recent results where abstraction is achieved by aggregation of variables, reflecting on the shortcomings in the state of the art and setting out challenges for future research., Comment: In Proceedings FORECAST 2016, arXiv:1607.02001

This volume contains the proceedings of the Thirteenth Workshop on Quantitative Aspects of Programming Languages and Systems (QAPL 2015), held in London, UK, on 11 and 12 April, 2015. QAPL 2015 was a satellite event of the European Joint Conferences on Theory and Practice of Software (ETAPS) focussing on quantitative aspects of computation. The Program Committee of QAPL 2015 selected 8 regular papers and 2 presentation-only papers. The workshop programme included two QAPL keynote presentations by Catuscia Palamidessi (Inria/LIX, France) on "Quantitative Aspects of Privacy and Information Flow," and Holger Hermanns (Saarland University, Germany) on "Optimal Continuous Time Markov Decisions."

We present two quantitative behavioral equivalences over species of a chemical reaction network (CRN) with semantics based on ordinary differential equations. Forward CRN bisimulation identifies a partition where each equivalence class represents the exact sum of the concentrations of the species belonging to that class. Backward CRN bisimulation relates species that have the identical solutions at all time points when starting from the same initial conditions. Both notions can be checked using only CRN syntactical information, i.e., by inspection of the set of reactions. We provide a unified algorithm that computes the coarsest refinement up to our bisimulations in polynomial time. Further, we give algorithms to compute quotient CRNs induced by a bisimulation. As an application, we find significant reductions in a number of models of biological processes from the literature. In two cases we allow the analysis of benchmark models which would be otherwise intractable due to their memory requirements., Comment: Extended version of the CONCUR 2015 paper

We study aggregations for ordinary differential equations induced by fluid semantics for Markovian process algebra which can capture the dynamics of performance models and chemical reaction networks. Whilst previous work has required perfect symmetry for exact aggregation, we present approximate fluid lumpability, which makes nearby processes perfectly symmetric after a perturbation of their parameters. We prove that small perturbations yield nearby differential trajectories. Numerically, we show that many heterogeneous processes can be aggregated with negligible errors., Comment: In Proceedings QAPL 2014, arXiv:1406.1567

We consider Markov models of large-scale networks where nodes are characterized by their local behavior and by a mobility model over a two-dimensional lattice. By assuming random walk, we prove convergence to a system of partial differential equations (PDEs) whose size depends neither on the lattice size nor on the population of nodes. This provides a macroscopic view of the model which approximates discrete stochastic movements with continuous deterministic diffusions. We illustrate the practical applicability of this result by modeling a network of mobile nodes with on/off behavior performing file transfers with connectivity to 802.11 access points. By means of an empirical validation against discrete-event simulation we show high quality of the PDE approximation even for low populations and coarse lattices. In addition, we confirm the computational advantage in using the PDE limit over a traditional ordinary differential equation limit where the lattice is modeled discretely, yielding speed-ups of up to two orders of magnitude.