Algorithms for exact and approximate linear abstractions of polynomial continuous systems

Boreale, Michele

doi:10.1145/3178126.3178137

Cited by 9 publications

(3 citation statements)

References 27 publications

(65 reference statements)

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“…Obtaining bounds on the approximation error is another aspect that deserves further investigation. Some preliminary progress on these issues is reported in [11]. 9.2.…”

Section: Conclusion Further and Related Workmentioning

confidence: 99%

Algebra, Coalgebra, and Minimization in Polynomial Differential Equations

Boreale

2017

Lecture Notes in Computer Science

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We consider reasoning and minimization in systems of polynomial ordinary differential equations (ode's). The ring of multivariate polynomials is employed as a syntax for denoting system behaviours. We endow this set with a transition system structure based on the concept of Lie derivative, thus inducing a notion of L-bisimulation. We prove that two states (variables) are L-bisimilar if and only if they correspond to the same solution in the ode's system. We then characterize L-bisimilarity algebraically, in terms of certain ideals in the polynomial ring that are invariant under Lie-derivation. This characterization allows us to develop a complete algorithm, based on building an ascending chain of ideals, for computing the largest L-bisimulation containing all valid identities that are instances of a user-specified template. A specific largest L-bisimulation can be used to build a reduced system of ode's, equivalent to the original one, but minimal among all those obtainable by linear aggregation of the original equations. A computationally less demanding approximate reduction and linearization technique is also proposed.

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“…Obtaining bounds on the approximation error is another aspect that deserves further investigation. Some preliminary progress on these issues is reported in [11]. 9.2.…”

Section: Conclusion Further and Related Workmentioning

confidence: 99%

Algebra, Coalgebra, and Minimization in Polynomial Differential Equations

Boreale

2017

Lecture Notes in Computer Science

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“…Analogously to the classical non-deterministic setting based on labeled transition systems, bisimulations for ODEs have been proposed for the related purposes of model comparison and model reduction [4], [5], [6], [7], with applications that have transcended computer science (e.g., [8], [9]). "Lumping" refers to a class of methods to reduce a system of ODEs onto lower-dimensional space such that each variable in the reduced ODE system represents an appropriate mapping of the set of original variables [10], [11].…”

Section: Introductionmentioning

confidence: 99%

Efficient Local Computation of Differential Bisimulations via Coupling and Up-to Methods

Bacci

Larsen

et al. 2021

2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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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 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.

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“…However, for purposes of verification of safety properties, it often suffices to compute an over-approximation of the reachable set of states-if the over-approximation does not intersect the set of bad states, then the original system will never reach a bad state. So far, the existing approaches are mainly based on approximate reachable set computations [7][8][9] and abstraction [10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

A New Approach to Nonlinear Invariants for Hybrid Systems Based on the Citing Instances Method

2020

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In generating invariants for hybrid systems, a main source of intractability is that transition relations are first-order assertions over current-state variables and next-state variables, which doubles the number of system variables and introduces many more free variables. The more variables, the less tractability and, hence, solving the algebraic constraints on complete inductive conditions by a comprehensive Gröbner basis is very expensive. To address this issue, this paper presents a new, complete method, called the Citing Instances Method (CIM), which can eliminate the free variables and directly solve for the complete inductive conditions. An instance means the verification of a proposition after instantiating free variables to numbers. A lattice array is a key notion in this paper, which is essentially a finite set of instances. Verifying that a proposition holds over a Lattice Array suffices to prove that the proposition holds in general; this interesting feature inspires us to present CIM. On one hand, instead of computing a comprehensive Gröbner basis, CIM uses a Lattice Array to generate the constraints in parallel. On the other hand, we can make a clever use of the parallelism of CIM to start with some constraint equations which can be solved easily, in order to determine some parameters in an early state. These solved parameters benefit the solution of the rest of the constraint equations; this process is similar to the domino effect. Therefore, the constraint-solving tractability of the proposed method is strong. We show that some existing approaches are only special cases of our method. Moreover, it turns out CIM is more efficient than existing approaches under parallel circumstances. Some examples are presented to illustrate the practicality of our method.

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Algorithms for exact and approximate linear abstractions of polynomial continuous systems

Cited by 9 publications

References 27 publications

Algebra, Coalgebra, and Minimization in Polynomial Differential Equations

Algebra, Coalgebra, and Minimization in Polynomial Differential Equations

Efficient Local Computation of Differential Bisimulations via Coupling and Up-to Methods

A New Approach to Nonlinear Invariants for Hybrid Systems Based on the Citing Instances Method

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