Symbolic computation of differential equivalences

Abstract

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 high-level 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 high-level 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.

Comparing chemical reaction networks: A categorical and algorithmic perspective

“…This notion, called emulation, has been recently introduced in [2], but no procedure to compute it was provided. Later, emulation has been related in [5] to a backward differential equivalence (BDE). This is an equivalence relation over the variables of an ODE system such that equivalent variables have the same ODE solutions whenever initialized equally.…”

“…A partition-refinement algorithm to compute the largest BDE (i.e., the unique BDE relation that contains any other BDE relation) has been provided in [5]. It cannot be used, however, to find emulations.…”

“…Obviously, however, this is an intractable task if not for trivial models. We avoid this naive exploration by exploiting a novel geometric interpretation of BDE and, interestingly, using the coarsest BDE refinement algorithm [4][5][6] as an inner step. Our key insight is that a BDE induces a linear space that can be spanned by an appropriate subset of the generalized eigenvectors for the Jacobian matrix.…”

“…In addition, we use BDE in this paper because the algorithm for computing all BDE partitions, as discussed, works for more general ODEs for which the coarsestrefinement algorithm of [5] can be used.…”

“…Section 2 sets the scene by providing the previously established notions of morphism, emulation and BDE [2,5]. Section 3 first introduces the notion of flux morphism and then generalizes all major results of [2,5].…”