Moment Semantics for Reversible Rule-Based Systems

Danos, Vincent; Heindel, Tobias; Honorato-Zimmer, Ricardo; Stucki, Sandro

doi:10.1007/978-3-319-20860-2_1

Cited by 8 publications

(20 citation statements)

References 36 publications

(44 reference statements)

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“…This open source platform offers a variety of high-performance simulation algorithms (for CTMCs based upon Kappa rewriting rules) as well as several variants of static analysis tools to analyze and verify biochemical models [12]. In view of the present paper, it is interesting to note that since the start of the Kappa development, the simulation-based algorithms have been augmented by differential semantics modules aimed at deriving ODE systems for the evolution of pattern-count observable average values [14,15,17,29]. In this section, we will experiment with a (re-)encoding of Kappa in terms of typed undirected graphs with certain structural constraints that permits to express such moment statistics ODEs via our general rule-algebraic stochastic mechanics formalism.…”

Section: Application Scenario 1: Biochemistry With Kappamentioning

confidence: 99%

Rewriting Theory for the Life Sciences: A Unifying Theory of CTMC Semantics

Behr

Krivine

2020

Lecture Notes in Computer Science

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The Kappa biochemistry and the MØD organo-chemistry frameworks are amongst the most intensely developed applications of rewriting theoretical methods in the life sciences to date. A typical feature of these types of rewriting theories is the necessity to implement certain structural constraints on the objects to be rewritten (a protein is empirically found to have a certain signature of sites, a carbon atom can form at most four bonds,. . .). In this paper, we contribute to the theoretical foundations of these types of rewriting theory a number of conceptual and technical developments that permit to implement a universal theory of continuous-time Markov chains (CTMCs) for stochastic rewriting systems. Our core mathematical concepts are a novel rule algebra construction for the relevant setting of rewriting rules with conditions, both in Double-and in Sesqui-Pushout semantics, augmented by a suitable stochastic mechanics formalism extension that permits to derive dynamical evolution equations for pattern-counting statistics.

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Section: Application Scenario 1: Biochemistry With Kappamentioning

confidence: 99%

Rewriting Theory for the Life Sciences: A Unifying Theory of CTMC Semantics

Behr

Krivine

2020

Lecture Notes in Computer Science

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“…For the convenience of the reader, we reproduce from Ref. [16] our basic definitions for the category Grph of directed multigraphs. Next, we briefly summarise the SPO approach and its stochastic semantics [31].…”

Section: Stochastic Graph Rewritingmentioning

confidence: 99%

“…4 is irreversible. In previous work, we have shown how to derive rate equations for graph transformation systems with only reversible derivations [13,15,16]. In §3, we overcome this limitation, giving a procedure that extends to the irreversible case.…”

Section: Graph Rewritingmentioning

confidence: 99%

“…The main finding is summarised in a single equation 15which we call the generalised rate equation for graphs (GREG). In previous work, we have published a solution to this problem for the subclass of reversible graph rewriting systems [15,16]. The solution presented here is valid for any such system, reversible or not.…”

Section: Introductionmentioning

confidence: 99%

“…As in Ref. [16], the somewhat informal approach of Ref. [15] is replaced with precise category-theoretical language with which the backward modularity Lemma finds a concise and natural formulation.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Rate Equations for Graphs

Danos¹,

Heindel²,

Honorato-Zimmer³

et al. 2020

Computational Methods in Systems Biology

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In this paper, we combine ideas from two different scientific traditions: 1) graph transformation systems (GTSs) stemming from the theory of formal languages and concurrency, and 2) mean field approximations (MFAs), a collection of approximation techniques ubiquitous in the study of complex dynamics. Using existing tools from algebraic graph rewriting, as well as new ones, we build a framework which generates rate equations for stochastic GTSs and from which one can derive MFAs of any order (no longer limited to the humanly computable). The procedure for deriving rate equations and their approximations can be automated. An implementation and example models are available online at https://rhz.github.io/fragger. We apply our techniques and tools to derive an expression for the mean velocity of a two-legged walker protein on DNA.

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Computing Continuous-Time Markov Chains as Transformers of Unbounded Observables

Danos

Heindel

Garnier

et al. 2017

Lecture Notes in Computer Science

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This paper provides broad sufficient conditions for the computability of time-dependent averages of stochastic processes of the form f (Xt) where Xt is a continuous-time Markov chain (CTMC), and f is a real-valued function (aka an observable). We consider chains with values in a countable state space S, and possibly unbounded f s. Observables are seen as generalised predicates on S and chains are interpreted as transformers of such generalised predicates, mapping each observable f to a new observable Ptf defined as (Ptf)(x) = Ex(f (Xt)), which represents the mean value of f at time time t as a function of the initial state x. We obtain three results. First, the well-definedness of this operator interpretation is obtained for a large class of chains and observables by restricting Pt to judiciously chosen rescalings of the basic Banach space C0(S) of S-indexed sequences which vanish at infinity. We prove, under appropriate assumptions, that the restricted family Pt forms a strongly continuous operator semigroup (equivalently the time evolution map t → Pt is continuous w.r.t. the usual topology on bounded operators). The computability of the time evolution map follows by generic arguments of constructive analysis. A key point here is that the assumptions are flexible enough to accommodate unbounded observables, and we give explicit examples of such using stochastic Petri nets and stochastic string rewriting. Thirdly, we show that if the rate matrix (aka the q-matrix) of the CTMC is locally algebraic on a subspace containing f , the time evolution of projections t → (Ptf)(x) is PTIME computable for each x. These results provide a functional analytic alternative to Monte Carlo simulation as test bed for mean-field approximations, moment closure, and similar techniques that are fast, but lack absolute error guarantees.

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Moment Semantics for Reversible Rule-Based Systems

Cited by 8 publications

References 36 publications

Rewriting Theory for the Life Sciences: A Unifying Theory of CTMC Semantics

Rewriting Theory for the Life Sciences: A Unifying Theory of CTMC Semantics

Rate Equations for Graphs

Computing Continuous-Time Markov Chains as Transformers of Unbounded Observables

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