Abstract Interpretation for Probabilistic Termination of Biological Systems

In a previous paper the authors applied the Abstract Interpretation approach for approximating the probabilistic semantics of biological systems, modeled specifically using the Chemical Ground Form calculus. The methodology is based on the idea of representing a set of experiments, which differ only for the initial concentrations, by abstracting the multiplicity of reagents present in a solution, using intervals. In this paper, we refine the approach in order to address probabilistic termination properties. More in details, we introduce a refinement of the abstract LTS semantics and we abstract the probabilistic semantics using a variant of Interval Markov Chains. The abstract probabilistic model safely approximates a set of concrete experiments and reports conservative lower and upper bounds for probabilistic termination

An Analysis for Proving Probabilistic Termination of Biological Systems

In this paper we apply the abstract interpretation approach for approximating the behavior of biological systems, modeled specifically using the Chemical Ground Form calculus, a simple stochastic calculus rich enough to model the dynamics of biochemical reactions. The analysis is based on the idea of representing a set of experiments, which differ only for the initial concentrations, by abstracting the multiplicity of reagents present in a solution, using intervals. For abstracting the probabilistic semantics, modeled as a Discrete-Time Markov Chain, we use a variant of Interval Markov Chains, where probabilistic and non-deterministic steps are combined together. The abstract probabilistic semantics is systematically derived from an abstract Labeled Transition System. The abstract probabilistic model safely approximates the set of concrete experiments and reports conservative lower and upper bounds for probabilistic termination

Code obfuscation against abstraction refinement attacks

Code protection technologies require anti reverse engineering transformations to obfuscate programs in such a way that tools and methods for program analysis become ineffective. We introduce the concept of model deformation inducing an effective code obfuscation against attacks performed by abstract model checking. This means complicating the model in such a way a high number of spurious traces are generated in any formal verification of the property to disclose about the system under attack.We transform the program model in order to make the removal of spurious counterexamples by abstraction refinement maximally inefficient. Because our approach is intended to defeat the fundamental abstraction refinement strategy, we are independent from the specific attack carried out by abstract model checking. A measure of the quality of the obfuscation obtained by model deformation is given together with a corresponding best obfuscation strategy for abstract model checking based on partition refinement

Causal static analysis for Brane Calculi

We present here a static analysis, based on Abstract Interpretation, obtained by defining an abstract version of the causal semantics for the Mate/Bud/Drip (MBD) version of Brane Calculi, proposed by Busi. Our analysis statically approximates the dynamic behaviour of MBD systems. More precisely, the analysis is able to describe the essential behaviour of the represented membranes, in terms of their possible interactions. Furthermore, our analysis is able to statically capture the possible causal dependencies among interactions, whose determination can be exploited to better understand the modelled biological phenomena. Finally, we apply our analysis to an abstract specification of the receptor-mediated endocytosis mechanism

Generalized contexts for reaction systems: definition and study of dynamic causalities

Reaction systems are a qualitative formalism for the modelling of systems of biochemical reactions. In their original formulation, a reaction system executes in an environment (or context) that can supply it with new objects at each evolution step. The context drives the behaviour of a reaction system: it can provide different inputs to the system that can lead to different behaviours. In order to more faithfully deal with open systems, in this paper we propose a more powerful notion of context having not only the capability to provide objects, but also to absorb (or remove) objects at each evolution step. For such reaction systems with generalized context we investigate properties of dynamic causality by revising the previously proposed concept of formula based predictor. A formula based predictor is a Boolean formula characterising all contexts that lead to the production of a certain object after a given number of steps. In this paper, we revise the theory of formula based predictors in order to deal with reaction systems executed in a context of the new kind. As applications, we show an example of interaction between biochemical pathways and a reaction system modelling cell metabolism and respiration

Characterization and computation of ancestors in reaction systems

AbstractIn reaction systems, preimages and nth ancestors are sets of reactants leading to the production of a target set of products in either 1 or n steps, respectively. Many computational problems on preimages and ancestors, such as finding all minimum-cardinality nth ancestors, computing their size or counting them, are intractable. In this paper, we characterize all nth ancestors using a Boolean formula that can be computed in polynomial time. Once simplified, this formula can be exploited to easily solve all preimage and ancestor problems. This allows us to directly relate the difficulty of ancestor problems to the cost of the simplification so that new insights into computational complexity investigations can be achieved. In particular, we focus on two problems: (i) deciding whether a preimage/nth ancestor exists and (ii) finding a preimage/nth ancestor of minimal size. Our approach is constructive, it aims at finding classes of reactions systems for which the ancestor problems can be solved in polynomial time, in exact or approximate way

Investigating dynamic causalities in reaction systems

Reaction systems are a qualitative formalism for modeling systems of biochemical reactions characterized by the non-permanency of the elements: molecules disappear if not produced by any enabled reaction. Moreover, reaction systems execute in an environment that provides new molecules at each step. Brijder, Ehrenfeucht and Rozenberg investigated dynamic causalities in reaction systems by introducing the idea of predictors. A predictor of a molecule s, for a given n, is the set of molecules to be observed in the environment in order to determine whether s is produced or not by the system at step n. In this paper, we continue the investigation on dynamic causalities by defining an abstract interpretation framework containing three different notions of predictor: Formula based predictors, that is a propositional logic formula that precisely characterizes environments that lead to the production of s after n steps; Multi-step based predictors, that consist of n sets of molecules to be observed in the environment, one for each step; and Set based predictors, that are those proposed by Brijder, Ehrenfeucht and Rozenberg, and consist of a unique set of molecules to be observed in all steps. For each kind of predictor we define an effective operator that allows predictors to be computed for any molecule s and number of steps n. The abstract interpretation framework allows us to compare the three notions of predictor in terms of precision, to relate the three defined operators and to compute minimal predictors. We also discuss a generalization of this approach that allows predictors to be defined independently of the value of n, and a tabling approach for the practical use of predictors on reaction systems models. As an application, we use predictors, generalization and tabling to give theoretical grounds to previously obtained results on a model of gene regulation