Completeness of Flat Coalgebraic Fixpoint Logics

Modal fixpoint logics traditionally play a central role in computer science, in particular in artificial intelligence and concurrency. The mu-calculus and its relatives are among the most expressive logics of this type. However, popular fixpoint logics tend to trade expressivity for simplicity and readability, and in fact often live within the single variable fragment of the mu-calculus. The family of such flat fixpoint logics includes, e.g., LTL, CTL, and the logic of common knowledge. Extending this notion to the generic semantic framework of coalgebraic logic enables covering a wide range of logics beyond the standard mu-calculus including, e.g., flat fragments of the graded mu-calculus and the alternating-time mu-calculus (such as alternating-time temporal logic ATL), as well as probabilistic and monotone fixpoint logics. We give a generic proof of completeness of the Kozen-Park axiomatization for such flat coalgebraic fixpoint logics.Comment: Short version appeared in Proc. 21st International Conference on Concurrency Theory, CONCUR 2010, Vol. 6269 of Lecture Notes in Computer Science, Springer, 2010, pp. 524-53

Probabilistic Interval Temporal Logic and Duration Calculus with Infinite Intervals: Complete Proof Systems

The paper presents probabilistic extensions of interval temporal logic (ITL) and duration calculus (DC) with infinite intervals and complete Hilbert-style proof systems for them. The completeness results are a strong completeness theorem for the system of probabilistic ITL with respect to an abstract semantics and a relative completeness theorem for the system of probabilistic DC with respect to real-time semantics. The proposed systems subsume probabilistic real-time DC as known from the literature. A correspondence between the proposed systems and a system of probabilistic interval temporal logic with finite intervals and expanding modalities is established too.Comment: 43 page

Parametric LTL on Markov Chains

This paper is concerned with the verification of finite Markov chains against parametrized LTL (pLTL) formulas. In pLTL, the until-modality is equipped with a bound that contains variables; e.g., $\Diamond_{\le x}\ \varphi$ asserts that $\varphi$ holds within $x$ time steps, where $x$ is a variable on natural numbers. The central problem studied in this paper is to determine the set of parameter valuations $V_{\prec p} (\varphi)$ for which the probability to satisfy pLTL-formula $\varphi$ in a Markov chain meets a given threshold $\prec p$, where $\prec$ is a comparison on reals and $p$ a probability. As for pLTL determining the emptiness of $V_{> 0}(\varphi)$ is undecidable, we consider several logic fragments. We consider parametric reachability properties, a sub-logic of pLTL restricted to next and $\Diamond_{\le x}$, parametric B\"uchi properties and finally, a maximal subclass of pLTL for which emptiness of $V_{> 0}(\varphi)$ is decidable.Comment: TCS Track B 201

Modular Verification of Biological Systems

Systems of interest in systems biology (such as metabolic pathways, signalling pathways and gene regulatory networks) often consist of a huge number of components interacting in different ways, thus exhibiting very complex behaviours. In biology, such behaviours are usually explored by means of simulation techniques applied to models defined on the basis of system observation and of hypotheses on its functioning. Model checking has also been recently applied to the analysis of biological systems. This analysis technique typically relies on a state space representation whose size, unfortunately, makes the analysis often intractable for realistic models. A method for trying to avoid the state space explosion problem is to consider a decomposition of the system, and to apply a modular verification technique. In particular, properties to be verified often concern only a small portion of the modelled system rather than the system as a whole. Hence, for each property it would be useful to be able to isolate a minimal fragment of the model that is necessary to verify such a property. In this thesis we introduce a modular verification technique in which the system of interest is described by means of an automata-based formalism, called sync-programs, that supports modular construction. Our modular verification technique is based on results of Grumberg et al.~and on their application to the theory of concurrent systems proposed by Attie and Emerson. In particular, we adapt Attie and Emerson's approach to deal with biological systems by allowing automata to synchronise by performing transitions simultaneously. Modular verification allows qualitative aspects of systems to be analysed with the guarantee that properties proved to hold in a suitable model fragment also hold in the whole model. The correctness of the verification technique is proved. The class of properties preserved is ACTL$^{-}$, the universal fragment of temporal logic CTL. The preservation holds only for positive answers and negative answers are not necessarily preserved. In order to verify properties we use the NuSMV model checker, which is a well-established and efficient instrument. We provide a formal translation of sync-programs to simpler automata, which can be given as input to NuSMV. We prove the correspondence of the verification problems. We show the application of our verification technique in some biological case studies. We compare the time required to verify the property on the whole model with the time needed to verify the same property by only considering those modules which are involved in the behaviour of the system related to the property. In order to handle modelling and verification of more realistic biological scenarios, we propose also a dynamic version of our formalism. It allows entities to be created dynamically, in particular by other already running entities, as it often happens in biological systems. Moreover, multiple copies of the same entities can be present at the same time in a system. We show a correspondence of our model with Petri Nets. This has a consequence that tools developed for Petri Nets could be used also for dynamic sync-programs. Modular verification allows properties expressed as DACTL- formulae (dynamic version of ACTL-) to be verified on a portion of the model. The results of analysis of the case study of the MAP kinase cascade activated by surface and internalised EGF receptors, which consists of 143 species and 80 reactions, suggest applicability and scalability of the approach. The results raise the prospect of rendering tractable problems that are currently intractable in the verification of biological systems. In addition, we expect that the techniques developed in the thesis could be applied with profit not only to models of biological systems, but more generally to models of concurrent systems