Dynamic-Epistemic reasoning on distributed systems

We propose a new logic designed for modelling and reasoning about information flow and information exchange between spatially located (but potentially mobile), interconnected agents witnessing a distributed computation. This is a major problem in the field of distributed systems, covering many different issues, with potential applications from Computer Science and Economy to Chemistry and Systems Biology. Underpinning on the dual algebraical-coalgebraical characteristics of process calculi, we design a decidable and completely axiomatizad logic that combines the processalgebraical/ equational and the modal/coequational features and is developed for process-algebraical semantics. The construction is done by mixing operators from dynamic and epistemic logics with operators from spatial logics for distributed and mobile systems. This is the preliminary version of a paper that will appear in Proceedings of the second Conference on Algebra and Coalgebra in Computer Science (CALCO2007), LNCS 4624, Springer, 2007. The original publication is available at www.springerlink.co

A Process Algebraical Approach to Modelling Compartmentalized Biological Systems

This paper introduces Protein Calculus, a special modeling language designed for encoding and calculating the behaviors of compartmentilized biological systems. The formalism combines, in a unified framework, two successful computational paradigms - process algebras and membrane systems. The goal of Protein Calculus is to provide a formal tool for transforming collected information from in vivo experiments into coded definition of the different types of proteins, complexes of proteins, and membrane-organized systems of such entities. Using this encoded information as input, our calculus computes, in silico, the possible behaviors of a living system. This is the preliminary version of a paper that was published in Proceedings of International Conference of Computational Methods in Sciences and Engineering (ICCMSE), American Institute of Physics, AIP Proceedings, N 2: 642-646, 2007 (http://scitation.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=963&Issue=2)

Model Checking Dynamic-Epistemic Spatial Logic

In this paper we focus on Dynamic Spatial Logic, the extension of Hennessy-Milner logic with the parallel operator. We develop a sound complete Hilbert-style axiomatic system for it comprehending the behavior of spatial operators in relation with dynamic/temporal ones. Underpining on a new congruence we define over the class of processes - the structural bisimulation - we prove the finite model property for this logic that provides the decidability for satisfiability, validity and model checking against process semantics. Eventualy we propose algorithms for validity, satisfiability and model checking

Probabilistic logics based on Riesz spaces

We introduce a novel real-valued endogenous logic for expressing properties of probabilistic transition systems called Riesz modal logic. The design of the syntax and semantics of this logic is directly inspired by the theory of Riesz spaces, a mature field of mathematics at the intersection of universal algebra and functional analysis. By using powerful results from this theory, we develop the duality theory of the Riesz modal logic in the form of an algebra-to-coalgebra correspondence. This has a number of consequences including: a sound and complete axiomatization, the proof that the logic characterizes probabilistic bisimulation and other convenient results such as completion theorems. This work is intended to be the basis for subsequent research on extensions of Riesz modal logic with fixed-point operators

Probabilistic Mu-Calculus: Decidability and Complete Axiomatization

We introduce a version of the probabilistic mu-calculus (PMC) built on top of a probabilistic modal logic that allows encoding n-ary inequational conditions on transition probabilities. PMC extends previously studied calculi and we prove that, despite its expressiveness, it enjoys a series of good meta-properties. Firstly, we prove the decidability of satisfiability checking by establishing the small model property. An algorithm for deciding the satisfiability problem is developed. As a second major result, we provide a complete axiomatization for the alternation-free fragment of PMC. The completeness proof is innovative in many aspects combining various techniques from topology and model theory

Timed Comparisons of Semi-Markov Processes

Semi-Markov processes are Markovian processes in which the firing time of the transitions is modelled by probabilistic distributions over positive reals interpreted as the probability of firing a transition at a certain moment in time. In this paper we consider the trace-based semantics of semi-Markov processes, and investigate the question of how to compare two semi-Markov processes with respect to their time-dependent behaviour. To this end, we introduce the relation of being "faster than" between processes and study its algorithmic complexity. Through a connection to probabilistic automata we obtain hardness results showing in particular that this relation is undecidable. However, we present an additive approximation algorithm for a time-bounded variant of the faster-than problem over semi-Markov processes with slow residence-time functions, and a coNP algorithm for the exact faster-than problem over unambiguous semi-Markov processes

Fixed-Points for Quantitative Equational Logics

We develop a fixed-point extension of quantitative equational logic and give semantics in one-bounded complete quantitative algebras. Unlike previous related work about fixed-points in metric spaces, we are working with the notion of approximate equality rather than exact equality. The result is a novel theory of fixed points which can not only provide solutions to the traditional fixed-point equations but we can also define the rate of convergence to the fixed point. We show that such a theory is the quantitative analogue of a Conway theory and also of an iteration theory; and it reflects the metric coinduction principle. We study the Bellman equation for a Markov decision process as an illustrative example

Partial Knowledge in Membrane Systems: A Logical Approach

Abstract. We propose a logic for specifying and proving properties of membrane systems. The main idea is to approach a membrane system by using the “point of view ” of an external observer. Observers (as epis-temic agents) accumulate their knowledge from the partial information they collect by observing subparts of the system and by applying logical reasoning to this information. We provide a formal framework to com-bine and interpret distributed knowledge in order to recover the complete knowledge about a membrane system. The proposed logic can be used to model biological situations where information concerning parts of the biological system is missing or incomplete.

Computing Probabilistic Bisimilarity Distances for Probabilistic Automata

The probabilistic bisimilarity distance of Deng et al. has been proposed as a robust quantitative generalization of Segala and Lynch's probabilistic bisimilarity for probabilistic automata. In this paper, we present a characterization of the bisimilarity distance as the solution of a simple stochastic game. The characterization gives us an algorithm to compute the distances by applying Condon's simple policy iteration on these games. The correctness of Condon's approach, however, relies on the assumption that the games are stopping. Our games may be non-stopping in general, yet we are able to prove termination for this extended class of games. Already other algorithms have been proposed in the literature to compute these distances, with complexity in $\textbf{UP} \cap \textbf{coUP}$ and \textbf{PPAD}. Despite the theoretical relevance, these algorithms are inefficient in practice. To the best of our knowledge, our algorithm is the first practical solution. The characterization of the probabilistic bisimilarity distance mentioned above crucially uses a dual presentation of the Hausdorff distance due to M\'emoli. As an additional contribution, in this paper we show that M\'emoli's result can be used also to prove that the bisimilarity distance bounds the difference in the maximal (or minimal) probability of two states to satisfying arbitrary $\omega$-regular properties, expressed, eg., as LTL formulas

Propositional Logics for the Lawvere Quantale

Lawvere showed that generalised metric spaces are categories enriched over $[0, \infty]$, the quantale of the positive extended reals. The statement of enrichment is a quantitative analogue of being a preorder. Towards seeking a logic for quantitative metric reasoning, we investigate three $[0,\infty]$-valued propositional logics over the Lawvere quantale. The basic logical connectives shared by all three logics are those that can be interpreted in any quantale, viz finite conjunctions and disjunctions, tensor (addition for the Lawvere quantale) and linear implication (here a truncated subtraction); to these we add, in turn, the constant $1$ to express integer values, and scalar multiplication by a non-negative real to express general affine combinations. Quantitative equational logic can be interpreted in the third logic if we allow inference systems instead of axiomatic systems. For each of these logics we develop a natural deduction system which we prove to be decidably complete w.r.t. the quantale-valued semantics. The heart of the completeness proof makes use of the Motzkin transposition theorem. Consistency is also decidable; the proof makes use of Fourier-Motzkin elimination of linear inequalities. Strong completeness does not hold in general, even (as is known) for theories over finitely-many propositional variables; indeed even an approximate form of strong completeness in the sense of Pavelka or Ben Yaacov -- provability up to arbitrary precision -- does not hold. However, we can show it for theories axiomatized by a (not necessarily finite) set of judgements in normal form over a finite set of propositional variables when we restrict to models that do not map variables to $\infty$; the proof uses Hurwicz's general form of the Farkas' Lemma