Bisimulation Research Articles

Fibrational bisimulations and quantitative reasoning: Extended version

Abstract Bisimulation and bisimilarity are fundamental notions in comparing state-based systems. Their extensions to a variety of systems have been actively pursued in recent years, a notable direction being quantitative extensions. In this paper we enhance a categorical framework for such extended (bi)simulation notions. We use coalgebras as system models and fibrations for organizing predicates—following the seminal work by Hermida and Jacobs. Endofunctor liftings are crucial predicate-forming ingredients; the first contribution of this work is to extend several extant lifting techniques from particular fibrations to $\textbf {CLat}_\wedge $-fibrations over $\textbf {Set}$. The second contribution of this work is to introduce endolifting morphisms as a mechanism for comparing predicates between fibrations. We apply these techniques by deriving some known properties of the Hausdorff pseudometric and approximate bisimulation in control theory.

Limited approximate bisimulations and the corresponding rough approximations

To measure the similarity of nodes in the neighboring subgraphs, Milner introduced the notion of k-limited bisimilarity. Recently, as a weaker version of k-limited bisimilarity, the notion of k-limited similarity was proposed and applied to graph pattern matching. (Bi)simulations have been widely used in comparing the behavior of fuzzy transition systems. In order to study the (bi)simulation semantics of labeled fuzzy transition systems in the residuated lattice-valued logic setting, we introduce an extension of labeled approximation spaces, called the quantitative fuzzy approximation spaces (QFASs), whose labels are equipped with a residuated lattice-valued equality relation. In a QFAS, we define a new notion of limited approximate similarity, to quantify to what extent one state is simulated by another in the neighboring subgraphs, and provide its properties. Based on the new notion, we give a definition of limited approximate simulation and discuss its properties. Then we introduce an ordered pair of relations, one on the state (vertex) set (limited approximate simulation) and one on the edge (transition) set induced by the relation on state set, called VE limited approximate simulation in this paper. We also present a new notion of limited approximate bisimilarity in a QFAS, to quantify to what extent two states are similar in the neighboring subgraphs, and give its properties. One main contribution of the paper is to give a condition for two states to be limited approximate bisimilar and investigate the degree of similarity between two states in a QFAS. Finally, we discuss the relationships between the rough approximations based on the underlying crisp relation induced by underlying labeled fuzzy relation and the rough approximations based on limited approximate bisimilarity.

Opacity of Nondeterministic Transition Systems: A (Bi)Simulation Relation Approach

In this paper, we propose several opacity-preserving (bi)simulation relations for general nondeterministic transition systems (NTS) in terms of initial-state opacity, current-state opacity, K-step opacity, and infinite-step opacity. We also show how one can leverage quotient construction to compute such relations. In addition, we use a two-way observer method to verify opacity of nondeterministic finite transition systems (NFTSs). As a result, although the verification of opacity for infinite NTSs is generally undecidable, if one can find such an opacity-preserving relation from an infinite NTS to an NFTS, the (lack of) opacity of the NTS can be easily verified over the NFTS which is decidable.

The metric linear-time branching-time spectrum on nondeterministic probabilistic processes

Behavioral equivalences were introduced as a simple and elegant proof methodology for establishing whether the behavior of two processes cannot be distinguished by an external observer. The knowledge of observers usually depends on the observations that they can make on process behavior. Furthermore, the combination of nondeterminism and probability in concurrent systems leads to several interpretations of process behavior. Clearly, different kinds of observations as well as different interpretations lead to different kinds of behavioral relations, such as (bi)simulations, traces and testing. If we restrict our attention to linear properties only, we can identify three main approaches to trace and testing semantics: the trace distributions, the trace-by-trace and the extremal probabilities approaches. In this paper, we propose novel notions of behavioral metrics that are based on the three classic approaches above, and that can be used to measure the disparities in the linear behavior of processes with respect to trace and testing semantics. We study the properties of these metrics, like compositionality (expressed in terms of the non-expansiveness property), and we compare their expressive powers. More precisely, we compare them also to (bi)simulation metrics, thus obtaining the first metric linear time – branching time spectrum.

Logical characterization of branching metrics for nondeterministic probabilistic transition systems

In this paper we propose a logical characterization of (bi)simulation metrics obtained by a probabilistic variant of Hennessy-Milner logic enriched with variables, whose semantics is defined following the equational μ-calculus approach. Our characterization is based on the novel notions of mimicking formulae and syntactical distance on formulae. The former ones are the quantitative analogous to characteristic formulae. The latter is a 1-bounded pseudometric on formulae measuring their syntactical disparities. The characterization is obtained by showing that the (bi)simulation distance between processes corresponds to the syntactical distance between their mimicking formulae.We also discuss the expressive power of mimicking formulae with respect to probabilistic (bi)simulations. We show that two processes are bisimilar if and only if their mimicking formulae are syntactically equivalent. Moreover, we obtain that mimicking formulae of processes coincide with their characteristic formulae for ready simulation and that negation free mimicking formulae coincide with the characteristic formulae for simulation.

Multi-valued bisimulation quotienting algorithms1

Multi-valued bisimulation quotienting algorithms1

Bisimulations of Boolean Control Networks

A framework for analyzing bisimulation relations of Boolean control networks (BCNs) is set up in this paper. Bisimulation relations are natural objectives in control systems theory: A bisimulation relation between a pair of control systems defines a relation on their state spaces or state sets explaining how a trajectory or transition of one system can be paired with a trajectory or transition of the other system, and vice versa. The paper first formalizes the notion of (bi)simulation in BCNs and presents detailed results characterizing (bi)simulation relations for BCNs. Then, as an application of the notion of bisimulation, it considers the propagation of the fundamental properties of (macro)controllability and stabilizability through a bisimulation relation. It thereby suggests the possibility that certain control properties of a BCN can be inferred by analyzing a potentially simpler BCN. The analysis developed in this paper is based on the semitensor product approach, which gives algorithms that have exponential time complexity. A question that naturally arises is if it is possible to check bisimulation relations for BCNs in polynomial time. This question is answered in the negative, by proving that the problem of deciding whether a relation between BCNs is a bisimulation relation is NP-hard.

Nonlinear norm-observability and simulation of control systems

The (bi)simulation relation has recently been attracting growing interest in the study of nonlinear control systems, in the hope that through such a relation, the behaviors and properties of a nonlinear system can be inferred from those of another system which is easier to handle. In this paper, we consider the propagation of the property of nonlinear norm-observability through a simulation relation. Given two control systems that are related by a graph simulation relation, we derive conditions under which the norm-observability of the simulating system implies the norm-observability of the simulated system. The obtained results are given in terms of set-valued functions. Several examples are included to illustrate various applications of our results.

Towards a bisimulation theory for open synchronized networks of automata

本文为分布式语言提出一个表达能力较强的模型-同步自动机的参数化网络。 通过定义开放式自动机给出了此模型的行为语义, 并在此基础上给出了一种特殊的等价关系--规范假设互模拟等价。 我们讨论了这种等价关系在规范假设条件下的可复合性和可判定性。

A Semantical Analysis of Second-Order Propositional Modal Logic

This paper is aimed as a contribution to the use of formal modal languages in Artificial Intelligence. We introduce a multi-modal version of Second-order Propositional Modal Logic (SOPML), an extension of modal logic with propositional quantification, and illustrate its usefulness as a specification language for knowledge representation as well as temporal and spatial reasoning. Then, we define novel notions of (bi)simulation and prove that these preserve the interpretation of SOPML formulas. Finally, we apply these results to assess the expressive power of SOPML.

Proving language inclusion and equivalence by coinduction

Language equivalence and inclusion can be checked coinductively by establishing a (bi)simulation on suitable deterministic automata. In this paper we present an enhancement of this technique called (bi)simulation-up-to. We give general conditions on language operations for which bisimulation-up-to is sound. These results are illustrated by a large number of examples, giving new proofs of classical results such as Arden's rule, and involving the regular operations of union, concatenation and Kleene star as well as language equations with complement and intersection, and shuffle (closure).

Model Theory of XPath on Data Trees. Part I: Bisimulation and Characterization

We investigate model theoretic properties of XPath with data (in)equality tests over the class of data trees, i.e., the class of trees where each node contains a label from a finite alphabet and a data value from an infinite domain. We provide notions of (bi)simulations for XPpath logics containing the child, parent, ancestor and descendant axes to navigate the tree. We show that these notions precisely characterize the equivalence relation associated with each logic. We study formula complexity measures consisting of the number of nested axes and nested subformulas in a formula; these notions are akin to the notion of quantifier rank in first-order logic. We show characterization results for fine grained notions of equivalence and (bi)simulation that take into account these complexity measures. We also prove that positive fragments of these logics correspond to the formulas preserved under (non-symmetric) simulations. We show that the logic including the child axis is equivalent to the fragment of first-order logic invariant under the corresponding notion of bisimulation. If upward navigation is allowed the characterization fails but a weaker result can still be established. These results hold both over the class of possibly infinite data trees and over the class of finite data trees. Besides their intrinsic theoretical value, we argue that bi-simulations are useful tools to prove (non)expressivity results for the logics studied here, and we substantiate this claim with examples.

The Psi-Calculi Workbench

Psi-calculi is a parametric framework for extensions of the pi-calculus with arbitrary data and logic. All instances of the framework inherit machine-checked proofs of the metatheory such as compositionality and bisimulation congruence. We present a generic analysis tool for psi-calculus instances, enabling symbolic execution and (bi)simulation checking for both unicast and broadcast communication. The tool also provides a library for implementing new psi-calculus instances. We provide examples from traditional communication protocols and wireless sensor networks. We also describe the theoretical foundations of the tool, including an improved symbolic operational semantics, with additional support for scoped broadcast communication.

Tableau-based Bisimulation Invariance Testing

A tableau procedure that tests bisimulation invariance of a given first-order formula, and therefore tests if that formula is equivalent to the standard translation of some modal formula, is presented. The test is sound and complete: a given formula is bisimulation invariant if and only if there is a closed tableau for that formula. The test generally does not terminate, but it does if a given formula is bisimulation invariant, i.e., the test is positive.

Scalable Minimization Algorithm for Partial Bisimulation

We present an efficient algorithm for computing the partial bisimulation preorder and equivalence for labeled transitions systems. The partial bisimulation preorder lies between simulation and bisimulation, as only a part of the set of actions is bisimulated, whereas the rest of the actions are simulated. Computing quotients for simulation equivalence is more expensive than for bisimulation equivalence, as for simulation one has to account for the so-called little brothers, which represent classes of states that can simulate other classes. It is known that in the absence of little brother states, (partial bi)simulation and bisimulation coincide, but still the complexity of existing minimization algorithms for simulation and bisimulation does not scale. Therefore, we developed a minimization algorithm and an accompanying tool that scales with respect to the bisimulated action subset.

Characterisations of testing preorders for a finite probabilistic π -calculus

Abstract We consider two characterisations of the may and must testing preorders for a probabilistic extension of the finite π -calculus: one based on notions of probabilistic weak simulations, and the other on a probabilistic extension of a fragment of Milner–Parrow–Walker modal logic for the π -calculus. We base our notions of simulations on similar concepts used in previous work for probabilistic CSP. However, unlike the case with CSP (or other non-value-passing calculi), there are several possible definitions of simulation for the probabilistic π -calculus, which arise from different ways of scoping the name quantification. We show that in order to capture the testing preorders, one needs to use the “earliest” simulation relation (in analogy to the notion of early (bi)simulation in the non-probabilistic case). The key ideas in both characterisations are the notion of a “characteristic formula” of a probabilistic process, and the notion of a “characteristic test” for a formula. As in an earlier work on testing equivalence for the π -calculus by Boreale and De Nicola, we extend the language of the π -calculus with a mismatch operator, without which the formulation of a characteristic test will not be possible.

Bisimulation quotients of Veltman models

Interpretability logic is a modal description of the interpretability predicate. The modal system IL is an extension of the provability logic GL (Godel–Lob). Bisimulation quotients and largest bisimulations have been well studied for Kripke models. We examine interpretability l

Computing Distances between Probabilistic Automata

We present relaxed notions of simulation and bisimulation on Probabilistic Automata (PA), that allow some error epsilon. When epsilon is zero we retrieve the usual notions of bisimulation and simulation on PAs. We give logical characterisations of these notions by choosing suitable logics which differ from the elementary ones, L with negation and L without negation, by the modal operator. Using flow networks, we show how to compute the relations in PTIME. This allows the definition of an efficiently computable non-discounted distance between the states of a PA. A natural modification of this distance is introduced, to obtain a discounted distance, which weakens the influence of long term transitions. We compare our notions of distance to others previously defined and illustrate our approach on various examples. We also show that our distance is not expansive with respect to process algebra operators. Although L without negation is a suitable logic to characterise epsilon-(bi)simulation on deterministic PAs, it is not for general PAs; interestingly, we prove that it does characterise weaker notions, called a priori epsilon-(bi)simulation, which we prove to be NP-difficult to decide.

Experimental and numerical study of ball indentation for evaluation of mechanical properties and fracture toughness of structural steel

Determination of mechanical properties of materials using non-conventional techniques has been an active area of research for a long time. Among various small specimen techniques to determine mechanical properties of materials, the ball indentation technique (BIT) has proved to be advantageous. BIT is used to measure material’s mechanical properties including fracture toughness when a tensile test cannot be performed: on welded joints or critical locations of components under service. The present work highlights the applicability of BIT for evaluating flow behaviour of engineering structural steels En24. Standard mechanical properties like ultimate tensile strength, yield stress, strain hardening coefficient evaluated for steels with varying microstructures generated through heat treatment. To determine fracture toughness from the flow behaviour, non linear damage models have been utilized. Using this model, fracture toughness of the En24 steel at different heat treated conditions has been computed using the results generated by the BIT. These results are verified with the already established correlation in literature. Numerical validation of the results generated by the BIT has been carried out by Finite Element Modelling (FEM) using standard ABAQUS software package. The FE model of the BI simulation helps to compute sub indenter stress-strain fields which determine the extent of pile-up in highly ductile materials.

Branching Bisimilarity with Explicit Divergence

We consider the relational characterisation of branching bisimilarity with explicit divergence. We prove that it is an equivalence and that it coincides with the original definition of branching bi...