BigraphER is a suite of open-source tools providing an effi-

cient implementation of rewriting, simulation, and visualisation for bigraphs,

a universal formalism for modelling interacting systems that

evolve in time and space and first introduced by Milner. BigraphER consists

of an OCaml library that provides programming interfaces for the

manipulation of bigraphs, their constituents and reaction rules, and a

command-line tool capable of simulating Bigraphical Reactive Systems

(BRSs) and computing their transition systems. Other features are native

support for both bigraphs and bigraphs with sharing, stochastic reaction

rules, rule priorities, instantiation maps, parameterised controls, predicate

checking, graphical output and integration with the probabilistic

model checker PRISM

DT Gillespie

G Bacci

GS Tseitin

J Ellson

J Krivine

L Birkedal

L Yu

M Calder

M Kwiatkowska

M Sevegnani

N Eén

R Milner

English

Sevegnani, Michele

Calder, Muffy

Enlighten

    Sevegnani, M. and Calder, M. (2016) BigraphER: rewriting and analysis engine for bigraphs. In: CAV 2016: 28th International Conference on Computer Aided Verification, Toronto, Canada, 17-23 Jul 2016, pp. 494-501. ISBN 9783319415390 (doi:10.1007/978-3-319-41540-6_27)  This is the author’s final accepted version.  There may be differences between this version and the published version. You are advised to consult the publisher’s version if you wish to cite from it.     http://eprints.gla.ac.uk/119384/            Deposited on: 24 May 2016              Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk BigraphER: rewriting and analysis engine forbigraphsMichele Sevegnani and Muffy CalderSchool of Computing Science, University of GlasgowAbstract. BigraphER is a suite of open-source tools providing an effi-cient implementation of rewriting, simulation, and visualisation for bi-graphs, a universal formalism for modelling interacting systems thatevolve in time and space and first introduced by Milner. BigraphER con-sists of an OCaml library that provides programming interfaces for themanipulation of bigraphs, their constituents and reaction rules, and acommand-line tool capable of simulating Bigraphical Reactive Systems(BRSs) and computing their transition systems. Other features are nativesupport for both bigraphs and bigraphs with sharing, stochastic reactionrules, rule priorities, instantiation maps, parameterised controls, pred-icate checking, graphical output and integration with the probabilisticmodel checker PRISM.1 IntroductionBigraphs were first introduced by Robin Milner as a universal mathematicalmodel for representing the spatial configuration of physical or virtual objects,their interaction capabilities and temporal evolution. They were subsequentlyextended to stochastic bigraphs [11] and bigraphs with sharing [16], and havebeen applied in areas such as wireless protocols, home network management,mixed reality systems, cloud computing, security and as meta-models to encodeprocess calculi (e.g. Mobile Ambients, CSS).BigraphER is a modelling and reasoning environment for bigraphs consistingof an OCaml library and a command-line tool. The functionality includes:– native support for both bigraphs and bigraphs with sharing;– a rewrite engine with support for stochastic reaction rules, rules with in-stantiation maps, rule priorities, (stochastic) simulation and exhaustive statespace exploration;– predicate checking;– efficient matching engine based on SAT (used to implement rewriting andpredicate checking);– support for parameterised controls and parameterised reaction rules;– export labelled transition systems to probabilistic model checker PRISM [12];– graphical output of bigraphs, reaction rules and transition systems (see Fig. 1(right) for an example bigraph and graphical layout).RouterInternetMachineRouter signalMachine signal0S Sl_in l_outIn In Out OutM RFig. 1. Left: wireless network with a router and a machine; signal coverage is repre-sented by coloured circles. Right: corresponding bigraphical representation automati-cally generated by BigraphER (S = signal, R = router, M = machine).Example Applications. While many early applications of bigraphs have beento meta-modelling, e.g. for encodings of the pi-calculus, λ-calculus, and CCS(Calculus of Communicating Systems), applications in other domains are re-cently beginning to emerge. Some examples are: security for cyber-physical sys-tems [18], quantitative analysis of biological processes [11], cloud computing [19],and a framework to control systems of networked mobile robotic systems [14].BigraphER has been used to specify and analyse a wide range of case studiesin many different application domains: wireless network protocols [6], wirelessmesh networks [4], run-time policy management for domestic networks [5], andhuman-computer interaction in mixed-reality systems [2]. Example analysis hasranged from detecting basic “programming” errors (e.g. through type checking)in [2], to generation of example state spaces, and run-time checking of invariants(i.e. predicates), implemented on a router in [5].Related Tools. BigMC [15] is an explicit-state model checking tool for BRSsbased on the BPL matching engine [3]. Currently, it does not support stochasticbigraphs nor bigraphs with sharing and can only check reachability properties.Big Red [9] is a visual editor for bigraphs and bigraphical reaction rules imple-mented as an Eclipse plugin; it does not implement rewriting. DBtk [1] is animplementation of matching for directed bigraphs, a variant of bigraphs with adirected link structure; there is no support for rewriting and BRS execution.2 Bigraphical Reactive Systems – OverviewA bigraph [13] is a pair of relations over the same set of nodes: a directed forest,called place graph, representing topological space in terms of node containmentand a hypergraph, called link graph, representing the interactions and (non-spatial) relationships among nodes. There is both an algebraic and graphicalform. The graphical representation of an example bigraph is in Fig. 1 (right);it models the simple network in Fig. 1 (left) with a router, a machine, and therange of their wireless signals.Nodes are indicated by circles and ovals and are assigned a type called con-trol indicated here by S (for signals), M (the machine), R (the router), etc. Theplace graph is specified by black arrows. Bigraphs with sharing [16] extend theoriginal theory by defining the place graph as a Directed Acyclic Graph (DAG),thus allowing a natural representation of overlapping or intersecting locations.For instance, the M-node in the example is contained by both nodes of controlS, meaning the machine is in a spatial location covered by both wireless sig-nals. The link graph is represented by green edges called links. Links may beonly partially specified, in which case they connect a name. Names are links(or potential links) to other bigraphs representing the external environment orcontext. By convention, names are drawn above the bigraph. In the example,names l in and l out are used to name incoming and outgoing (potential) linksto remote resources. The number of links of a node, also called arity, dependson its control, i.e. entities with the same control have the same number of links.Dashed rectangles denote regions of adjacent parts of the system and sites areused to model parts of the model that have been abstracted away (see Fig. 3(top)). A bigraph with node identifiers is said to be concrete. When all the iden-tifiers are ignored, we obtain an abstract bigraph which can be interpreted as anequivalence class of bigraphs with the same structure.A BRS consists of a set of reaction rules together with an initial bigraphon which the rules operate. In stochastic bigraphs [11], a rate is associated witheach rule.3 BigraphER Specification LanguageThe BigraphER specification language almost corresponds to the standard alge-braic notation for bigraphical expressions [13,16]. In the following, we highlightsome of the distinctive features of the BigraphER language by presenting a simplemodel for wireless networks inspired by [5]. The model is specified by the codein Fig. 2. A valid BRS model consists of four separate blocks of definitions: asignature containing all the controls in the model, a set of bigraphs, a set ofreaction rules and a reactive system specifying the initial state of the BRS, thepriority hierarchy among reaction rules and a set of predicates.Controls are defined in lines 1-2 by using keyword ctrl. The integer onthe right-hand side of each definition indicates the arity of each control. Thekeyword atomic specifies that a node may not contain other nodes. Bigraphdefinitions are in lines 4-8. Line 5 defines bigraph s0. Expression M{w,s} de-notes a node of control M with names w and s. Operators . and | denote nestingand merge product, respectively. Nesting is the operation allowing to place abigraph inside another one; merge product is the operation placing two bigraphsside-by-side inside the same region. Closures like /s0 indicate that a link hasno names (see link between M and S). Sharing is introduced by ternary opera-tor share ... by ... in .... The first argument specifies the entities to be1 ctrl M = 2; ctrl R = 2; ctrl S = 1;2 atomic ctrl In = 1; atomic ctrl Out = 1; atomic ctrl Block = 1;34 big links = In{l_in} | Out{l_out };5 big s0 = /s0 /s1 (share (/w (M{w,s0}.links || R{w,s1}.links))6 by ([{0,1}, {0,1}], 2)7 in (id{s0 ,s1,l_in ,l_out} | S{s0} | S{s1}));8 big is_in_blocked = M{w,s}.(In{l} | Block{l} | id);910 react block_in =11 M{w,s}.(In{l} | id) --> M{w,s}.(In{l} | Block{l} | id);1213 react leave_net =14 /s (share (M{w,s} || id) by ([{0, 1}, {1}], 2) in (id(1,{w,s}) || S{s}))15 --> ({w} || 1 || 1 || 0 || 0);1617 brs18 init s0;19 rules = [ { block_in , leave_net } ];20 preds = { is_in_blocked };21 endbrsFig. 2. Specification of a BRS in the BigraphER language.0Mls wIn0I0Ml s wInBlock0Fig. 3. Reaction rule block in for blocking a machine’s incoming traffic.shared, e.g. machine M and router R. The second argument specifies how theyare shared: {0, 1} indicates that M is shared by the first and the second signals(counting from left to right). The third argument specifies the entities containingthe shared entities, e.g. signals Ss0 and Ss1. The graphical representation of s0,automatically generated by BigraphER is shown in Fig. 1 (right).The code in lines 10-11 defines reaction rule block in. Operator --> is usedto separate the left-hand side from the right-hand side of the rule. Expression idindicates the identity bigraph, i.e. the bigraph with one site inside one region.This reaction rule models a firewall rule blocking a machine’s incoming traffic.The corresponding graphical representation is in Fig. 3. Reaction rule leave netdefined in lines 13-15 models a machine leaving the network.Finally, lines 17-21 contain the reactive system definition. A BRS is definedby construct brs ... endbrs. Keyword init specifies the initial state of thesystem. In the example, this is bigraph s0. Construct rules = [...] defines alist of priority classes in descending order of priority. A priority class is specifiedby construct {...} and may only contain reaction rules identifiers. Constructpreds = {...} defines a set of predicates. Predicate is_in_blocked (definedin line 8) can be used to tag states in which there are machines with blockedincoming traffic. In a more extensive model like in [5], this predicate can be usedto verify network invariants after network policies are enforced by users.This simple example highlights the main features; more complex examplesincluding stochastic reaction rules, reducible priority classes and instantiationmaps can be accessed at http://www.dcs.gla.ac.uk/~michele/bigrapher.html.4 Components and FeaturesThe BigraphER command-line tool is composed of three distinct modules: thecompiler, the matching engine and the rewriting engine. All are coded in OCaml.Compiler. The compiler translates an input source file in the BigraphER lan-guage into a run-time representation of the model. Each declaration specifies thebinding of an identifier to a data type representing either a control, a bigraphor a (stochastic) reaction rule. Each bigraph is stored in memory as a pair ofspecialised data structures: a sparse boolean matrix encoding the DAG’s adja-cency matrix of the place graph, and a set of hyperedges (i.e. multisets withnodes and names as elements) for the representation of the link graph. Althoughthe BigraphER language only defines abstract bigraphs, the compiler operateson the corresponding concrete bigraphs by assigning arbitrary node labellings.This is required to allow the enumeration of all distinct occurrences of a reactionrule and thus to compute exit rates in stochastic BRSs. Additional features are:type-checking of parameterised definitions, combinatorial generation of paramet-ric reaction rules, graphical representation of all the bigraph defined in the inputmodel (useful for debugging).Matching Engine. The bigraph matching problem determines whether a bi-graph, called pattern, occurs in another bigraph, called target. The BigraphERmatching engine implements the algorithm introduced in [16]: a SAT encodingof a specialisation of the sub-graph isomorphism problem. For each instance ofthe problem, the matching engine generates a set of constraints (formulas inConjunctive Normal Form (CNF)) encoding the instance. Solutions are then ob-tained by passing all the constraints through the OCaml bindings for the MiniSatsolver [7]. Solutions are expressed as total maps from the nodes of the pattern tosub-sets of the nodes of target. Because the matching problem is NP-complete,two techniques to optimise performance have been adopted in the implementa-tion. The first is to reduce the size of the SAT instances by applying Tseitintransformation [17] to constraints. The second is to minimise instances by ex-ploiting the symmetries in the structure of the pattern: when enumerating all theoccurrences, the automorphisms of the pattern are used to generate all the sym-metric solutions starting from a computed solution. The matching engine alsoimplements specialised constraints to support bigraph equality and predicatechecking.Rewrite Engine. This component computes the dynamic evolution of the(stochastic) BRS specified in the input file by iteratively applying all the reac-tion rules to each bigraph (state) until either a fixpoint or a user-defined boundon the number of states is reached.1 The transition system generated by a BRSis represented internally by BigraphER as a directed graph; the Continuous TimeMarkov Chain resulting from a stochastic BRS as a labelled directed graph. Ruleapplication consists of two steps: first the matching engine is queried for occur-rences of the left-hand side of a reaction rule, then, for each distinct occurrence,a new state is computed by replacing the occurrence with the right-hand side ofthe rule (see Fig. 3). BigraphER also supports reaction rules with instantiationmaps2 allowing to easily duplicate or discard parts of a bigraph when a reac-tion rule is applied. The rewriting engine incrementally builds the state space ina breadth-first search (BFS). Support for simulation is obtained by computingonly one random path of the transition system. Simulation for stochastic BRSsimplements Gillespie’s Stochastic Simulation Algorithm (SSA) [10]. Besides stan-dard rule priorities, BigraphER admits reducible classes3 i.e. priority classes inwhich rules are treated like rewriting within an equivalence class. This meansthat after applying all possible rules in an arbitrary order only a canonical formis stored. This feature allows, for instance, to reduce the number of intermediatestates generated by the application of instantaneous stochastic reaction rules.Predicates expressed as matches are checked during the generation of the tran-sition system: every time a new state is discovered, all the predicates specifiedin the input model are checked against it and the labelling function is updated.The rewriting engine can return either a textual or a graphical representationof the (labelled) transition system and its states. Graphical output is computedby the open-source graph layout generator Graphviz [8]. Textual output is com-patible with the PRISM probabilistic model checker, thus enabling quantitativeverification for BRSs.OCaml Library. This component provides programming interfaces for the datastructures used internally by the BigraphER command-line tool. For instance, itallows manipulation of bigraphs and their constituents by providing implementa-tion for the following operations: composition, tensor product, parallel product,merge product and nesting. The library also provides APIs to check predicates,construct reaction rules and apply them to rewrite bigraphs. The full library doc-umentation can be accessed at http://www.dcs.gla.ac.uk/~michele/docs/bigraph/index.html.Technical Details and Availability. BigraphER is free and open source (BSD)and runs on all major operating systems. It is available for download from http://www.dcs.gla.ac.uk/~michele/bigrapher.html.1 Note that a model may have an infinite state space.2 An instantiation map is a function associating each site in the right-hand side tosites in the left-hand side.3 The reaction rules belonging to a reducible class are assumed confluent i.e. theyyield the same result regardless of the order in which they are applied.Acknowledgments. This work was supported by EPSRC project Homework(EP/F064225/1) and an EPSRC Doctoral Prize Research Fellowship.References1. Bacci, G., Grohmann, D., Miculan, M.: DBtk: A toolkit for directed bigraphs.LNCS 5728, 413–422 (2009)2. Benford, S., Rodden, T., Calder, M., Sevegnani, M.: On lions, impala, and bigraphs:modelling interactions in physical/virtual spaces. ACM Transactions on Computer-Human Interaction (2016), in press3. Birkedal, L., Damgaard, T.C., Glenstrup, A.J., Milner, R.: Matching of bigraphs.ENTCS 175(4), 3 – 19 (2007)4. Boucebsi, R., Belala, F.: Towards a channels allocation scheme model for WMNsbased on SBRS with sharing. In: Proceedings of MeMo 2015. p. 5 (2015)5. Calder, M., Koliousis, A., Sevegnani, M., Sventek, J.: Real-time verification ofwireless home networks using bigraphs with sharing. Science of Computer Pro-gramming 80, Part B, 288–310 (2014)6. Calder, M., Sevegnani, M.: Modelling IEEE 802.11 CSMA/CA RTS/CTS withstochastic bigraphs with sharing. 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Pereira, E., Kirsch, C., Sengupta, R., Sousa, J.: BigActors - a model for structure-aware computation. In: 4th International Conference on Cyber-Physical Systems.pp. 199–208. ACM/IEEE (2013)15. Perrone, G., Debois, S., Hildebrandt, T.T.: A model checker for bigraphs. In: Pro-ceedings of SAC ’12. pp. 1320–1325. ACM (2012)16. Sevegnani, M., Calder, M.: Bigraphs with sharing. Theoretical Computer Science577, 43 – 73 (2015)17. Tseitin, G.S.: On the complexity of derivation in propositional calculus. Studies inConstructive Mathematics and Mathematical Logic 2(115-125), 10–13 (1968)18. Tsigkanos, C., Pasquale, L., Ghezzi, C., Nuseibeh, B.: Ariadne: Topology awareadaptive security for cyber-physical systems. In: ICSE 2015. vol. 2, pp. 729–732(2015)19. Yu, L., Tsai, W.T., Wei, X., Gao, J., Hildebrandt, T., Guo, X.Q.: Modeling andanalysis of mobile cloud computing based on bigraph theory. In: MobileCloud 2014.pp. 67–76 (2014)

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