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Abstraction and Abstraction Refinement in the Verification Of Abstraction and Abstraction Refinement in the Verification of Graph Transformation Systems Vom Fachbereich Ingenieurwissenschaften Abteilung Informatik und angewandte Kognitionswissenschaft der Unversit¨at Duisburg-Essen zur Erlangung des akademischen Grades eines Doktor der Naturwissenschaften (Dr.-rer. nat.) genehmigte Dissertation von Vitaly Kozyura aus Magadan, Russland Referent: Prof. Dr. Barbara K¨onig Korreferent: Prof. Dr. Arend Rensink Tag der m¨undlichen Pr¨ufung: 05.08.2009 Abstract Graph transformation systems (GTSs) form a natural and convenient specification language which is used for modelling concurrent and distributed systems with dynamic topologies. These can be, for example, network and Internet protocols, mobile processes with dynamic behavior and dynamic pointer structures in programming languages. All this, together with the possibility to visualize and explain system behavior using graphical methods, makes GTSs a well-suited formalism for the specification of complex dynamic distributed systems. Under these circumstances the problem of checking whether a certain property of GTSs holds – the verification problem – is considered to be a very important question. Unfortunately the verification of GTSs is in general undecidable because of the Turing- completeness of GTSs. In the last few years a technique for analysing GTSs based on approximation by Petri graphs has been developed. Petri graphs are Petri nets having additional graph structure. In this work we focus on the verification techniques based on counterexample-guided abstraction refinement (CEGAR approach). It starts with a coarse initial over-approxi- mation of a system and an obtained counterexample. If the counterexample is spurious then one starts a refinement procedure of the approximation, based on the structure of the counterexample. The CEGAR approach has proved to be very successful for the verification of systems based on their over-approximations. This thesis investigates a counterexample-guided abstraction refinement approach for systems modelled with GTSs. Starting with a given spurious counterexample, we describe here how to construct a more exact approximation (by separating merged nodes) for which this counterexamples disappears. This procedure can be performed repeatedly for any number of spurious counterexamples. Furthermore, an incremental coverability approach for Petri nets is developed, which allows one to speed-up the construction of over-approximations of GTSs. A well-known approach is to extend a modelling language with the possibility of de- scribing attributes as values of some data types. The approximation-based verification technique, including a counterexample-guided abstraction refinement, is hence also gen- eralized in this work to attributed GTSs (AGTSs), where the attributes are abstracted in the framework of abstract interpretation. In the practical part, a verification tool Augur 2 is developed, which supports the whole verification process for GTSs and AGTSs. A number of case studies (both at- tributed and non-attributed GTSs) were successfully solved with Augur 2. Acknowledgement First of all, I would like to thank my doctoral adviser, Barbara K¨onig, for being my supervisor, for encouraging and challenging me throughout the academic program. This work would not have been possible without her. I want to express my gratitude to Javier Esparza, whose works on Petri net unfolding have awakened my interests to the related areas of computer science. I thank him also for his support and advices on various aspects of my work. I also would like to thank my colleagues, Sander Bruggink, Tobias Heindel, Stefan Kiefer, Michael Luttenberger, Claus Schr¨oter, Alin Stefanescu and Dejvuth Suwimon- teerabuth, for their help and many interesting discussions on the topics of this thesis. Also I thank a lot all students have being involved in the development of the verification tool Augur. It was a very interesting and pleasant team work experience. I am very grateful to my family, to my parents for their lifelong support, and to my wife for being my friend and companion. Finally my thanks go to the reviewers of this thesis for reading the result of my efforts. Contents 1 Introduction 3 1.1 Verification of Graph Transformation Systems . ....... 3 1.2 Summary and Contribution of the Thesis . ... 5 1.3 RelatedWork.................................. 7 1.4 Publications................................... 10 1.5 Contributions.................................. 12 2 Petri Nets 13 2.1 IntroductiontoPetriNets . .. 13 2.2 CoverabilityGraphs .............................. 14 2.3 Incremental Calculation of Coverability Graphs . ......... 18 2.4 BackwardCoverabilityAlgorithm. .... 23 3 Graph Transformation Systems 26 3.1 Introduction to Graph Transformation Systems . ....... 26 3.2 Verification of Graph Transformation Systems . ....... 28 4 Attributed Petri Nets and Graph Transformation Systems 37 4.1 AlgebrasandAbstraction . 37 4.2 AttributedPetriNets ............................. 40 4.3 Attributed Graph Transformation Systems . ...... 42 4.4 Verification of Attributed Graph Transformation Systems ......... 45 5 Counterexample-Guided Abstraction Refinement 53 5.1 Abstraction Refinement of Graph Transformation Systems ......... 53 5.1.1 SpuriousRuns ............................. 54 5.1.2 Relations on Nodes for Refining Abstract Runs . ... 55 5.1.3 Elimination of Spurious Runs . 58 5.1.4 Correctness ............................... 60 5.2 Abstraction Refinement of Attributed Graph TransformationSystems . 63 6 Augur – a Tool for the Analysis of Graph Transformation Systems 69 6.1 Brief Description of Augur 1......................... 69 6.2 Software Design of Augur 2 ......................... 70 6.3 System Architecture of Augur 2 ....................... 72 6.4 FunctionalityandUsage . 77 1 7 Case Studies 83 7.1 Public-PrivateServers . .. 84 7.2 Firewall ..................................... 87 7.3 Experiments with the Incremental Coverability Approach ......... 91 7.4 Random Graph Transformation Systems - Statistical Results ....... 93 7.5 VerifyingRed-BlackTrees . .. 96 7.6 LeaderElectionProtocol. 102 7.7 Needham-SchroederProtocol . 104 8 Conclusion and Future Work 108 A Basic Category Theory for Graph Rewriting 118 B Example in GTXL format 122 C Example in new GTXL format 124 D Example in GXL format 127 E Example of an SPL program 129 F Verification Example 131 2 Chapter 1 Introduction 1.1 Verification of Graph Transformation Systems In the last decades, the development of computers and networks, the appearance and the rapid expansion of Internet technologies have led to increasing complexity and strong interconnection of computer systems. Under these circumstances, the problem of mod- elling and analyzing complex distributed systems is considered to be a very important question. Unfortunately, many of today’s methods describe distributed and mobile sys- tems with formalisms that either have an infinite state space, making many interesting questions undecidable, or a very a large state space, leading to the so-called state explo- sion problem. This makes the important problem of modelling and verification of mobile and distributed systems a very difficult one. The approach in this thesis is based on a rather natural and expressive modelling language, namely graph transformation systems (GTSs) [95]. GTSs have their origin in the late 1960’s and were developed up to now into a rich theory with many different branches. The development was originally motivated by such areas of computer science as pattern recognition, description of data types and compiler construction. Later, more modern branches of computer science, such as computer networks with different topolo- gies, mobile processes, UML diagrams and web services have also been specified and analysed with GTSs. Being a natural generalization of formal language theory, based on string rewriting and the theory of term rewriting, GTSs are also interesting from a theoretical point of view. For many structures in computer science, a static description of system states can be obtained with the help of graphs in a rather natural way. As an example, we consider computer networks, where stations can be represented as nodes of graphs and the network connections as edges. As a more specific example, we mention here pointer structures on the heap in a programming language, where objects are nodes and the values of pointers are modelled with edges. These can be, for example, memory allocation structures in the programming language C, but the same structures arise also for object-oriented languages, such as Java. Graph transformations provide also the possibility of describing the dynamical be- havior of systems. Using GTSs one can model in a natural way dynamic creation and deletion of objects, for example, computer stations entering and leaving the network or mobile processes moving between different stations. GTSs are also a very convenient modelling language for systems with evolving topologies, where not only the number of the components but also the structure of the connections between them is dynamic. This is often the case in network and Internet protocols. Also in the evaluation of dynamic pointer structures on the heap in a programming language mentioned above, the topology 3 of the structures is a subject of constant changes. The language of GTSs is also very helpful in the modelling of distributed and con- current systems. Such systems appear as a natural result of using networks, and the research on their specification and verification
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