On the Representation of Distributed Behavior Christopher Shaver Electrical Engineering and Computer Sciences University of California at Berkeley Technical Report No. UCB/EECS-2016-206 http://www2.eecs.berkeley.edu/Pubs/TechRpts/2016/EECS-2016-206.html December 16, 2016 Copyright © 2016, by the author(s). All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission. On the Representation of Distributed Behavior by Christopher Daniel Shaver A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Computer Science in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Edward A. Lee, Chair Professor Marjan Sirjani Adjunct Professor Stavros Tripakis Professor John Steel Fall 2016 On the Representation of Distributed Behavior Copyright 2016 by Christopher Daniel Shaver 1 Abstract On the Representation of Distributed Behavior by Christopher Daniel Shaver Doctor of Philosophy in Computer Science University of California, Berkeley Professor Edward A. Lee, Chair Technologies pervasive today have enabled a plethora of diverse networked devices to prolif- erate in the market. Among these devices are sensors, wearables, mobile devices, and many other embedded systems, in addition to conventional computers. This diverse tapestry of networked devices, often termed the Internet of Things (IoT) and the Swarm, has the po- tential to serve as a platform for a new breed of sophisticated distributed applications that leverage the ubiquity, concurrency, and flexibility of these devices, often to integrate the sim- ilarly diverse information to which these devices have access. These kinds of applications, which we are calling Highly Dynamic Distributed Applications (HDDAs), are particularly important in the domain of automated environments such as smart homes, buildings, and even cities. In contrast with the kind of concurrent computation that can be represented with petri nets, synchronous systems, and other rigid models of concurrent computation, applications running on networks of devices such as the IoT demand models that are more dynamic, semantically heterogeneous, and composable. Because these devices are more than just peripherals to central servers, themselves capable of significant amounts of computation, HDDAs can involve the establishment of direct connections between these devices to share and process information. From the perspective of the platform this will look like an ever- evolving network of dynamic configuration and communication that may have no clear sense of a center, or even a central point of observation. Computation in this fashion begins to look less like a sequence of coherent states and more like a physical interaction in space. Designing and reasoning about these kinds of applications in a rigorous and scalable fashion will require the development of new programming language semantics, specification logics, verification methods, and synthesis algorithms. At the core of all of these components of a robust programming model will be a need for an appropriate mathematical represen- tation of behavior, specifying precisely what happens in these spaces of devices during the execution of a HDDA. This behavioral representation must characterize an application in as much detail as is necessary, without having to introduce details regarding the realization of the execution on a particular platform, bound to a specific architecture of concrete pro- 2 cess and communication relationships. This representation must also be itself as scalable, modular, and composable as the distributed computations it describes. The aim of this thesis is to identify the mathematical representation of behavior best fit to meet the challenges HDDAs pose to formal semantics and formal verification. To this end, this thesis proposes a new mathematical representation of concurrent computational behavior called an Ontological Event Graph (OEG), which generalizes, subsumes, or improves upon other representations of concurrent behavior such as Mazurkiewicz Traces, Event Structures, and Actor Event Diagrams. In contrast with many other representations of concurrent behavior, which are either graphs or partial orders, OEGs are, in essence, acyclic ported block diagrams. In these diagrams, each ported block represents an event and each connection between ports represents a direct dependency between events, while each port around a block represents the specific kind of information or influence consumed or produced in the event. OEGs have the advantage over other representations of concurrent behavior of being both modular and composable, as well as possessing a clear concept of hierarchy and abstraction. The motivations and reasoning behind this choice of representation will be developed from two directions. The first will be an exploration of the context, consisting of networks of devices such as the IoT. The potential of these networks to serve as a distributed compu- tational platform for HDDAs will be understood through the construction of a conceptual application model that will be called a Process Field. The demands of this application model and some concrete examples will be used to construct an intuitive picture of how the corre- sponding behavioral representation must look, and what it would need to serve the purposes of constructing semantics and performing verification in the context of HDDAs. The second direction will reflect on the mathematical details of two classes of sequential representations of behavior, sequences and free monoids, and look at the ways these two classes generalize from the sequential case to the concurrent one. This will suggest, as po- tential generalizations of the two classes, generalized sequences and free monoidal categories. The former of these classes already has many instances prevalent in the literature. These existing representations based on generalized sequences, along with some other similar es- tablished means to represent concurrent behavior, will be reviewed in detail, and reasons will be given for why they do not meet the demands for modularity and composability needed for reasoning about HDDAs. It will be concluded that the latter, far less investigated class of free monoidal categories offers a richer mathematical structure that is more appropriate for the demands of construct- ing formal models of HDDAs. In particular, free monoidal categories offer a very general form of composition involving both a parallel and a sequential product. These products, along with a collection of distinguished constants, form an algebra that can be used to con- struct semantics or to formulate specifications and prove properties of behaviors. Time will be taken to first review the concepts from category theory needed to define these structures, particularly focusing on monoidal categories and the concept of free structures and universal properties. The construction of free monoidal categories, and the particular variant of free symmetric monoidal categories, will then be detailed, and several important properties of these structures will be shown and discussed. 3 It will ultimately be established that spaces of OEGs defined in this thesis, along with their compositions, form free symmetric monoidal categories. Consequently, the study of OEGs can leverage all of the mathematical tools of monoidal category theory to study and better understand distributed behavior. A chapter will be devoted to showing specifically how the π-calculus can be given a particularly elegant and powerful behavioral semantics using OEGs. i To my Kitty. ii Contents Contents ii List of Figures iv 1 Introduction 1 1.1 Behavioral Representations . 2 1.2 Outline of This Thesis . 5 1.3 Structures and Notations . 8 2 Highly Dynamic Distributed Applications 12 2.1 The Process Field Platform . 13 2.2 The Actor Model . 15 2.3 A MoC for the Process Field . 17 2.4 Examples of HDDAs . 19 2.5 Process Field Theory . 28 3 Generalizing Representations of Sequential Behavior 34 3.1 Representing Sequential Systems . 37 3.2 Generalizing . 54 4 Existing Representations of Distributed Behavior 62 4.1 Which Kinds of Systems? . 63 4.2 Traces . 64 4.3 True Concurrency . 69 4.4 Other Representations . 78 5 Key Concepts from Category Theory 84 5.1 Monoidal Categories . 85 5.2 Free Objects and Derived Structures . 97 6 Free Monoidal Categories 110 6.1 Monoidal Schemes . 111 6.2 Free Monoidal Categories . 115 iii 6.3 Properties of Monoidal Schemes and Free Monoidal Categories . 126 7 Ontological Event Graphs 135 7.1 An Event Ontology . 136 7.2 Ontological Event Schemes . 141 7.3 The Formalism of Ontological Event Graphs . 155 8 OEG Semantics for Process Calculi 190 8.1 The π-calculus .................................. 192 8.2 OEG Semantics . 199 9 Conclusions and Further Work 216 9.1 More Semantics . 218 9.2 Mathematical Developments . 220 9.3 Logic and Formal Verification . 222 Bibliography 224 iv List of Figures 1.1 The relationship between programs, formulae, and behaviors. 3 1.2 Flow of the thesis. 6 2.1 The Process Field . 14 2.2 Climate control HDDA. 19 2.3 Initial configuration in the Process Field. 21 2.4 A home automation swarmlet. 23 2.5 Final configuration in the Process Field. 24 2.6 Initial configuration in the Process Field. 25 2.7 Simple system with one managing actor P ...................... 26 2.8 A more complex system with two independent processes managing the doors. 28 2.9 A possible set of connections between the two processes. 28 2.10 Execution in a Process Field. 30 2.11 Feynman diagram of electron scattering (polarized). 31 2.12 The same interaction from three frames of reference. 32 2.13 Events as collisions of computational data such as messages or states. 33 3.1 Sequential behavioral representation. 38 3.2 Free monoidal behavioral representation. 40 3.3 Generalized sequence. 54 3.4 Diagram in a monoidal category.