Modeling Time in Computing: A Taxonomy and a Comparative Survey

Modeling Time in Computing: A Taxonomy and a Comparative Survey Carlo A. Furia, Dino Mandrioli, Angelo Morzenti, and Matteo Rossi October 9, 2018 Abstract The increasing relevance of areas such as real-time and embedded sys- tems, pervasive computing, hybrid systems control, and biological and social systems modeling is bringing a growing attention to the temporal aspects of computing, not only in the computer science domain, but also in more traditional fields of engineering. This article surveys various approaches to the formal modeling and analysis of the temporal features of computer-based systems, with a level of detail that is also suitable for nonspecialists. In doing so, it provides a unifying framework, rather than just a comprehensive list of formalisms. The article first lays out some key dimensions along which the various formalisms can be evaluated and compared. Then, a significant sample of formalisms for time modeling in computing are presented and discussed according to these dimensions. The adopted perspective is, to some ex- tent, historical, going from “traditional” models and formalisms to more modern ones. 1 arXiv:0807.4132v3 [cs.GL] 11 Oct 2010 Contents 1 Introduction 3 2 Languages and Interpretations 5 3 Dimensions of the Time Modeling Problem 8 3.1 Discrete vs. Dense Time Domains . . . . . . . . . . . . . . . . . . 8 3.2 Ordering vs. Metric . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.3 Linear vs. Branching Time Models . . . . . . . . . . . . . . . . . 13 3.4 Implicit vs. Explicit Time Reference . . . . . . . . . . . . . . . . 14 3.5 The Time Advancement Problem . . . . . . . . . . . . . . . . . . 15 3.6 Concurrency and Composition . . . . . . . . . . . . . . . . . . . 17 3.7 Analysis and Verification Issues . . . . . . . . . . . . . . . . . . . 19 4 Historical Overview 20 4.1 Traditional Dynamical Systems . . . . . . . . . . . . . . . . . . . 21 4.2 The Hardware View . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.3 The Software View . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5 Temporal Models in Modern Theory and Practice 28 5.1 Operational Formalisms . . . . . . . . . . . . . . . . . . . . . . . 30 5.1.1 Synchronous Abstract Machines . . . . . . . . . . . . . . 30 5.1.2 Asynchronous Abstract Machines: Petri nets . . . . . . . 40 5.2 Descriptive Formalisms . . . . . . . . . . . . . . . . . . . . . . . . 47 5.2.1 Temporal Logics . . . . . . . . . . . . . . . . . . . . . . . 48 5.2.2 Explicit-Time Logics . . . . . . . . . . . . . . . . . . . . . 56 5.2.3 Algebraic Formalisms . . . . . . . . . . . . . . . . . . . . 59 5.3 Dual Language Approaches . . . . . . . . . . . . . . . . . . . . . 63 6 Conclusions 66 2 1 Introduction In many fields of science and engineering, the term dynamics is intrinsically bound to a notion of time. In fact, in classical physics a mathematical model of a dynamical system most often consists of a set of equations that state a relation between a time variable and other quantities characterizing the system, often referred to as system state. In the theory of computation, conversely, the notion of time does not always play a major role. At the root of the theory, a problem is formalized as a function from some input domain to an output range. An algorithm is a process aimed at computing the value of the function; in this process, dynamic aspects are usually abstracted away, since the only concern is the result produced. Timing aspects, however, are quite relevant in computing too, for many rea- sons; let us recall some of them by adopting a somewhat historical perspective. • First, hardware design leads down to electronic devices where the physical world of circuits comes back into play, for instance when the designer must verify that the sequence of logical gate switches that is necessary to execute an instruction can be completed within a clock’s tick. The time models adopted here are borrowed from physics and electronics, and range from differential equations on continuous time for modeling devices and circuits, to discrete time (coupled with discrete mathematics) for describing logical gates and digital circuits. • When the level of description changes from hardware to software, physical time is progressively disregarded in favor of more “coarse-grained” views of time, where a time unit represents a computational step, possibly in a high-level programming language; or it is even completely abstracted away when adopting a purely functional view of software, as a mapping from some input to the computed output. In this framework, computational complexity theory was developed as a natural complement of computabil- ity theory: it was soon apparent that knowing an algorithm to solve a problem is not enough if the execution of such an algorithm takes an unaffordable amount of time. As a consequence, models of abstract ma- chines have been developed or refined so as to measure the time needed for their operations. Then, such an abstract not

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