Several probabilistic simulation relations for probabilistic systems are defined and evaluated according to two criteria: compositionality and preservation of "interesting" properties. Here, the interesting properties of a system are identified with those that are expressible in an untimed version of the Timed Probabilistic concurrent Computation Tree Logic (TPCTL) of Hansson. The definitions are made, and the evaluations carried out, in terms of a general labeled transition system model for concurrent probabilistic computation. The results cover weak simulations, which abstract from internal computation, as well as strong simulations, which do not.

. We present algorithms for computing similarity relations of labeled graphs. Similarity relations have applications for the refinement and verification of reactive systems. For finite graphs, we present an O(mn) algorithm for computing the similarity relation of a graph with n vertices and m edges (assuming m n). For effectively presented infinite graphs, we present a symbolic similarity-checking procedure that terminates if a finite similarity relation exists. We show that 2D rectangular automata, which model discrete reactive systems with continuous environments, define effectively presented infinite graphs with finite similarity relations. It follows that the refinement problem and the 8CTL model-checking problem are decidable for 2D rectangular automata. 1 Introduction A labeled graph G = (V; E;A; hh\Deltaii) consist of a (possibly infinite) set V of vertices, a set E ` V 2 of edges, a set A of labels, and a function hh\Deltaii : V ! A that maps each vertex v to a label hh...

A new solution to the Generalized Railroad Crossing problem, based on timed automata, invariants and simulation mappings, is presented and evaluated. The solution shows formally the correspondence between four system descriptions: an axiomatic specification, an operational specification, a discrete system implementation, and a system implementation that works with a continuous gate model.

When proving the correctness of algorithms in distributed systems, one generally considers safety conditions and liveness conditions. The Input/Output (I/O) automaton model and its timed version have been used successfully, but have focused on safety conditions and on a restricted form of liveness called fairness. In this paper we develop a new I/O automaton model, and a new timed I/O automaton model, that permit the verification of general liveness properties on the basis of existing verification techniques. Our models include a notion of environment-freedom which generalizes the idea of receptiveness of other existing formalisms, and enables the use of compositional verification techniques.

A general automaton model for timing-based systems is presented and is used as the context for developing a variety of simulation proof techniques for such systems. These techniques include (1) refinements, (2) forward and backward simulations, (3) hybrid forward-backward and backward-forward simulations, and (4) history and prophecy relations. Relationships between the different types of simulations, as well as soundness and completeness results, are stated and proved. These results are (with one exception) analogous to the results for untimed systems in Part I of this paper. In fact, many of the results for the timed case are obtained as consequences of the analogous results for the untimed case.

Abstract. Lamport showed that a replicated deterministic state machine is a general way to implement a highly available system, given a consensus algorithm that the replicas can use to agree on each input. His Paxos algorithm is the most fault-tolerant way to get consensus without real-time guarantees. Because general consensus is expensive, practical systems reserve it for emergencies and use leases (locks that time out) for most of the computing. This paper explains the general scheme for efficient highly available computing, gives a general method for understanding concurrent and fault-tolerant programs, and derives the Paxos algorithm as an example of the method. 1

. This paper develops a new I/O automaton model called the Clock General Timed Automaton (Clock GTA) model. The Clock GTA is based on the General Timed Automaton (GTA) of Lynch and Vaandrager. The Clock GTA provides a systematic way of describing timingbased systems in which there is a notion of "normal" timing behavior, but that do not necessarily always exhibit this "normal" behavior. It can be used for practical time performance analysis based on the stabilization of the physical system. We use the Clock GTA automaton to model, verify and analyze the paxos algorithm. The paxos algorithm is an efficient and highly faulttolerant algorithm, devised by Lamport, for reaching consensus in a distributed system. Although it appears to be practical, it is not widely known or understood. This paper contains a new presentation of the paxos algorithm, based on a formal decomposition into several interacting components. It also contains a correctness proof and a time performance and fault-tole...

The timed automaton model of [LV92, LV93] is a general model for timing-based systems. A notion of timed action transducer is here defined as an automata-theoretic way of representing operations on timed automata. It is shown that two timed trace inclusion relations are substitutive with respect to operations that can be described by timed action transducers. Examples are given of operations that can be described in this way, and a preliminary proposal is given for an appropriate language of operators for describing timing-based systems.

Safety critical computers increasingly a#ect nearly every aspect of our lives. Computers control the planes we #y on, monitor our health in hospitals and do our work in hazardous environments. Computers with software de#ciencies that fail to meet stringent timing constraints have resulted in catastrophic failures. This paper surveys formal methods for specifying, designing and verifying real-time systems, so as to improve their safety and reliability. # To appear in Journal of Systems and Software,Vol. 18, Number 1, pages 33#60, April 1992. Jonathan Ostro# is with the Department of Computer Science, York University 4700 Keele Street, North York, Ontario, Canada, M3J 1P3. This work is supported by the Natural Sciences and Engineering Research Council of Canada. 1 CONTENTS 2 Contents 1 Introduction 3 2 De#ning the terms 6 2.1 Major issues that formal theories must address ::::::: 13 3 Real-Time Programming Languages 14 4 Structured Methods and#or Graphical Languages 15 4.1 Str...

This paper reports the results of a case study on the feasibility of developing and applying mechanical methods, based on the proof system PVS, to prove propositions about real-time systems specified in the Lynch-Vaandrager timed automata model. In using automated provers to prove propositions about systems described by a specific mathematical model, both the proofs and the proof process can be simplified by exploiting the special properties of the mathematical model. Because both specifications and methods of reasoning about them tend to be repetitive, the use of a standard template for specifications, accompanied by standard shared theories and standard proof strategies or tactics, is often feasible. Presented are the PVS specification of three theories that underlie the timed automata model, a template for specifying timed automata models in PVS, and an example of its instantiation. Both hand proofs and the corresponding PVS proofs of two propositions are provided to illustrate h...