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Two-thirds simulation topology
Abstract
Two-thirds simulation provides a kind of abstract description for the implementation of satisfying its specification. In order to characterize the implementation closer to its specification more and more, based on the definition of limit and two-thirds simulation, the limit and topology theories of two-thirds simulation are proposed. Firstly, two-thirds limit simulation and two-thirds simulation limit are defined. The limit theory of two-thirds simulation is established. Secondly, topological structure of two-thirds simulation is constructed, including subnet closure, tail closure, natural extension and iteration. Two-thirds simulation limit is demonstrated to form a convergence class, which induces a topology. Finally, pre-congruence property of two-thirds simulation limit is showed, which states the continuity of various combined operators under two-thirds simulation limit.
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Correctness is a key attribute of software trustworthiness. Abstractly, software correctness can be represented as whether or not the implementation of software satisfies its specification. However, in the real world, it is difficult to get the satisfaction absolutely. In the course of developing and designing software, implementation is often modified in order to satisfy its specification. This means that the software is more and more close to correctness, i.e. software correctness is a dynamic course. In order to describe the dynamic correctness of software, in this paper, the abstract characterization and the limit theory of dynamic correctness based on parameterized bisimulation are proposed. Firstly, the infinite evolution mechanism of parameterized bisimulation is established. Parameterized limit bisimulation is defined in order to characterize the relation between a series of software implementations obtained in the real design and its specification, and some special examples are shown. Secondly, parameterized bisimulation limit is given, and the recursive characterization of parameterized bisimulation limit is stated. Finally, some algebraic properties are proved, such as the uniqueness of parameterized bisimulation limit and the consistence between parameterized bisimulation limit and parameterized bisimulation.
Since a nondeterministic and concurrent program may, in general, communicate repeatedly with its environment, its meaning cannot be presented naturally as an input/output function (as is often done in the denotational approach to semantics). In this paper, an alternative is put forth. First, a definition is given of what it is for two programs or program parts to be equivalent for all observers; then two program parts are said to be observation congruent if they are, in all program contexts, equivalent. The behavior of a program part, that is, its meaning, is defined to be its observation congruence class.
The paper demonstrates, for a sequence of simple languages expressing finite (terminating) behaviors, that in each case observation congruence can be axiomatized algebraically. Moreover, with the addition of recursion and another simple extension, the algebraic language described here becomes a calculus for writing and specifying concurrent programs and for proving their properties.
Bisimulation expresses the equivalence of processes whose external actions are identical. Sometimes we may meet two processes which are not exactly bisimilar but more or less bisimilar in the sense that whenever a process makes an action the other can make an action different from but very similar to the action performed by the first one. To describe this kind of looser bisimulations we propose the concept of bisimulation index in a labelled transition system and give its various properties, especially those properties related to the operations of transition systems. Furthermore, we establish a modal logical characterization of bisimulation indexes. This characterization is a generalization of Hennessy–Milner logic. We study strong and weak bisimulation indexes in the basic asynchronous process calculus, and some of their fundamental properties are derived. Bisimulation indexes are not substitutive under composition. To overcome this defect we introduce an approximate communication rule to replace the original rule in process calculus. This enables us to recover some useful properties of composition with respect to bisimulation indexes. Finally, we present three examples in timed CCS and real time ACP to demonstrate the usage of bisimulation indexes in the analysis of real time systems. These examples show that bisimulation indexes are suitable formal tools for describing approximate implementations of real time systems.
Boolean notions of correctness are formalized by preorders on systems. Quantitative measures of correctness can be formalized by real-valued distance functions between systems, where the distance between implementation and specification provides a measure of "fit" or "desirability". We extend the simulation preorder to the quantitative setting by making each player of a simulation game pay a certain price for her choices. We use the resulting games with quantitative objectives to define three different simulation distances. The correctness distance measures how much the specification must be changed in order to be satisfied by the implementation. The coverage distance measures how much the implementation restricts the degrees of freedom offered by the specification. The robustness distance measures how much a system can deviate from the implementation description without violating the specification. We consider these distances for safety as well as liveness specifications. The distances can be computed in polynomial time for safety specifications, and for liveness specifications given by weak fairness constraints. We show that the distance functions satisfy the triangle inequality, that the distance between two systems does not increase under parallel composition with a third system, and that the distance between two systems can be bounded from above and below by distances between abstractions of the two systems. These properties suggest that our simulation distances provide an appropriate basis for a quantitative theory of discrete systems. We also demonstrate how the robustness distance can be used to measure how many transmission errors are tolerated by error correcting codes.