Relating Alternating Relations for Conformance and Refinement

• Published in: Integrated Formal Methods Vol. 11918; pp. 246 - 264
• Main Authors: Janssen, Ramon, Vaandrager, Frits, Tretmans, Jan
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2019; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Alternating refinement; Interface automata; ioco; uioco
• ISBN: 9783030349677; 3030349675
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-030-34968-4_14

Various relations have been defined to express refinement and conformance for state-transition systems with inputs and outputs, such as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{ioco}_{\text {}}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{uioco}}$$\end{document} in the area of model-based testing, and alternating simulation and alternating-trace containment originating from game theory and formal verification. Several papers have compared these independently developed relations, but these comparisons make assumptions (e.g., input-enabledness), pose restrictions (e.g., determinism – then they all coincide), use different models (e.g., interface automata and Kripke structures), or do not deal with the concept of quiescence. In this paper, we present the integration of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{ioco}_{\text {}}}$$\end{document}/\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{uioco}}$$\end{document} theory of model-based testing and the theory of alternating refinements, within the domain of non-deterministic, non-input-enabled interface automata. A standing conjecture is that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{ioco}_{\text {}}}$$\end{document} and alternating-trace containment coincide. Our main result is that this conjecture does not hold, but that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{uioco}}$$\end{document} coincides with a variant of alternating-trace containment, for image finite interface automata and with explicit treatment of quiescence. From the comparison between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{ioco}_{\text {}}}$$\end{document} theory and alternating refinements, we conclude that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{ioco}_{\text {}}}$$\end{document} and the original relation of alternating-trace containment are too strong for realistic black-box scenarios. We present a refinement relation which can express both \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{uioco}}$$\end{document} and refinement in game theory, while being simpler and having a clearer observational interpretation.