A domain equation for refinement of partial systems

  • Authors:

  • Michael R. A. Huth;Radha Jagadeesan;David A. Schmidt

  • Affiliations:

  • Department of Computing, Imperial College London, South Kensington campus, London SW7 2AZ, England Email: M.Huth@doc.imperial.ac.uk;School of Computer Science, Telecommunications, and Information Sciences, DePaul University, 243 S. Wabash Avenue, Chicago, Illinois 60604-2287 Email: rjagadeesan@cs.depaul.edu;Department of Computing and Information Sciences, Kansas State University, 234 Nichols Hall, Manhattan, Kansas 66506 Email: schmidt@cis.ksu.edu

  • Venue:
  • Mathematical Structures in Computer Science
  • Year:
  • 2004

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Abstract

A reactive system can be specified by a labelled transition system, which indicates static structure, along with temporal-logic formulas, which assert dynamic behaviour. But refining the former while preserving the latter can be difficult, because:(i) Labelled transition systems are ‘total’ – characterised up to bisimulation – meaning that no new transition structure can appear in a refinement.(ii) Alternatively, a refinement criterion not based on bisimulation might generate a refined transition system that violates the temporal properties.In response, Larsen and Thomson proposed modal transition systems, which are ‘partial’, and defined a refinement criterion that preserved formulas in Hennessy–Milner logic. We show that modal transition systems are, up to a saturation condition, exactly the mixed transition systems of Dams that meet a mix condition, and we extend such systems to non-flat state sets. We then solve a domain equation over the mixed powerdomain whose solution is a bifinite domain that is universal for all saturated modal transition systems and is itself fully abstract when considered as a modal transition system. We demonstrate that many frameworks of partial systems can be translated into the domain: partial Kripke structures, partial bisimulation structures, Kripke modal transition systems, and pointer-shape-analysis graphs.