Is observational congruence on μ-expressions axiomatisable in equational Horn logic?

  • Authors:

  • Michael Mendler;Gerald Lüttgen

  • Affiliations:

  • University of Bamberg, Faculty of Information Systems and Applied Computer Sciences, 96045 Bamberg, Germany;University of Bamberg, Faculty of Information Systems and Applied Computer Sciences, 96045 Bamberg, Germany

  • Venue:
  • Information and Computation
  • Year:
  • 2010

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Abstract

It is well known that bisimulation on @m-expressions cannot be finitely axiomatised in equational logic. Complete axiomatisations such as those of Milner and Bloom/Esik necessarily involve implicational rules. However, both systems rely on features beyond pure equational Horn logic: either rules that are impure by involving non-equational side-conditions, or rules that are schematically infinitary like the congruence rule which is not Horn. It is an open question whether these complications cannot be avoided in the proof-theoretically and computationally clean and powerful setting of second-order equational Horn logic. This paper presents a positive and a negative result regarding the axiomatisability of observational congruence in equational Horn logic. Firstly, we show how Milner's impure rule system can be reworked into a pure Horn axiomatisation that is complete for guarded processes. Secondly, we prove that for unguarded processes, both Milner's unique fixed point rule and Bloom/Esik's GA rule are incomplete without the congruence rule, and neither system has a complete extension in rank-1 equational axioms. It remains open whether there are higher-rank equational axioms or other Horn rules which would render Milner's or Bloom/Esik's axiomatisations complete.