A Menagerie of Non-Finitely Based Process Semantics over BPA*: From Ready Simulation Semantics to Completed Traces
Authors: Aceto Luca , Fokkink Willem Jan , Ingólfsdóttir Anna 
Published: February 2016

<p>Fokkink and Zantema ((1994) Computer Journal 37:259-267) have shown that<br />bisimulation equivalence has a finite equational axiomatization over the language<br />of Basic Process Algebra with the binary Kleene star operation (BPA*). In light<br />of this positive result on the mathematical tractability of bisimulation equivalence<br />over BPA*, a natural question to ask is whether any other (pre)congruence relation<br />in van Glabbeek's linear time/branching time spectrum is finitely (in)equationally<br />axiomatizable over it. In this paper, we prove that, unlike bisimulation equivalence,<br />none of the preorders and equivalences in van Glabbeek's linear time/branching time<br />spectrum, whose discriminating power lies in between that of ready simulation and<br />that of completed traces, has a finite equational axiomatization. This we achieve by<br />exhibiting a family of (in)equivalences that holds in ready simulation semantics, the<br />finest semantics that we consider, whose instances cannot all be proven by means of<br />any finite set of (in)equations that is sound in completed trace semantics, which is<br />the coarsest semantics that is appropriate for the language BPA*. To this end, for<br />every finite collection of (in)equations that are sound in completed trace semantics, we<br />build a model in which some of the (in)equivalences of the family under consideration<br />fail. The construction of the model mimics the one used by Conway ((1971) Regular<br />Algebra and Finite Machines, page 105) in his proof of a result, originally due to<br />Redko, to the effect that infinitely many equations are needed to axiomatize equality<br />of regular expressions.</p><p>Our non-finite axiomatizability results apply to the language BPA* over an arbitrary<br />non-empty set of actions. In particular, we show that completed trace equivalence<br />is not finitely based over BPA* even when the set of actions is a singleton.<br />Our proof of this result may be easily adapted to the standard language of regular expressions to yield a solution to an open problem posed by Salomaa ((1969) Theory<br />of Automata, page 143).<br />Another semantics that is usually considered in process theory is trace semantics.<br />Trace semantics is, in general, not preserved by sequential composition, and is<br />therefore inappropriate for the language BPA*. We show that, if the set of actions<br />is a singleton, trace equivalence and preorder are preserved by all the operators in<br />the signature of BPA, and coincide with simulation equivalence and preorder, respectively.<br />In that case, unlike all the other semantics considered in this paper, trace<br />semantics have nite, complete equational axiomatizations over closed terms.</p><p> </p><p>AMS Subject Classification (1991): 08A70, 03C05, 68Q10, 68Q40, 68Q45,<br />68Q55, 68Q68, 68Q70.<br />CR Subject Classification (1991): D.3.1, F.1.1, F.1.2, F.3.2, F.3.4, F.4.1.<br />Keywords and Phrases: Concurrency, process algebra, Basic Process Algebra<br />(BPA*), Kleene star, bisimulation, ready simulation, simulation, completed trace semantics,<br />ready trace semantics, failure trace semantics, readiness semantics, failures<br />semantics, trace semantics, equational logic, complete axiomatizations.</p><p> </p>