## On deciding trace equivalences for processes.(English)Zbl 0783.68043

Summary: We investigate the complexity of deciding trace, maximal trace and $$\omega$$-equivalences for finite state processes and recursively defined processes specified by normed context-free grammars (CFGs) in Greibach normal form (GNF). The main results are as follows:
(1) Trace, maximal trace and $$\omega$$-trace equivalences for processes specified by normed GNF CFGs are all undecidable. For this class of processes, the regularity problem with respect to trace, maximal trace or $$\omega$$-trace equivalence is also undecidable. Moreover, all these undecidability results hold even for locally unary processes. For processes specified by unary GNF CFGs, the maximal trace equivalence is $$\prod^ p_ 2$$-complete while the $$\omega$$-trace equivalence is $$NL$$- complete and the trace equivalence is decidable in polynomial time by a dynamic programming algorithm.
(2) Trace, maximal trace and $$\omega$$-trace equivalences for finite state processes are PSPACE-complete. This holds even for locally unary finite state processes. For unary finite state processes, the maximal trace equivalence is co-$$NP$$-complete while trace and $$\omega$$-trace equivalences are $$NL$$-complete.

### MSC:

 68Q10 Modes of computation (nondeterministic, parallel, interactive, probabilistic, etc.) 68Q55 Semantics in the theory of computing
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