Uppsala University
Department of Information Technology

### Technical Report 2008-016

# R-automata

### Parosh Aziz Abdulla, Pavel Krcal, and Wang Yi

June 2008

**Abstract:**We introduce *R-automata* - a model for analysis of systems with resources which are consumed in small parts but which can be replenished at once. An R-automaton is a finite state machine which operates on a finite number of unbounded counters (modeling the resources). The values of the counters can be incremented, reset to zero, or left unchanged along the transitions. We define the language accepted by an R-automaton relative to a natural number *D* as the set of words allowing a run along which no counter value exceeds *D*. As the main result, we show decidability of the universality problem, i.e., the problem whether there is a number *D* such that the corresponding language is universal. The decidability proof is based on a reformulation of the problem in the language of finite monoids and solving it using the factorization forest theorem. This approach extends the way in which the factorization forest theorem was used to solve the limitedness problem for distance automata in Simon, 1994. We also show decidability of the non-emptiness problem and the limitedness problem, i.e., whether there is a natural number *D* such that the corresponding language is non-empty resp. all the accepted words can also be accepted with counter values smaller than *D*. Finally, we extend the decidability results to R-automata with Büchi acceptance conditions.

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