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We investigate a small fragment, FCPL, of the fluent calculus. FCPL can be derived from the fluent calculus by allowing a domain description to contain a finite number of actions and fluents, only. Consequently, it is just powerful enough for specifying certain resource sensitive actions. In this paper, we contribute to the research about the fluent calculus (1) by proving that even in this small fragment the entailment problem for a fairly restricted class of formulas is undecidable. (2) We show decidability of a class of formulas which has interesting applications in resource planning. We achieve our results by establishing a tight correspondence between models of FCPL-theories and Petri nets. Then, many problems concerning FCPL-theories can be reduced to problems of the well developed Petri net theory. As a consequence of the correspondence on the structural level, we also expect strong relationships between more general fluent calculus fragments and more general net classes, e.g. coloured Petri nets or predicate transition systems.