Chapter III - Events, Languages, and Acceptors

Abstract

This chapter focuses on probabilistic languages and probabilistic events (PEs). The chapter discusses the closure properties of those languages and events, and their relation to regular events. Some particular cases such as definite, quasi-definite, and exclusive events are investigated, and the problem of approximating PEs by non-probabilistic ones is reviewed. A PE f is a regular event if the underlying probabilistic automaton is deterministic and the function f can assume only the values 0 or 1. By definition, the class of PEs contains, as a proper subclass, the class of regular events. The class of pseudo probabilistic events (SPEs) includes, as a proper subclass, the class of PEs as the values of a PE f are bounded while the values of an SPE may increase beyond and bound. On the other hand, not every event is an SPE. Various theorems are proven in the chapter. The closure properties of probabilistic cut-point events (PCEs) and their relation to general cut-point event (GCEs) are also discussed.