Recursive Automata on In nite

The present paper gives a thorough characterization of classes of !-languages deened by several classes of recursive automata and elementary acceptance conditions in terms of the arithmetical hierarchy. Here it is interesting to note that nitely branching nondeterministic looping or CoB uchi-accepting automata are only as powerful as their de-terministic counterparts whereas nondeterministic B uchi-accepting au-tomata are more powerful than deterministic ones. These results are used to estimate the complexity of veriication problems for programs and speciications deening recursive automata. This paper is concerned with sets of innnite sequences (so-called !-languages) deenable by recursive automata. It is motivated by the fact that the study of innnite computations has gained increasing interest in computer science, especially in the area of speciication and veriication of concurrent programs (cf. e.g. the brief surveys contained in Kl90], Th90] or Va87]). A framework based on recursive automata that uniies a large number of trends in this area, such as temporal logic, model checking, automata theory, and fair termination , was presented by Vardi Va87]. He suggests the use of eeective not necessarily nite-state (so-called recursive) automata on innnite words. a similar approach is contained in Chap. 7 of Kl90]. Earlier investigations on recursive devices on innnite words are contained in the papers CG78] and WS77] (cf. also St86a]) dealing with innnite computations on Turing machines. Regard a program P as deening an automaton P which describes the possible execution sequences (computations of the program) T(P). Similarly, a spec-iication S deenes an automaton S describing the allowed computations T(S). To verify that P satisses S amounts to checking whether every computation of P is a computation of S, that is, whether the behaviour T(P) is contained in T(S) (cf. VW86]). Hence the veriication problem reduces to the containment problem \T(P) T(S)?" This problem is intrinsically diicult for nondeterministic recursive automata, because the question \T(P) T(S)?" depends on the one hand on a universal quantiier over an innnite object, and on the other hand the description of the behaviour of a nondeterministic recursive automaton involves an existential quantiier over an innnite object (cf. Va87] and SW77]), therefore it is 1 2-complete, as observed e.g. in DY92] or Si89]. For some classes of recursive automata, e.g. for deterministic ones which do not involve this existential quantiier, however, the veriication problem is simpler. In this paper we investigate in more detail which reasons lead to a sim-pliication of the …