Abstract We prove that every Turing degree a bounding some non-GL2 degree is recursively enumerable in and above (r.e.a.) some 1-generic degree.

In the representation approach (TTE) to computable analysis, the representations of an algebraic or topological structure for which the basic predicates and functions become computable are of particular interest. There are, however, many predicates (like equality of real numbers) and functions that areabsolutely non-computable, that is, not computable for any representation. Many of these results can be deduced from a simple lemma. In this article we prove this lemma for multi-representations and apply it to a number of examples. As applications, we show that various predicates and functions on computable measure spaces are absolutely non-computable. Since all the arguments are topological, we prove that the predicates are not relatively open and the functions are not relatively continuous for any multi-representation.

Abstract. We prove that there is no sw-complete c.e. real, negatively answering a question in [6].

This paper investigates the consistency property of FC -normal logic program and presents an equivalent deciding condition whether a logic program P is an FC -normal program. The deciding condition describes the characterizations of FC -normal program. By the Petri-net presentation of a logic program, the characterizations of stratification of FC -normal program are investigated. The stratification of FC -normal program motivates us to introduce a new kind of stratification, extended stratification, over logic program. It is shown that an extended (locally) stratified logic program is an FC -normal program. Thus, an extended (locally) stratified logic program has at least one stable model. Finally, we have presented algorithms about computation of consistency property and a few equivalent deciding methods of the finite FC -normal program.

In this paper we give two alternative notions of possibility that satisfy Halpern's two conditions. One of the two notions, for the logic S4n, seems to behave in the same way as Halpern's original one; and the other has quite different properties from those of Halpern's. They exemplify that the answers are negative to the two questions proposed by Halpern, that is, whether his two conditions are sufficient to determine the notion of 'possibility' uniquely for a given logic and whether the results he proved for his three notions of possibility hold for any of those satisfying the two conditions.