This paper is divided in two parts. In the rst one we analyse in great generality data types in relation to partial morphisms. We introduce partial function spaces, partial cartesian closed categories and complete objects, motivate their introduction and show some of their properties. In the second part we dene the (partial) cartesian closed category GEN of generalized numbered sets, prove that it is a good extension of the category of numbered sets and show how it is related to the recursive topos. Introduction By data type one usually means a set of objects of the same kind, suitable for manipulation by a computer program. Of course, computers actually manipulate formal representations of objects. The purpose of the mathematical semantics of programming languages, however, is to characterize data types (and functions on them) in a way which is independent of any specic representation mechanism. So the objects one deals with are mostly elements of structures borrowed fro...

INTRODUCTION By an S-valued distribution on a topos E bounded over a base topos S it is meant here a cocontinuous S-indexed functor : E ! S. Since introduced by F. W. Lawvere in 1983, considerable progress has been made in the study of distributions on toposes from a variety of viewpoints [19, 15, 24, 5, 6, 12, 7, 8, 9]. However, much work still remains to be done in this area. The purpose of this paper is to deepen our understanding of topos distributions by exploring a (dual) lattice-theoretic notion of distribution algebra. We characterize the distribution algebras in E relative to S as the S-bicomplete S-atomic Heyting algebras in E . As an illustration, we employ distribution algebras explicitly in order to give an alternative description of the display locale (complete spread) of a distribution [10, 12, 7].

We continue the investigation of the extension into the topos realm of the concepts introduced by R.H. Fox [10] and E. Michael [22] in connection with topological singular coverings. In particular, we construct an analogue of the Michael completion of a spread and compare it with the analogue of the Fox completion obtained earlier by the first two named authors [4]. Two ingredients are present in our analysis of geometric morphisms ': F ! E between toposes bounded over a base topos S. The first is the nature of the domain of ', which need only be assumed to be a "definable dominance" over S, a condition that is trivially satisfied if S is a Boolean topos. The Heyting algebras arising from the object S of truth values in the base topos play a special role in that they classify the de nable monomorhisms in those toposes. The geometric morphisms F ! F 0 over E which preserve these Heyting algebras (and that are not typically complete) are said to be strongly pure. The second is the nature of ' itself, which is assumed to be some kind of a spread. Applied to a spread, the (strongly pure, weakly entire) factorization obtained here gives what we call the "Michael completion" of the given spread. Whereas the Fox complete spreads over a topos E correspond to the S-valued Lawvere distributions on E [21] and relate to the distribution algebras [7], the Michael complete spreads seem to correspond to some sort of "S-additive measures" on E whose analysis we do not pursue here. We close the paper with several other open questions and directions for future work.

In program synthesis, we transform a specification into a system that is guaranteed to satisfy the specification. When the system is open, then at each moment it reads input signals and writes output signals, which depend on the input signals and the history of the computation so far. The specification considers all possible input sequences. Thus, if the specification is linear, it should hold in every computation generated by the interaction, and if the specification is branching, it should hold in the tree that embodies all possible input sequences.

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