Dieter Spreen

On Some Decision Problems in Programming

One of the central problems in programming is the correctness problem, i.e., the question of whether a program computes a given function. We choose a rather general formal semantical framework, effectively given topological T0-spaces, and study the problem to decide whether an element of the space is equal to a fixed element. Moreover, we consider the problems of deciding for two elements, whether they are equal and whether one approximates the other in the specialization order. These are one-one equivalent for a large class of spaces, including effectively given Scott domains. All these problems are undecidable. In most cases they are complete on some level of the arithmetical and/or the Boolean hierarchy. The complexity respectively depends on whether the fixed element is not finite and whether the space contains a nonfinite element. The problem of deciding whether an element is not finite is potentially ?02-complete and for domain-like spaces the membership problem of any nonempty set of nonfinite elements that intersects the effective closure of its complement is ?02-hard. If the given element is finite or the space contains only finite elements, the complexity also depends on the location of the given element in the specialization order and/or the boundedness of the set of lengths of all decreasing chains of basic open sets.

Information systems revisited – the general continuous case

In this paper a new notion of continuous information system is introduced. It is shown that the information systems of this kind generate exactly the continuous domains. The new information systems are of the same logic-oriented style as the information systems first introduced by Scott in 1982: they consist of a set of tokens, a consistency predicate and an entailment relation satisfying a set of natural axioms. It is shown that continuous information systems are closely related to abstract bases. Indeed, both categories are equivalent. Since it is known that the categories of abstract bases and/or continuous domains are equivalent, it follows that the category of continuous information systems is also equivalent to that of continuous domains. In applications, mostly subclasses of continuous domains are considered. For example, the domains have to be pointed, algebraic, bounded-complete or FS. Conditions are presented that, when fulfilled by a continuous information system, force the generated domain to belong to the required subclass. In most cases the requirements are not only sufficient but also necessary.

Effective inseparability in a topological setting

Abstract Effective inseparability of pairs of sets is an important notion in logic and computer science. We study the effective inseparability of sets which appear as index sets of subsets of an effectively given topological T0-space and discuss its consequences. It is shown that for two disjoint subsets X and Y of the space one can effectively find a witness that the index set of X cannot be separated from the index set of Y by a recursively enumerable set, if X intersects the topological closure of an effectively enumerable subset of Y. As a consequence of a more general parametric inseparability result a theorem of Rice-Shapiro type is obtained. Moreover, under some additional requirements it follows that nonopen subsets have productive index sets. This implies a generalized Rice theorem: Connected spaces have only trivial completely recursive subsets. As application some decision problems in computable analysis and domain theory are studied. It follows that the complement of the halting problem can be reduced to the problem to decide of a number whether it is a computable irrational. The same is true for the problems to decide whether two numbers are equal, whether one is not greater than the other, and whether a number is equal to a given number. In the case of an effectively given continuous complete partial order the complexity of the last problem depends on whether the given element is the smallest element, in which case the complement of the halting problem is reducible to it, whether it is a base element and maximal, then the decision problem is recursively isomorphic to the halting problem, or whether it is none of these. In this case, both the halting problem and its complement are reducible to the problem. The same is true in nontrivial cases for the problems whether an element belongs to the basis, whether two elements of the partial order are equal, or whether one approximates the other. In general, for any nonempty proper subset of the partial order either the halting problem or its complement can be reduced to the membership problem of the subset.

Representations versus numberings: on the relationship of two computability notions

Abstract This paper gives an answer to Weihrauchs (Computability, Springer, Berlin, 1987) question whether and, if not always, when an effective map between the computable elements of two represented sets can be extended to a (partial) computable map between the represented sets. Examples are known showing that this is not possible in general. A condition is introduced and for countably based topological T 0 -spaces it is shown that exactly the (partial) effective maps meeting the requirement are extendable. For total effective maps the extra condition is satisfied in the standard cases of effectively given separable metric spaces and continuous directed-complete partial orders, in which the extendability is already known. In the first case a similar result holds also for partial effective maps, but not in the second.

Can Partial Indexings Be Totalized

In examples like the total recursive functions or the computable real numbers the canonical indexings are only partial maps. It is even impossible in these cases to nd an equivalent total numbering. We consider eeectively given topological T0-spaces and study the problem in which cases the canonical numberings of such spaces can be totalized, i. e., have an equivalent total indexing. Moreover, we show under very natural assumptions that such spaces can eeectively and eeectively homeomorphically be embedded into a totally indexed algebraic constructively directed-complete partial order.

On the Continuity of Effective Multifunctions

If one wants to compute with infinite objects like real numbers or data streams, continuity is a necessary requirement: better and better (finite) approximations of the input are transformed in better and better (finite) approximations of the output. In case the objects are constructively generated, they can be represented by a finite description of the generating procedure. By effectively transforming such descriptions for the generation of the input (respectively, their codes) in (the code of) a description for the generation of the output another type of computable operation is obtained. Such operations are also called effective. The relationship of both classes of operations has always been a question of great interest and well known theorems such as those of Myhill and Shepherdson, Kreisel, Lacombe and Shoenfield, Ceitin, and/or Moschovakis present answers for important special cases. A general, unifying approach has been developed by the present author in [D. Spreen. On effective topological spaces. The Journal of Symbolic Logic, 63 (1998), 185-221. Corrections ibid., 65 (2000), 1917-1918]. In this paper the approach is extended to the case of multifunctions. Such functions appear very naturally in applied mathematics, logic and theoretical computer science. Various ways of coding (indexing) sets are discussed and effective versions of several continuity notions for multifunctions are introduced. For each of these notions an indexing system for sets is exhibited so that the multifunctions that are effective with respect to this indexing system and possess certain witness functions are exactly the multifunction which are effectively continuous with respect to the continuity notion under consideration. Important special cases are discussed where such witnessing functions always exist.

Logic Colloquium 2005: On some problems in computable topology

Computations in spaces like the real numbers are not done on the points of the space itself but on some representation. If one considers only computable points, i.e., points that can be approximated in a computable way, finite objects as the natural numbers can be used for this. In the case of the real numbers such an indexing can e.g. be obtained by taking the Godel numbers of those total computable functions that enumerate a fast Cauchy sequence of rational numbers. Obviously, the numbering is only a partial map. It will be seen that this is not a consequence of a bad choice, but is so by necessity. The paper will discuss some consequences. All is done in a rather general topological framework.

On functions preserving levels of approximation: a refined model construction for various lambda calculi

Abstract In this paper dI-domains are enriched by a family of projections which assign to each point a sequence of canonical approximations. The morphisms are stable maps that preserve the levels of approximation generated by the projections. For the computation of an approximation of a given level of the output they require, in addition, only information about the input of at most the same level of approximation. It is shown that the category of these domains and maps is Cartesian closed. The set of morphisms between two such domains is a domain of the same kind, but turns out not to be an exponent in the category. Domains D are constructed which are isomorphic to the exponent D D . Moreover, it is proved that the space of retractions in an exponent D D is a retract of this. Both results together provide new models of Amadio-Longos extension λβp of the λ-calculus. As has been shown by Amadio and Longo, strong type theories which incorporate the Type: Type assumption can be syntactically interpreted in this calculus.

On r.e. inseparability of CPO index sets

In this paper the r.e. inseparability and the effective r.e. inseparability of index sets under certain indexings of the computable elements of an effective cpo are studied. As a consequence of the main result on effective r.e. inseparability we obtain a generalization of a theorem by McNaughton. As a further application we obtain generalizations of results by Myhill/Dekker on the productivity of certain index sets. From this we infer the generalization of theorems by Rice/Shapiro/McNaughton/Myhill and Myhill/Shepherdson. This demonstrates the importance of the r.e. inseparability notion.

A note on partial numberings

The different behaviour of total and partial numberings with respect to the reducibility preorder is investigated. Partial numberings appear quite naturally in computability studies for topological spaces. The degrees of partial numberings form a distributive lattice which in the case of an infinite numbered set is neither complete nor contains a least element. Friedberg numberings are no longer minimal in this situation. Indeed, there is an infinite descending chain of non-equivalent Friedberg numberings below every given numbering, as well as an uncountable antichain. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)