Computing with Infinite Data: Topological and Logical Foundations Edited by (original) (raw)

Computing with Infinite Data: Topological and Logical Foundations (Dagstuhl Seminar 11411)

There is a large gap between mathematical structures and the structures computer implementations are based on. To stimulate research to overcome this-especially for infinitary structureshighly non-trivial problem the Dagstuhl Seminar 11411 "Computing with Infinite Data: Topological and Logical Foundations" was held. This report collects the ideas that were presented and discussed during the course of the seminar.

07441 Abstracts Collection--Algorithmic-Logical Theory of Infinite Structures}

Abstract From 28.10. to 02.11. 2007, the Dagstuhl Seminar 07441``Algorithmic-Logical Theory of Infinite Structures''was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper.

On problems of databases over a fixed infinite universe

Banach Center Publications, 1999

In the relational model of databases a database state is thought of as a nite collection of relations between elements. For many applications it is convenient to pre-x an in nite domain where the nite relations are going to be de ned. Often, we also x a set of domain functions and/or relations. These functions/relations are in nite by their nature. Some special problems arise if we use such an approach. In the paper we discuss some of the problems. We show that there exists a recursive domain with decidable theory in which (1) there is no recursive syntax for nite queries, and in which (2) the state-safety problem is undecidable. We provide very general conditions on the FO theory of an ordered domain that ensure collapse of order-generic extended FO queries to pure order queries over this domain: the Pseudo-nite Homogeneity Property and a stronger Isolation Property. We further distinguish one broad class of ordered domains satisfying the Isolation Property, the so-called quasi-o-minimal domains. This class includes all o-minimal domains, but also the ordered group of integer numbers and the ordered semigroup of natural numbers, and some other domains. We generalize all the notions to the case of nitely representable database states | as opposed to nite states | and develop a general lifting technique that, essentially, allows us to extend any result of the kind we are interested in, from nite to nitely-representable states. We show, however, that these results cannot be transferred to arbitrary in nite states. We prove that safe Datalog :;<z-programs do not have any e ective syntax.

Mathematical Foundations of the Infinity Computer

All the existing computers are able to execute arithmetical operations only with finite numerals. Operations with infinite and infinitesimal quantities could not be realized. The paper describes a new positional system with infinite radix allowing us to write down finite, in finite, and infinitesimal numbers as particular cases of a unique framework. The new approach both gives possibilities to execute calculations of a new type and simplifies fields of mathematics where usage of infinity and/o r infinitesimals is required. Usage of the numeral system described in the paper gives possibility to introduc e a new type of computer - Infinity Computer - able to operate not only with finite numbers but also with infinite a nd infinitesimal ones.

08271 Abstracts Collection--Topological and Game-Theoretic Aspects of Infinite Computations}

From June 29, 2008, to July 4, 2008, the Dagstuhl Seminar 08271 Topological and Game-Theoretic Aspects of Innite Computations was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, many participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper.

Recursion and topology on 2⩽ω for possibly infinite computations

Theoretical Computer Science, 2004

In the context of possibly infinite computations yielding finite or infinite (binary) outputs, the space 2 ≤ω = 2 * ∪ 2 ω appears to be one of the most fundamental spaces in Computer Science. Though underconsidered, next to 2 ω , this space can be viewed (§3.5.2) as the simplest compact space native to computer science. In this paper we study some of its properties involving topology and computability. Though 2 ≤ω can be considered as a computable metric space in the sense of computable analysis, a direct and self-contained study, based on its peculiar properties related to words, is much illuminating. It is well known that computability for maps 2 ω → 2 ω reduces to continuity with recursive modulus of continuity. With 2 ≤ω , things get less simple. Maps 2 ω → 2 ≤ω or 2 ≤ω → 2 ≤ω induced by input/output behaviors of Turing machines on finite or infinite words-which we call semicomputable maps-are not necessarily continuous but merely lower semicontinuous with respect to the prefix partial ordering on 2 ≤ω. Continuity asks for a stronger notion of computability. We prove for (semi)continuous and (semi)computable maps F : I → O with I, O ∈ {2 ω , 2 ≤ω } a detailed representation theorem (Thm.82) via functions f : 2 * → 2 * following two approaches: bottom-up from f to F and top-down from F to f .

Unbounded computational structures

Software: Practice and Experience, 1978

The concept of suspended evaluation is used as an approach to co-routines. Problems from the literature involving infinite data structures are solved in a LISP-like applicative language to demonstrate that simple new semantics can enrich old and 'friendly' control structures. It appears that the very nature of these problems draws control structure and data structure together, so that issues of style may be studied at once for both. KEY WORDS Co-routines LISP Data structures Control structures Programming style * Research reported herein was supported (in part) by the National Science Foundation under grants numbered DCR75-06678 and MCS75-08145.

CID - Computing with Infinite Data

Bull. EATCS, 2017

CID is a new international research project with about 20 participating institutions, mainly from Europe, but also from Chile, Japan, New Zealand, Russia, Singapore, South Africa, South Korea, and the USA. It is funded by the European Union as well as several national funding organisations, and is running for four years. The joint research will study important aspects—both theoretical as well as applied—of computing with infinite o bjects. A central aim is laying the grounds for the generation of efficient and verified software in engineering applications. A prime example for infinite data is provided by the real numbers, most commonly conceived as infinite sequences of d igits. Since the reals are fundamental in mathematics, any attempt to compute objects of mathematical interest has to be based on an implementation of real numbers. While most applications in science and engineering substitute the reals with floating p oint n umbers o f fi xed finite precision and thus have to deal...

Multidimensional infinite data in the language Lucid

Mathematical Structures in Computer Science, 2014

Although the language Lucid was not originally intended to support computing with infinite data structures, the notion of (infinite) sequence quickly came to the fore, together with a demand-driven computation model in which demands are propagated for the values of particular values at particular index points. This naturally generalized to sequences of multiple dimensions so that a programmer could, for example, write a program that could be understood as a (nonterminating) loop in which one of the loop variables is an infinite vector.Programmers inevitably found use for more and more dimensions, which led to a problem which is fully solved for the first time in this paper. The problem is that the implementation's cache requires some estimate of the dimensions actually used to compute a value being fetched. This estimate can be difficult or (if dimensions are passed as parameters) impossible to obtain, and the demand-driven evaluation model for Lucid breaks down.We outline the e...