Products of effective topological spaces and a uniformly computable Tychonoff Theorem

This article is a fundamental study in computable analysis. In the framework of Type-2 effectivity, TTE, we investigate computability aspects on finite and infinite products of effective topological spaces. For obtaining uniform results we introduce natural multi-representations of the class of all effective topological spaces, of their points, of their subsets and of their compact subsets. We show that the binary, finite and countable product operations on effective topological spaces are computable. For spaces with non-empty base sets the factors can be retrieved from the products. We study computability of the product operations on points, on arbitrary subsets and on compact subsets. For the case of compact sets the results are uniformly computable versions of Tychonoff's Theorem (stating that every Cartesian product of compact spaces is compact) for both, the cover multi-representation and the "minimal cover" multi-representation

The Computable Universe Hypothesis

When can a model of a physical system be regarded as computable? We provide the definition of a computable physical model to answer this question. The connection between our definition and Kreisel's notion of a mechanistic theory is discussed, and several examples of computable physical models are given, including models which feature discrete motion, a model which features non-discrete continuous motion, and probabilistic models such as radioactive decay. We show how computable physical models on effective topological spaces can be formulated using the theory of type-two effectivity (TTE). Various common operations on computable physical models are described, such as the operation of coarse-graining and the formation of statistical ensembles. The definition of a computable physical model also allows for a precise formalization of the computable universe hypothesis--the claim that all the laws of physics are computable.Comment: 33 pages, 0 figures; minor change

The Rice-Shapiro theorem in Computable Topology

We provide requirements on effectively enumerable topological spaces which guarantee that the Rice-Shapiro theorem holds for the computable elements of these spaces. We show that the relaxation of these requirements leads to the classes of effectively enumerable topological spaces where the Rice-Shapiro theorem does not hold. We propose two constructions that generate effectively enumerable topological spaces with particular properties from wn--families and computable trees without computable infinite paths. Using them we propose examples that give a flavor of this class

What Isn’t Obvious about ‘obvious’: A Data-driven Approach to Philosophy of Logic

It is often said that ‘every logical truth is obvious’ (Quine 1970: 82), that the ‘axioms and rules of logic are true in an obvious way’ (Murawski 2014: 87), or that ‘logic is a theory of the obvious’ (Sher 1999: 207). In this chapter, I set out to test empirically how the idea that logic is obvious is reflected in the scholarly work of logicians and philosophers of logic. My approach is data-driven. That is to say, I propose that systematically searching for patterns of usage in databases of scholarly works, such as JSTOR, can provide new insights into the ways in which the idea that logic is obvious is reflected in logical and philosophical practice, i.e., in the arguments that logicians and philosophers of logic actually make in their published work

On the Continuity of Effective Multifunctions

AbstractIf 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, Ceĭtin, 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

Computability of probability measures and Martin-Lof randomness over metric spaces

In this paper we investigate algorithmic randomness on more general spaces than the Cantor space, namely computable metric spaces. To do this, we first develop a unified framework allowing computations with probability measures. We show that any computable metric space with a computable probability measure is isomorphic to the Cantor space in a computable and measure-theoretic sense. We show that any computable metric space admits a universal uniform randomness test (without further assumption).Comment: 29 page

Multi-representation associated to the numbering of a subbasis and formal inclusion relations

We show how the use of a formal inclusion relation associated to a topological (sub)basis, as introduced by Dieter Spreen to study Type 1 computable topological spaces, is also beneficial to the study of represented spaces. We show that different definitions of the multi-representation of a topological space associated to a numbering of a (sub)basis, as considered for instance by Grubba, Weihrauch and Schr\"oder, can be seen as special cases of a more general definition which uses a formal inclusion relation. We show that the use of an appropriate formal inclusion relation guarantees that the representation associated to a computable metric space seen as a topological space always coincides with the Cauchy representation. We also show how the use of a formal inclusion relation guarantees that when defining multi-representations on a set and on one of its subsets, the obtained multi-representations will be compatible, i.e. inclusion will be a computable map. The proposed definitions are also more robust under change of equivalent bases.Comment: 19 page

A generalization of Markov's approach to the continuity problem for Type 1 computable functions

We axiomatize and generalize Markov's approach to the continuity problem for Type 1 computable functions, i.e. the problem of finding sufficient conditions on a computable topological space to obtain a theorem of the form "computable functions are (effectively) continuous". In a computable topological space, a point $x$ is called effectively adherent to a set $A$ if there is an algorithm that on input a neighborhood of $x$ produces a point of $A$ in that neighborhood. We say that a space $X$ satisfies a Markov condition if, whenever a point $x$ of $X$ is effectively adherent to a subset $A$ of $X$, the singleton $\{x\}$ is not a semi-decidable subset of $A\cup\{x\}$. We show that this condition prevents functions whose domain is $X$ from having effective discontinuities, provided that their codomain is a space where points have neighborhood bases of co-semi-decidable sets. We then show that results that forbid effective discontinuities can be turned into (abstract) continuity results on spaces where the closure and effective closure of semi-decidable sets naturally agree -this happens for instance on spaces which admit a dense and computable sequence. This work is motivated by the study of the space of marked groups, for which the author has shown that most known continuity results (of Ceitin, Moschovakis, Spreen) do not apply.Comment: 21 page

Irreversible computable functions

International audienceThe strong relationship between topology and computations has played a central role in the development of several branches of theoretical computer science: foundations of functional programming, computational geometry, computability theory, computable analysis. Often it happens that a given function is not computable simply because it is not continuous. In many cases, the function can moreover be proved to be non-computable in the stronger sense that it does not preserve computability: it maps a computable input to a non-computable output. To date, there is no connection between topology and this kind of non-computability, apart from Pour-El and Richards ''First Main Theorem'', applicable to linear operators on Banach spaces only. In the present paper, we establish such a connection. We identify the discontinuity notion, for the inverse of a computable function, that implies non-preservation of computability. Our result is applicable to a wide range of functions, it unifies many existing ad hoc constructions explaining at the same time what makes these constructions possible in particular contexts, sheds light on the relationship between topology and computability and most importantly allows us to solve open problems. In particular it enables us to answer the following open question in the negative: if the sum of two shift-invariant ergodic measures is computable, must these measures be computable as well? We also investigate how generic a point with computable image can be. To this end we introduce a notion of genericity of a point w.r.t. a function, which enables us to unify several finite injury constructions from computability theory

Effective representations of the space of linear bounded operators

[EN] Representations of topological spaces by infinite sequences of symbols are used in computable analysis to describe computations in topological spaces with the help of Turing machines. From the computer science point of view such representations can be considered as data structures of topological spaces. Formally, a representation of a topological space is a surjective mapping from Cantor space onto the corresponding space. Typically, one is interested in admissible, i.e. topologically well-behaved representations which are continuous and characterized by a certain maximality condition. We discuss a number of representations of the space of linear bounded operators on a Banach space. Since the operator norm topology of the operator space is nonseparable in typical cases, the operator space cannot be represented admissibly with respect to this topology. However, other topologies, like the compact open topology and the Fell topology (on the operator graph) give rise to a number of promising representations of operator spaces which can partially replace the operator norm topology. These representations reflect the information which is included in certain data structures for operators, such as programs or enumerations of graphs. We investigate the sublattice of these representations with respect to continuous and computable reducibility. Certain additional conditions, such as finite dimensionality, let some classes of representations collapse, and thus, change the corresponding graph. Altogether, a precise picture of possible data structures for operator spaces and their mutual relation can be drawn.Work partially supported by DFG Grant BR 1807/4-1Brattka, V. (2003). Effective representations of the space of linear bounded operators. Applied General Topology. 4(1):115-131. https://doi.org/10.4995/agt.2003.20141151314