Related papers: On some computational properties of open sets
Computational models are quantitative representations of systems. By analyzing and comparing the outputs of such models, it is possible to gain a better understanding of the system itself. Though as the complexity of model outputs…
quest for processing speed potential. In fact, we always get a fraction of the technically available computing power (so-called {\em theoretical peak}), and the gap is likely to go hand-to-hand with the hardware complexity of the target…
We revisit Dieter Spreen's notion of a representation associated to a numbered basis equipped with a strong inclusion relation. We show that by relaxing his requirements, we obtain different classically considered representations as…
Continuous first-order logic is used to apply model-theoretic analysis to analytic structures (e.g. Hilbert spaces, Banach spaces, probability spaces, etc.). Classical computable model theory is used to examine the algorithmic structure of…
We develop a general theory of jump operators, which is intended to provide an abstraction of the notion of "limit-computability" on represented spaces. Jump operators also provide a framework with a strong categorical flavor for…
Cost functions provide a framework for constructions of sets Turing below the halting problem that are close to computable. We carry out a systematic study of cost functions. We relate their algebraic properties to their expressive…
In this note, we present a characterization of sets definable in Skolem arithmetic, i.e., the first-order theory of natural numbers with multiplication. This characterization allows us to prove the decidability of the theory. The idea is…
This paper considers the problem of minimizing the ordered weighted average (or ordered median) function of finitely many rational functions over compact semi-algebraic sets. Ordered weighted averages of rational functions are not, in…
We give a selection of major open problems involving selective properties, diagonalizations, and covering properties for sets of real numbers. This is a revision of the version published as a chapter in the book \textbf{Open Problems in…
In previous papers on this project a general static logical framework for formalizing and mechanizing set theories of different strength was suggested, and the power of some predicatively acceptable theories in that framework was explored.…
When faced with the question of how to represent properties in a formal proof system any user has to make design decisions. We have proved three of the theorems from Maskin's 2004 survey article on Auction Theory using the Isabelle/HOL…
Cohesive powers of computable structures are effective analogs of ultrapowers, where cohesive sets play the role of ultrafilters. Let $\omega$, $\zeta$, and $\eta$ denote the respective order-types of the natural numbers, the integers, and…
The problem of finding a (continuous) utility function for a semiorder has been studied since in 1956 R.D. Luce introduced in \emph{Econometrica} the notion. There was almost no results on the continuity of the representation. A similar…
It is here proposed an analysis of symbolic and sub-symbolic models for studying cognitive processes, centered on emergence and logical openness notions.The Theory of Logical Openness connects the Physics of system/environment relationships…
Given a real dataset and a computation family, we wish to encode and store the dataset in a distributed system so that any computation from the family can be performed by accessing a small number of nodes. In this work, we focus on the…
We develop a synthesis of Turing's paradigm of computation and von Neumann's quantum logic to serve as a model for quantum computation with recursion, such that potentially non-terminating computation can take place, as in a quantum Turing…
The theory of the call-by-value lambda-calculus relies on weak evaluation and closed terms, that are natural hypotheses in the study of programming languages. To model proof assistants, however, strong evaluation and open terms are…
In this note we address various algorithmic problems that arise in the computation of the operator norm in unitary representations of a group on Hilbert space. We show that the operator norm in the universal unitary representation is…
The game of SET is a popular card game in which the objective is to form Sets using cards from a special deck. In this paper we study single- and multi-round variations of this game from the computational complexity point of view and…
A notable feature of the TTE approach to computability is the representation of the argument values and the corresponding function values by means of infinitistic names. Two ways to eliminate the using of such names in certain cases are…