A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature

A Swiss Pocket Knife for Computability

This research is about operational- and complexity-oriented aspects of classical foundations of computability theory. The approach is to re-examine some classical theorems and constructions, but with new criteria for success that are natural from a programming language perspective. Three cornerstones of computability theory are the S-m-ntheorem; Turing's "universal machine"; and Kleene's second recursion theorem. In today's programming language parlance these are respectively partial evaluation, self-interpretation, and reflection. In retrospect it is fascinating that Kleene's 1938 proof is constructive; and in essence builds a self-reproducing program. Computability theory originated in the 1930s, long before the invention of computers and programs. Its emphasis was on delimiting the boundaries of computability. Some milestones include 1936 (Turing), 1938 (Kleene), 1967 (isomorphism of programming languages), 1985 (partial evaluation), 1989 (theory implementation), 1993 (efficient self-interpretation) and 2006 (term register machines). The "Swiss pocket knife" of the title is a programming language that allows efficient computer implementation of all three computability cornerstones, emphasising the third: Kleene's second recursion theorem. We describe experiments with a tree-based computational model aiming for both fast program generation and fast execution of the generated programs.Comment: In Proceedings Festschrift for Dave Schmidt, arXiv:1309.455

Cinnamons: A Computation Model Underlying Control Network Programming

We give the easily recognizable name "cinnamon" and "cinnamon programming" to a new computation model intended to form a theoretical foundation for Control Network Programming (CNP). CNP has established itself as a programming paradigm combining declarative and imperative features, built-in search engine, powerful tools for search control that allow easy, intuitive, visual development of heuristic, nondeterministic, and randomized solutions. We define rigorously the syntax and semantics of the new model of computation, at the same time trying to keep clear the intuition behind and to include enough examples. The purposely simplified theoretical model is then compared to both WHILE-programs (thus demonstrating its Turing-completeness), and the "real" CNP. Finally, future research possibilities are mentioned that would eventually extend the cinnamon programming into the directions of nondeterminism, randomness, and fuzziness.Comment: 7th Intl Conf. on Computer Science, Engineering & Applications (ICCSEA 2017) September 23~24, 2017, Copenhagen, Denmar

Algorithmic complexity of quantum capacity

Recently the theory of communication developed by Shannon has been extended to the quantum realm by exploiting the rules of quantum theory. This latter stems on complex vector spaces. However complex (as well as real) numbers are just idealizations and they are not available in practice where we can only deal with rational numbers. This fact naturally leads to the question of whether the developed notions of capacities for quantum channels truly catch their ability to transmit information. Here we answer this question for the quantum capacity. To this end we resort to the notion of semi-computability in order to approximately (by rational numbers) describe quantum states and quantum channel maps. Then we introduce algorithmic entropies (like algorithmic quantum coherent information) and derive relevant properties for them. Finally we define algorithmic quantum capacity and prove that it equals the standard one

Kolmogorov Complexity in perspective. Part II: Classification, Information Processing and Duality

We survey diverse approaches to the notion of information: from Shannon entropy to Kolmogorov complexity. Two of the main applications of Kolmogorov complexity are presented: randomness and classification. The survey is divided in two parts published in a same volume. Part II is dedicated to the relation between logic and information system, within the scope of Kolmogorov algorithmic information theory. We present a recent application of Kolmogorov complexity: classification using compression, an idea with provocative implementation by authors such as Bennett, Vitanyi and Cilibrasi. This stresses how Kolmogorov complexity, besides being a foundation to randomness, is also related to classification. Another approach to classification is also considered: the so-called "Google classification". It uses another original and attractive idea which is connected to the classification using compression and to Kolmogorov complexity from a conceptual point of view. We present and unify these different approaches to classification in terms of Bottom-Up versus Top-Down operational modes, of which we point the fundamental principles and the underlying duality. We look at the way these two dual modes are used in different approaches to information system, particularly the relational model for database introduced by Codd in the 70's. This allows to point out diverse forms of a fundamental duality. These operational modes are also reinterpreted in the context of the comprehension schema of axiomatic set theory ZF. This leads us to develop how Kolmogorov's complexity is linked to intensionality, abstraction, classification and information system.Comment: 43 page