Logical closure properties of propositional proof systems - (Extended abstract)

In this paper we define and investigate basic logical closure properties of propositional proof systems such as closure of arbitrary proof systems under modus ponens or substitutions. As our main result we obtain a purely logical characterization of the degrees of schematic extensions of EF in terms of a simple combination of these properties. This result underlines the empirical evidence that EF and its extensions admit a robust definition which rests on only a few central concepts from propositional logic

Logic in the Tractatus

I present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein’s “form-series” device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named. There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is countably infinite, then the property of being a tautology is \Pi^1_1-complete. But third, it is only granted the assumption of countability that the class of tautologies is \Sigma_1-definable in set theory. Wittgenstein famously urges that logical relationships must show themselves in the structure of signs. He also urges that the size of the universe cannot be prejudged. The results of this paper indicate that there is no single way in which logical relationships could be held to make themselves manifest in signs, which does not prejudge the number of objects

Automatic Generation of Proof Tactics for Finite-Valued Logics

A number of flexible tactic-based logical frameworks are nowadays available that can implement a wide range of mathematical theories using a common higher-order metalanguage. Used as proof assistants, one of the advantages of such powerful systems resides in their responsiveness to extensibility of their reasoning capabilities, being designed over rule-based programming languages that allow the user to build her own `programs to construct proofs' - the so-called proof tactics. The present contribution discusses the implementation of an algorithm that generates sound and complete tableau systems for a very inclusive class of sufficiently expressive finite-valued propositional logics, and then illustrates some of the challenges and difficulties related to the algorithmic formation of automated theorem proving tactics for such logics. The procedure on whose implementation we will report is based on a generalized notion of analyticity of proof systems that is intended to guarantee termination of the corresponding automated tactics on what concerns theoremhood in our targeted logics

Superconducting electronics

During the last decades superconducting electronics has been the most prominent area of research for small scale applications of superconductivity. It has experienced quite a stormy development, from individual low frequency devices to devices with high integration density and pico second switching time. Nowadays it offers small losses, high speed and the potential for large scale integration and is superior to semiconducting devices in many ways ¿ apart from the need for cooling by liquid helium for devices based on classical superconductors like niobium, or cooling by liquid nitrogen or cryocoolers (40K to 77K) for high-Tc superconductors like YBa2Cu3O7. This article gives a short overview over the current state of the art on typical devices out of the main application areas of superconducting electronics

In search of an evolutionary coding style

In the near future, all the human genes will be identified. But understanding the functions coded in the genes is a much harder problem. For example, by using block entropy, one has that the DNA code is closer to a random code then written text, which in turn is less ordered then an ordinary computer code; see \cite{schmitt}. Instead of saying that the DNA is badly written, using our programming standards, we might say that it is written in a different style -- an evolutionary style. We will suggest a way to search for such a style in a quantified manner by using an artificial life program, and by giving a definition of general codes and a definition of style for such codes.Comment: 14 pages, 7 postscript figure

An Operational Definition of Topological Order

The unrivaled robustness of topologically ordered states of matter against perturbations has immediate applications in quantum computing and quantum metrology, yet their very existence poses a challenge to our understanding of phase transitions. In particular, topological phase transitions cannot be characterized in terms of local order parameters, as it is the case with conventional symmetry-breaking phase transitions. Currently, topological order is mostly discussed in the context of nonlocal topological invariants or indirect signatures like the topological entanglement entropy. However, a comprehensive understanding of what actually constitutes topological order is still lacking. Here we show that one can interpret topological order as the ability of a system to perform topological error correction. We find that this operational approach corresponding to a measurable observable does not only lay the conceptual foundations for previous classifications of topological order, but it can also be applied to hitherto inaccessible problems, such as the question of topological order for mixed quantum states arising in open quantum systems. We demonstrate the existence of topological order in open systems and their phase transitions to topologically trivial states, including topological criticality. Our results demonstrate the viability of topological order in nonequilibrium quantum systems and thus substantially broaden the scope of possible technological applications. We therefore expect our work to be a starting point for many future theoretical and experimental investigations, such as the application of our approach to fracton or Floquet topological order, or the direct experimental realization of the error correction protocol presented in our work for the development of future quantum technological devices.Comment: 7 pages, 5 figure