Towards A Theory Of Quantum Computability

We propose a definition of quantum computable functions as mappings between superpositions of natural numbers to probability distributions of natural numbers. Each function is obtained as a limit of an infinite computation of a quantum Turing machine. The class of quantum computable functions is recursively enumerable, thus opening the door to a quantum computability theory which may follow some of the classical developments

Quantum Turing Machines Computations and Measurements

Contrary to the classical case, the relation between quantum programming languages and quantum Turing Machines (QTM) has not being fully investigated. In particular, there are features of QTMs that have not been exploited, a notable example being the intrinsic infinite nature of any quantum computation. In this paper we propose a definition of QTM, which extends and unifies the notions of Deutsch and Bernstein and Vazirani. In particular, we allow both arbitrary quantum input, and meaningful superpositions of computations, where some of them are "terminated" with an "output", while others are not. For some infinite computations an "output" is obtained as a limit of finite portions of the computation. We propose a natural and robust observation protocol for our QTMs, that does not modify the probability of the possible outcomes of the machines. Finally, we use QTMs to define a class of quantum computable functions---any such function is a mapping from a general quantum state to a probability distribution of natural numbers. We expect that our class of functions, when restricted to classical input-output, will be not different from the set of the recursive functions.Comment: arXiv admin note: substantial text overlap with arXiv:1504.02817 To appear on MDPI Applied Sciences, 202

Propositional computability logic I

In the same sense as classical logic is a formal theory of truth, the recently initiated approach called computability logic is a formal theory of computability. It understands (interactive) computational problems as games played by a machine against the environment, their computability as existence of a machine that always wins the game, logical operators as operations on computational problems, and validity of a logical formula as being a scheme of "always computable" problems. The present contribution gives a detailed exposition of a soundness and completeness proof for an axiomatization of one of the most basic fragments of computability logic. The logical vocabulary of this fragment contains operators for the so called parallel and choice operations, and its atoms represent elementary problems, i.e. predicates in the standard sense. This article is self-contained as it explains all relevant concepts. While not technically necessary, however, familiarity with the foundational paper "Introduction to computability logic" [Annals of Pure and Applied Logic 123 (2003), pp.1-99] would greatly help the reader in understanding the philosophy, underlying motivations, potential and utility of computability logic, -- the context that determines the value of the present results. Online introduction to the subject is available at http://www.cis.upenn.edu/~giorgi/cl.html and http://www.csc.villanova.edu/~japaridz/CL/gsoll.html .Comment: To appear in ACM Transactions on Computational Logi

From truth to computability I

The recently initiated approach called computability logic is a formal theory of interactive computation. See a comprehensive online source on the subject at http://www.cis.upenn.edu/~giorgi/cl.html . The present paper contains a soundness and completeness proof for the deductive system CL3 which axiomatizes the most basic first-order fragment of computability logic called the finite-depth, elementary-base fragment. Among the potential application areas for this result are the theory of interactive computation, constructive applied theories, knowledgebase systems, systems for resource-bound planning and action. This paper is self-contained as it reintroduces all relevant definitions as well as main motivations.Comment: To appear in Theoretical Computer Scienc

Weighted Branching Simulation Distance for Parametric Weighted Kripke Structures

This paper concerns branching simulation for weighted Kripke structures with parametric weights. Concretely, we consider a weighted extension of branching simulation where a single transitions can be matched by a sequence of transitions while preserving the branching behavior. We relax this notion to allow for a small degree of deviation in the matching of weights, inducing a directed distance on states. The distance between two states can be used directly to relate properties of the states within a sub-fragment of weighted CTL. The problem of relating systems thus changes to minimizing the distance which, in the general parametric case, corresponds to finding suitable parameter valuations such that one system can approximately simulate another. Although the distance considers a potentially infinite set of transition sequences we demonstrate that there exists an upper bound on the length of relevant sequences, thereby establishing the computability of the distance.Comment: In Proceedings Cassting'16/SynCoP'16, arXiv:1608.0017