Classical Turing Machine Research Articles

'Quantum supremacy' revisited: low-complexity, deterministic solutions of the original Deutsch-Jozsa problem in classical physical systems.

The original Deutsch-Jozsa (oDJ) problem is for an oracle (realized here as a database) of size N, where, according to their claim, the deterministic solution of the problem on a classical Turing computer requires O(N) computational complexity. They produced the famous Deutsch-Jozsa quantum algorithm that offered an exponential speed-up over the classical computer, namely O[log(N)] complexity for the solution in a quantum computer. In this paper, the problem is implemented on an instantaneous noise-based logic processor. It is shown that, similarly to the quantum algorithm, the oDJ problem can deterministically be solved with O[log(N)] complexity. The implication is that by adding a truly random coin to a classical Turing machine and using this classical-physical algorithm can also speed up the deterministic solution of the Deutsch-Jozsa problem exponentially, similarly to the quantum algorithm. Then it is realized that the same database and the solution of the Deutsch-Jozsa problem can also be realized by using an identical algorithmic structure in a simpler way, even without noise/random coin. The only lost function in this new system, as compared with noise-based logic, is the ability to do generic parallel logic operations over the whole database. As the latter feature is not needed for the oDJ problem, it is concluded that the problem can be solved on a classical computer with O[log(N)] complexity even without a random coin. Therefore, while the oDJ algorithm is a historically important step in the developments of quantum computers, it is insufficient to prove quantum supremacy. Note, there is also a simplified Deutsch-Jozsa problem proposed later, which is more popular in the field; however, it is irrelevant for the present paper.

Simulation of Closed Timelike Curves in a Darwinian Approach to Quantum Mechanics

Closed timelike curves (CTCs) are non-intuitive theoretical solutions of general relativity field equations. The main paradox associated with the physical existence of CTCs, the so-called grandfather paradox, can be satisfactorily solved by a quantum model named Deutsch-CTC. An outstanding theoretical result that has been demonstrated in the Deutsch-CTC model is the computational equivalence of a classical and a quantum computer in the presence of a CTC. In this article, in order to explore the possible implications for the foundations of quantum mechanics of that equivalence, a fundamental particle is modelled as a classical-like system supplemented with an information space in which a randomizer and a classical Turing machine are stored. The particle could then generate quantum behavior in real time in case it was controlled by a classical algorithm coding the rules of quantum mechanics and, in addition, a logical circuit simulating a CTC was present on its information space. The conditions that, through the action of evolution under natural selection, might produce a population of such particles with both elements on their information spaces from initial sheer random behavior are analyzed.

Simple circuit simulations of classical and quantum Turing machines

We construct reversible Boolean circuits efficiently simulating reversible Turing machines. Both the circuits and the simulation proof are rather simple. Then we give a fairly straightforward generalization of the circuits and the simulation proof to the quantum case.

How Hard Is It to Detect Surveillance? A Formal Study of Panopticons and Their Detectability Problem

The Panopticon (which means “watcher of everything”) is a well-known prison structure of continuous surveillance and discipline studied by Bentham in 1785. Today, where persistent, massive scale, surveillance is immensely facilitated by new technologies, the term Panopticon vaguely characterizes institutions with a power to acquire and process, undetectably, personal information. In this paper we propose a theoretical framework for studying Panopticons and their detectability status. We show, based on the Theory of Computation, that detecting Panopticons, modelled either as a simple Turing Machine or as an Oracle Turing Machine, is an undecidable problem. Furthermore, we show that for each sufficiently expressive formal system, we can effectively construct a Turing Machine for which it is impossible to prove, within the formal system, its Panopticon status. Finally, we discuss how Panopticons can be physically detected by the heat they dissipate each time they acquire, effortlessly, information in the form of an oracle and we investigate their detectability status with respect to a more powerful computational model than classical Turing Machines, the Infinite Time Turing Machines (ITTMs).

Prefix-free quantum Kolmogorov complexity

We introduce quantum-K (QK), a measure of the descriptive complexity of density matrices using classical prefix-free Turing machines and show that the initial segments of weak Solovay random and quantum Schnorr random states are incompressible in the sense of QK. Many properties enjoyed by prefix-free Kolmogorov complexity (K) have analogous versions for QK; notably a counting condition.Several connections between Solovay randomness and K, including the Chaitin type characterization of Solovay randomness, carry over to those between weak Solovay randomness and QK. We work towards a Levin–Schnorr type characterization of weak Solovay randomness in terms of QK.Schnorr randomness has a Levin–Schnorr characterization using KC; a version of K using a computable measure machine, C. We similarly define QKC, a version of QK. Quantum Schnorr randomness is shown to have a Levin–Schnorr and a Chaitin type characterization using QKC. The latter implies a Chaitin type characterization of classical Schnorr randomness using KC.

Riemann Hypothesis from the Physicist’s Point of View

This article presents an attempt to comprehend the evolution of the ideas underlying the physical approach to the proof of one of the problems of the century - the Riemann hypothesis regarding the location of non-trivial zeros of the Riemann zeta function. Various formulations of this hypothesis are presented, which make it possible to clarify its connection with the distribution of primes in the set of natural numbers. A brief overview of the main directions of this approach is given. The probable cause of their failures is indicated - the solution of the problem within the framework of the classical Turing paradigm. A successful proof of the Riemann hypothesis based on the use of a relativistic computation model that allows one to overcome the Turing barrier is presented. This model has been previously applied to solve another problem not computable on the classical Turing machine - the calculation of the sums of divergent series for the Riemann zeta function of the real argument. The possibility of using relativistic computing for the development of artificial intelligence systems is noted.

Multivaluedness in Cellular Automata with Strong Anticipation and Some Prospects for Computation Theory

It is considered the general formulations and properties of cellular automata with cells which have the strong anticipatory property (introduced by D. Dubois). Multivalued behavior (hyperincursion) of solutions of such CA is described. It was posed new research problems of computation theory related to presumable multivaluedness of cellular automata with strong anticipation property. Extending of classical automata, Turing machine and algorithms had been proposed. Also some relation of such cellular automata and quantum mechanics are discussed.

Quantum-Inspired Acromyrmex Evolutionary Algorithm

Obtaining efficient optimisation algorithms has become the focus of much research interest since current developing trends in machine learning, traffic management, and other cutting-edge applications require complex optimised models containing a huge number of parameters. At present, computers based on the classical Turing-machine are inefficient when intent to solve optimisation tasks in complex and wicked problems. As a solution, quantum computers that should satisfy the Deutsch-Church-Turing principle have been proposed but this technology is still at an early stage. quantum-inspired algorithms (QIA) have emerged trying to fill-up an existing gap between the theoretical advances in quantum computation and real quantum computers. QIA use classical computers to simulate some physical phenomena such as superposition and entanglement to perform quantum computations. This paper proposes the quantum-inspired Acromyrmex evolutionary algorithm (QIAEA) as a highly efficient global optimisation method for complex systems. We present comparative statistical analyses that demonstrate how this nature-inspired proposal outperforms existing outstanding quantum-inspired evolutionary algorithms when testing benchmark functions.

Bell inequality violation in the framework of a Darwinian approach to quantum mechanics

A fundamental particle in physical space subject to conservation of momentum and energy, and characterized by its average mass and its position is methodologically supplemented with an information processor – a classical Turing machine – and a randomizer both defined on an information space localized on every particle. In this way the particle can be considered a generalized Darwinian system on which natural selection could act steering the evolution on the information space of the algorithms that govern the behaviour of the particles, giving rise plausibly to emergent quantum behaviour from initial randomness. This theory is applied to an EPR-Bohm experiment for electrons in order to analyse Bell inequality violation. A model for the entanglement of two particles has been considered. The model includes shared randomness – each particle stores its own randomizer and that of its partner – and the mutual transfer of their algorithms – sharing programs – that contain their respective anticipation modules. This fact enables every particle to anticipate not only the possible future configurations of its surrounding systems, but also those of the surrounding systems of its entangled partner. Thus, while preserving locality and realism, this theory implies outcome dependence – through shared randomness – and parameter dependence – through shared anticipation – for entangled states and, as a consequence, the violation of the Bell inequality in an EPR-Bohm experiment.

Some improvements in fuzzy turing machines

In this paper, we improve some previous definitions of fuzzy-type Turing machines to obtain degrees of accepting and rejecting in a computational manner. We apply a BFS-based search method and some level’s upper bounds to propose a computational process in calculating degrees of accepting and rejecting. Next, we introduce the class of Extended Fuzzy Turing Machines equipped with indeterminacy states. These states are used to characterize the loops of classical Turing machines in a mathematical sense. In the sequel, as well as the notions of acceptable and decidable languages, we define the new notion of indeterminable language. An indeterminable language corresponds to non-halting runs of a machine. Afterwards, we show that there is not any universal extended machine; which concludes that these machines cannot solve the halting problem. Also, we show that our extended machines and classical Turing machines have the same computational power. Then, we define the new notion of semi-universality and prove that there exists a semi-universal extended machine. This machine can indeterminate the complement of classical halting problem. Moreover, to each r.e or co-r.e language, we correspond a language that is related to some extended fuzzy Turing machines.

Verifiable Stream Computation and Arthur--Merlin Communication

In the setting of streaming interactive proofs (SIPs), a client (verifier) needs to compute a given function on a massive stream of data, arriving online, but is unable to store even a small fraction of the data. It outsources the processing to a third party service (prover) but is unwilling to blindly trust answers returned by this service. Thus, the service cannot simply supply the desired answer; it must convince the verifier of its correctness via a short interaction after the stream has been seen. In this work we study “barely interactive” SIPs. Specifically, we show that one or two rounds of interaction suffice to solve several query problems---including index, median, nearest neighbor search, pattern matching, and range counting---with polylogarithmic space and communication costs. Such efficiency with $O(1)$ rounds of interaction was thought to be impossible based on previous work. On the other hand, we initiate a formal study of the limitations of constant-round SIPs by introducing a new hierarchy of communication models called online interactive proofs (OIPs). The online nature of these models is analogous to the streaming restriction placed upon the verifier in a SIP. We give upper and lower bounds that (1) characterize, up to quadratic blowups, every finite level of the OIP hierarchy in terms of other well-known communication complexity classes, (2) separate the first four levels of the hierarchy, and (3) reveal that the hierarchy collapses to the fourth level. Our study of OIPs reveals marked contrasts and some parallels with the classic Turing machine theory of interactive proofs, establishes limits on the power of existing techniques for developing constant-round SIPs, and provides a new characterization of (nononline) Arthur--Merlin communication in terms of an online model.

On defining and modeling context-awareness

PurposeThis paper aims to present a methodology for defining and modeling context-awareness and describing efficiently the interactions between systems, applications and their context. Also, the relation of modern context-aware systems with distributed computation is investigated.Design/methodology/approachOn this purpose, definitions of context and context-awareness are developed based on the theory of computation and especially on a computational model for interactive computation which extends the classical Turing Machine model. The computational model proposed here encloses interaction and networking capabilities for computational machines.FindingsThe definition of context presented here develops a mathematical framework for working with context. Also, the modeling approach of distributed computing enables us to build robust, scalable and detailed models for systems and application with context-aware capabilities. Also, it enables us to map the procedures that support context-aware operations providing detailed descriptions about the interactions of applications with their context and other external sources.Practical implicationsA case study of a cloud-based context-aware application is examined using the modeling methodology described in the paper so as to demonstrate the practical usage of the theoretical framework that is presented.Originality/valueThe originality on the framework presented here relies on the connection of context-awareness with the theory of computation and distributed computing.

Efficient classical simulation of the Deutsch\u2013Jozsa and Simon\u2019s algorithms

A long-standing aim of quantum information research is to understand what gives quantum computers their advantage. This requires separating problems that need genuinely quantum resources from those for which classical resources are enough. Two examples of quantum speed-up are the Deutsch–Jozsa and Simon’s problem, both efficiently solvable on a quantum Turing machine, and both believed to lack efficient classical solutions. Here we present a framework that can simulate both quantum algorithms efficiently, solving the Deutsch–Jozsa problem with probability 1 using only one oracle query, and Simon’s problem using linearly many oracle queries, just as expected of an ideal quantum computer. The presented simulation framework is in turn efficiently simulatable in a classical probabilistic Turing machine. This shows that the Deutsch–Jozsa and Simon’s problem do not require any genuinely quantum resources, and that the quantum algorithms show no speed-up when compared with their corresponding classical simulation. Finally, this gives insight into what properties are needed in the two algorithms and calls for further study of oracle separation between quantum and classical computation.

Anomalous Diffusion within the Transcriptome as a Bio-Inspired Computing Framework for Resilience

Much of biology-inspired computer science is based on the Central Dogma, as implemented with genetic algorithms or evolutionary computation. That 60-year-old biological principle based on the genome, transcriptome and proteasome is becoming overshadowed by a new paradigm of complex ordered associations and connections between layers of biological entities, such as interactomes, metabolomics, etc. We define a new hierarchical concept as the “Connectosome”, and propose new venues of computational data structures based on a conceptual framework called “Grand Ensemble” which contains the Central Dogma as a subset. Connectedness and communication within and between living or biology-inspired systems comprise ensembles from which a physical computing system can be conceived. In this framework the delivery of messages is filtered by size and a simple and rapid semantic analysis of their content. This work aims to initiate discussion on the Grand Ensemble in network biology as a representation of a Persistent Turing Machine. This framework adding interaction and persistency to the classic Turing-machine model uses metrics based on resilience that has application to dynamic optimization problem solving in Genetic Programming.

On relationships between complexity classes of Turing machines

Classes of time and space complexity of Turing machines are defined, and relationships between them are discussed. New relationships between the defined complexity classes are described.

Physical microscopic free‐choice model in the framework of a Darwinian approach to quantum mechanics

A compatibilistic model of free choice for a fundamental particle is built within a general framework that explores the possibility that quantum mechanics be the emergent result of generalised Darwinian evolution acting on the abstract landscape of possible physical theories. The central element in this approach is a probabilistic classical Turing machine –basically an information processor plus a randomiser– methodologically associated with every fundamental particle. In this scheme every system acts not under a general law, but as a consequence of the command of a particular, evolved algorithm. This evolved programme enables the particle to algorithmically anticipate possible future world configurations in information space, and as a consequence, without altering the natural forward causal order in physical space, to incorporate elements to the decision making procedure that are neither purely random nor strictly in the past, but in a possible future.

On Computing Discrete Logarithms in Bulk and Randomness Extractors

We prove several results of independent interest related to the problem of computing de- terministically discrete logarithms in a finite field. The motivation was to give a number-theoretic construction of a non-malleable extractor improving the solution from the recent paper Privacy Am- plification and Non-Malleable Extractors via Character Sums by Dodis et al. There, the authors provide the first explicit example of a non-malleable extractor - a cryptographic primitive that sig- nificantly strengthens the notion of a classical randomness extractor. In order to make the extractor robust, so that it runs in polynomial time and outputs a linear number of bits, they rely on a certain conjecture on the least prime in a residue class. In this work we present a modification of their construction that allows to remove that dependency and address an issue we identified in the original development. Namely, it required an additional assumption about feasibility of finding a primitive element of a finite field. As an auxiliary result, we show an efficiently computable bijection between any order M subgroup of the multiplicative group of a finite field and a set of integers modulo M with the provision that M is a smooth number. Also, we provide a version of the baby-step giant- step method for solving multiple instances of the discrete logarithm problem in the multiplicative group of a prime field. It performs better than the generic algorithm when run on a machine without constant-time access to each memory cell, e.g., on a classical Turing machine.

Quantum information, cognition, and music.

Parallelism represents an essential aspect of human mind/brain activities. One can recognize some common features between psychological parallelism and the characteristic parallel structures that arise in quantum theory and in quantum computation. The article is devoted to a discussion of the following questions: a comparison between classical probabilistic Turing machines and quantum Turing machines.possible applications of the quantum computational semantics to cognitive problems.parallelism in music.

Executable Behaviour and the π-Calculus (extended abstract)

Reactive Turing machines extend classical Turing machines with a facility to model observable interactive behaviour. We call a behaviour executable if, and only if, it is behaviourally equivalent to the behaviour of a reactive Turing machine. In this paper, we study the relationship between executable behaviour and behaviour that can be specified in the pi-calculus. We establish that all executable behaviour can be specified in the pi-calculus up to divergence-preserving branching bisimilarity. The converse, however, is not true due to (intended) limitations of the model of reactive Turing machines. That is, the pi-calculus allows the specification of behaviour that is not executable up to divergence-preserving branching bisimilarity. Motivated by an intuitive understanding of executability, we then consider a restriction on the operational semantics of the pi-calculus that does associate with every pi-term executable behaviour, at least up to the version of branching bisimilarity that does not require the preservation of divergence.

Continuous Turing Machine: Real Function Computability and Iteration Issues

Contemporary computer theory is governed by the discretization of continuous problems. Classical Turing machines (TMs) are originally built to solve computation and computability problems, which main feature is discreteness. However, even some simple numerical calculations problems, e.g., iterations in R n , generate difficulties to be described or solved by constructing a TM. This paper explores the computability of continuous problems by proposing a class of continuous Turing machines (CTMs) that are an extension of TMs. CTMs can be applied to the standard for the precision of algorithms. First, computable real numbers are precisely defined by CTMs and their computations are regarded as the running of the CTMs. CTMs introduce the coded recursive descriptions, machine states, and operations with the characters of computer instructions in esse nce compared with usual computable continuous models. Hence, they can precisely present continuous computations with the form of processes. Second, the concepts of CTM computable and CTM handleable are proposed. Moreover, the basic concepts on approximation theory such as convergency, metric space, and fixed-point in R n are defined in a new space CTM Rn . Finally, an iterative algorithm is shown by constructing a CTM to solve linear