Turing Machine Tape Research Articles

NP-Difficult Tasks: Automatic Proof of Theorems and Turings Machine

It is offered to use for solving NP-complete (difficult) tasks a modification of methods for satisfying the constraint (CS) by including automatic proof of theorems (APT), and programming in constraints — generation of Turing's machine (TM). Currently, CS uses AP in a truncated form (logical programming), and it is suggested using the method of invariant transformations (MIT), which is a full-fledged APT. In addition, it is offered to use CS methods to generate TM solving NP-difficult tasks recorded on the TM tape, which is an extension of programming capabilities in constraints.

Statistical Complexity Analysis of Turing Machine tapes with Fixed Algorithmic Complexity Using the Best-Order Markov Model.

Sources that generate symbolic sequences with algorithmic nature may differ in statistical complexity because they create structures that follow algorithmic schemes, rather than generating symbols from a probabilistic function assuming independence. In the case of Turing machines, this means that machines with the same algorithmic complexity can create tapes with different statistical complexity. In this paper, we use a compression-based approach to measure global and local statistical complexity of specific Turing machine tapes with the same number of states and alphabet. Both measures are estimated using the best-order Markov model. For the global measure, we use the Normalized Compression (NC), while, for the local measures, we define and use normal and dynamic complexity profiles to quantify and localize lower and higher regions of statistical complexity. We assessed the validity of our methodology on synthetic and real genomic data showing that it is tolerant to increasing rates of editions and block permutations. Regarding the analysis of the tapes, we localize patterns of higher statistical complexity in two regions, for a different number of machine states. We show that these patterns are generated by a decrease of the tape’s amplitude, given the setting of small rule cycles. Additionally, we performed a comparison with a measure that uses both algorithmic and statistical approaches (BDM) for analysis of the tapes. Naturally, BDM is efficient given the algorithmic nature of the tapes. However, for a higher number of states, BDM is progressively approximated by our methodology. Finally, we provide a simple algorithm to increase the statistical complexity of a Turing machine tape while retaining the same algorithmic complexity. We supply a publicly available implementation of the algorithm in C++ language under the GPLv3 license. All results can be reproduced in full with scripts provided at the repository.

Photochromic molecular implementations of universal computation

Unconventional computing is an area of research in which novel materials and paradigms are utilised to implement computation. Previously we have demonstrated how registers, logic gates and logic circuits can be implemented, unconventionally, with a biocompatible molecular switch, NitroBIPS, embedded in a polymer matrix. NitroBIPS and related molecules have been shown elsewhere to be capable of modifying many biological processes in a manner that is dependent on its molecular form. Thus, one possible application of this type of unconventional computing is to embed computational processes into biological systems. Here we expand on our earlier proof-of-principle work and demonstrate that universal computation can be implemented using NitroBIPS. We have previously shown that spatially localised computational elements, including registers and logic gates, can be produced. We explain how parallel registers can be implemented, then demonstrate an application of parallel registers in the form of Turing machine tapes, and demonstrate both parallel registers and logic circuits in the form of elementary cellular automata. The Turing machines and elementary cellular automata utilise the same samples and same hardware to implement their registers, logic gates and logic circuits; and both represent examples of universal computing paradigms. This shows that homogenous photochromic computational devices can be dynamically repurposed without invasive reconfiguration. The result represents an important, necessary step towards demonstrating the general feasibility of interfacial computation embedded in biological systems or other unconventional materials and environments.

Computable Diagonalizations and Turing’s Cardinality Paradox

A. N. Turing’s 1936 concept of computability, computing machines, and computable binary digital sequences, is subject to Turing’s Cardinality Paradox. The paradox conjoins two opposed but comparably powerful lines of argument, supporting the propositions that the cardinality of dedicated Turing machines outputting all and only the computable binary digital sequences can only be denumerable, and yet must also be nondenumerable. Turing’s objections to a similar kind of diagonalization are answered, and the implications of the paradox for the concept of a Turing machine, computability, computable sequences, and Turing’s effort to prove the unsolvability of the Entscheidungsproblem, are explained in light of the paradox. A solution to Turing’s Cardinality Paradox is proposed, positing a higher geometrical dimensionality of machine symbol-editing information processing and storage media than is available to canonical Turing machine tapes. The suggestion is to add volume to Turing’s discrete two-dimensional machine tape squares, considering them instead as similarly ideally connected massive three-dimensional machine information cells. Three-dimensional computing machine symbol-editing information processing cells, as opposed to Turing’s two-dimensional machine tape squares, can take advantage of a denumerably infinite potential for parallel digital sequence computing, by which to accommodate denumerably infinitely many computable diagonalizations. A three-dimensional model of machine information storage and processing cells is recommended on independent grounds as better representing the biological realities of digital information processing isomorphisms in the three-dimensional neural networks of living computers.

Complexity at large

Complexity at large

Chromatin Computation

In living cells, DNA is packaged along with protein and RNA into chromatin. Chemical modifications to nucleotides and histone proteins are added, removed and recognized by multi-functional molecular complexes. Here I define a new computational model, in which chromatin modifications are information units that can be written onto a one-dimensional string of nucleosomes, analogous to the symbols written onto cells of a Turing machine tape, and chromatin-modifying complexes are modeled as read-write rules that operate on a finite set of adjacent nucleosomes. I illustrate the use of this “chromatin computer” to solve an instance of the Hamiltonian path problem. I prove that chromatin computers are computationally universal – and therefore more powerful than the logic circuits often used to model transcription factor control of gene expression. Features of biological chromatin provide a rich instruction set for efficient computation of nontrivial algorithms in biological time scales. Modeling chromatin as a computer shifts how we think about chromatin function, suggests new approaches to medical intervention, and lays the groundwork for the engineering of a new class of biological computing machines.

Improved simulation of nondeterministic Turing machines

The standard simulation of a nondeterministic Turing machine (NTM) by a deterministic one essentially searches a large bounded-degree graph whose size is exponential in the running time of the NTM. The graph is the natural one defined by the configurations of the NTM. All methods in the literature have required time linear in the size S of this graph. This paper presents a new simulation method that runs in time O ̃ ( S ) . The search savings exploit the one-dimensional nature of Turing machine tapes. In addition, we remove most of the time dependence on nondeterministic choices of states and tape head movements.

Chomskian Hierarchies of Families of Sets of Piecewise Continuous Functions

The time-honored Chomsky hierarchy has long shown its value as a structural tool in formal languages and automata theory, and gained followers in various areas. We show here how very similar hierarchies can be obtained for families of sets of piecewise continuous functions. We use systems of ordinary differential equations in the same way that automata are used in establishing the traditional Chomsky hierarchy. A functional memory is provided by state-dependent delays which are used in a novel way, paired with certain state components, giving memory structures similar to push-down stores and Turing machine tapes. The resulting machine model may be viewed as a “functional computing machine’’, with functional input, functional memory and, though this is not emphasized here, functional output.

New Real-Time Simulations of Multihead Tape Units

Single-head tapes are used to simulate multihead Turing machine tape units without loss. For one-dimensional tapes, the new simulations are simpler and use far fewer tapes than prevj known simulations. For multidimensional tapes, the results are entirely new.