Abstract: We investigate numerically Turing patterns in the Lengyel-Epstein model in three dimensions. In a bulk homogeneous system under periodic boundary conditions, we obtain not only lamellar, cylindrical, and spherical structures but also several interconnected periodic structures including the Schwartz P-surface structure. In order to examine Turing patterns in the conditions accessible experimentally, we consider inhomogeneous systems where a parameter in the reaction-diffusion equations depends on the space coordinate with either Dirichlet or Neumann boundary conditions. In this situation, we find that a perforated-lamellar structure and an Fddd structure, both of which have a uniaxial symmetry, appear depending on the boundary conditions.

Abstract: We study the problem of finding small universal reversible Turing machines URTMs with four symbols or less. Here, we present two models of URTMs: a 24-state 4-symbol URTM, and a 32-state 3-symbol URTM. Both of them are designed so that they can simulate cyclic tag systems, a kind of string rewriting systems that are universal. By converting the 24-state 4-symbol URTM into a 2-symbol machine, we also obtain a 138-state 2-symbol URTM.

Abstract: The foundational work of Alan Turing and contemporaries on computability marked a turning point in the development of mathematical sciences: It clarified in a rather absolute sense what is computable in the setting of symbolic computation, and it also opened the way to computer science where the use of algorithms and the discussion on their nature was enriched by many new facets The present essay is an attempt to address both aspects: We review the historical development of the concept of algorithm up to Turing, emphasizing the essential role that logic played in this context, and we discuss the subsequent widening of understanding of “algorithm” and related “machines”, much in the spirit of Turing whose visions we see realized today

Abstract: Models play an important role in the development of computerscience and information technology applications. Turing machine isone of the most popular model of computing devices andcomputations. This model, or more exactly, a family of models,provides means for exploration of capabilities of informationtechnology. However, a Turing machine stops after giving a result.In contrast to this, computers, networks and their software, suchas an operating system, very often work without stopping but givevarious results. There are different modes of such functioning andTuring machines do not provide adequate models for theseprocesses. One of the closest to halting computation isstabilizing computation when the output has to stabilize in orderto become the result of a computational process. Such stabilizingcomputations are modeled by inductive Turing machines. Incomparison with Turing machines, inductive Turing machinesrepresent the next step in the development of computer scienceproviding better models for contemporary computers and computernetworks. At the same time, inductive Turing machines reflectpivotal traits of stabilizing computational processes. In thispaper, we study relations between different modes of inductiveTuring machines functioning. In particular, it is demonstratedthat acceptation by output stabilizing and acceptation by statestabilizing are linguistically equivalent.

Abstract: To investigate the evolution of syntax, we need to ascertain the evolutionary r\^ole of syntax and, before that, the very nature of syntax. Here, we will assume that syntax is computing. And then, since we are computationally Turing complete, we meet an evolutionary anomaly, the anomaly of sytax: we are syntactically too competent for syntax. Assuming that problem solving is computing, and realizing that the evolutionary advantage of Turing completeness is full problem solving and not syntactic proficiency, we explain the anomaly of syntax by postulating that syntax and problem solving co-evolved in humans towards Turing completeness. Examining the requirements that full problem solving impose on language, we find firstly that semantics is not sufficient and that syntax is necessary to represent problems. Our final conclusion is that full problem solving requires a functional semantics on an infinite tree-structured syntax. Besides these results, the introduction of Turing completeness and problem solving to explain the evolution of syntax should help us to fit the evolution of language within the evolution of cognition, giving us some new clues to understand the elusive relation between language and thinking.

Abstract: Based upon the principles of Turing incomputability, connectedness and novel properties of the Active Element Machine, a malware-resistant computing machine is constructed. This new computing machine is a non-Turing, non-register machine (non von-Neumann), called an active element machine (AEM). AEM programs are designed so that the purpose of the AEM computations are difficult to apprehend by an adversary and hijack with malware. These methods can also be used to help thwart reverse engineering of proprietary algorithms, hardware design and other areas of intellectual property. Using quantum randomness, the AEM can deterministically execute a universal Turing machine (universal digital computer program) with active element firing patterns that are Turing incomputable. In an embodiment, a more powerful computational procedure is created than Turing's computational procedure (equivalent to a digital computer procedure). Current digital computer algorithms and procedures can be derived or designed with a Turing machine computational procedure. A novel computer is invented so that a program's execution is difficult to apprehend.

Abstract: We improve the results by Siegelmann & Sontag (1995) by providing a novel and parsimonious constructive mapping between Turing Machines and Recurrent Artificial Neural Networks, based on recent developments of Nonlinear Dynamical Automata. The architecture of the resulting R-ANNs is simple and elegant, stemming from its transparent relation with the underlying NDAs. These characteristics yield promise for developments in machine learning methods and symbolic computation with continuous time dynamical systems. A framework is provided to directly program the R-ANNs from Turing Machine descriptions, in absence of network training. At the same time, the network can potentially be trained to perform algorithmic tasks, with exciting possibilities in the integration of approaches akin to Google DeepMind's Neural Turing Machines.

Abstract: The notion of nondeterminism has disappeared from the current definition of NP, which has led to ambiguities in understanding NP, and caused fundamental difficulties in studying the relation P versus NP. In this paper, we question the equivalence of the two definitions of NP, the one defining NP as the class of problems solvable by a nondeterministic Turing machine in polynomial time, and the other defining NP as the class of problems verifiable by a deterministic Turing machine in polynomial time, and reveal cognitive biases in this equivalence. Inspired from a famous Chinese paradox white horse is not horse, we further analyze these cognitive biases. The work shows that these cognitive biases arise from the confusion between different levels of nondeterminism and determinism, due to the lack of understanding about the essence of nondeterminism. Therefore, we argue that fundamental difficulties in understanding P versus NP lie firstly at cognition level, then logic level.

Abstract: In this paper, we make a preliminary interpretation of Cook's theorem presented in [1]. This interpretation reveals cognitive biases in the proof of Cook's theorem that arise from the attempt of constructing a formula in CNF to represent a computation of a nondeterministic Turing machine. Such cognitive biases are due to the lack of understanding about the essence of nondeterminism, and lead to the confusion between different levels of nondeterminism and determinism, thus cause the loss of nondeterminism from the NP-completeness theory. The work shows that Cook's theorem is the origin of the loss of nondeterminism in terms of the equivalence of the two definitions of NP, the one defining NP as the class of problems solvable by a nondeterministic Turing machine in polynomial time, and the other defining NP as the class of problems verifiable by a deterministic Turing machine in polynomial time. Therefore, we argue that fundamental difficulties in understanding P versus NP lie firstly at cognition level, then logic level.

Abstract: Against Maudlin, I argue that machines which merely reproduce a pre-programmed series of changes ought to be classed with Turing’s O-Machines even if they would counterfactually show Turing Machine-like activity. This can be seen on an interventionist picture of computational architectures, on which basic operations are the primitive loci for interventions. While constructions like Maudlin’s Olympia still compute, then, claims about them do not threaten philosophical arguments that depend on Turing Machine architectures and their computational equivalents.

Abstract: In this article, we formalize polynomially bounded sequences that plays an important role in computational complexity theory. Class P is a fundamental computational complexity class that contains all polynomial-time decision problems [11], [12]. It takes polynomially bounded amount of computation time to solve polynomial-time decision problems by the deterministic Turing machine. Moreover we formalize polynomial sequences [5].

Abstract: This paper introduces a new computing model based on the cooperation among Turing machines called orchestrated machines. Like universal Turing machines, orchestrated machines are also designed to simulate Turing machines but they can also modify the original operation of the included Turing machines to create a new layer of some kind of collective behavior. Using this new model we can define some interested notions related to cooperation ability of Turing machines such as the intelligence quotient or the emotional intelligence quotient for Turing machines.

Abstract: Given a strong bimonoid P, we introduce P-valued non-deterministic Turing machines (P-NTM), P-valued non-deterministic Turing machines with classical transition (P-NTMc), P-valued deterministic Turing machines (P-DTM) and P-valued non-deterministic Turing machines with the tape-head no remove (P-NTMS). We adapt depth-first and width-first methods for defining the weight of acceptance of languages recognized by Turning machines. And we study some basic properties of P-NTM, P-NTMc, P-NTMS. Moreover, We prove that variants of P-NTM with changing its P-valued initial function have the same power as P-NTM. These conclusions still hold for P-NTMc and P-NTMS.

Abstract: Turing’s (1950) article on the Turing test is often interpreted as supporting the behaviouristic view that human intelligence can be represented in terms of input/output functions, and thus emulated by a machine. We show that the properties of functions are not decidable in practice by the behaviour they exhibit, a result, we argue, of which Turing was likely aware. Given that the concept of a function is strictly a Platonic ideal, the question of whether or not the mind is a program is a pointless one, because it has no demonstrable implications. Instead, the interesting question is what intelligence means in practice. We suggest that Turing was introducing the novel idea that intelligence can be reliably evidenced in practice by finite interactions. In other words, although intelligence is not decidable in practice, it is testable in practice. We explore the connections between Turing’s idea of testability and subsequent developments in computational complexity theory.

Abstract: The micro-Extended Analog Computer (uEAC) is an electronic implementation inspired by Rubel’s EAC model. In this study, a fully connected uEACs array is proposed to overcome the limitations of a single uEAC, within which each uEAC unit is connected to all the other units by some weights. Then its computational capabilities are investigated by proving that a Turing machine can be simulated with uEAC-computable functions, even in the presence of bounded noise.

Abstract: In this paper we consider a bio-inspired description of the polynomial space Turing machines. For this purpose we use membrane computing, a formalism inspired by the way living cells are working. We define and use logarithmic space systems with active membranes, employing a binary representation in order to encode the positions on the Turing machine tape.

Abstract: We study the problem of designing small universal reversible Turing machines (URTMs). So far, six kinds of small URTMs that simulate cyclic tag systems have been constructed. In addition, applying general conversion methods to some of these small URTMs, 2-symbol, and 3or 4-state URTMs have been obtained. Here, we give precise descriptions of these URTMs, and show the computer simulation results to see how they work. The description files of the constructed URTMs are also given as attachment files. 2

Abstract: Turing machines are equivalent to modern electronic computers at a certain theoretical level, but differ in many details. In the analogy with a computer, the “tape” of the Turing machine is the computer memory, idealized to extend infinitely in each direction. The remarkable fact is that certain Turing machines are “universal”, in the sense that with appropriate input, they can be made to perform any ordinary computation. Not every Turing machine has this property; many can only behave in very simple ways. In effect, they can only do specific computations; they cannot act as “general-purpose computers”. Turing machine can process different types of strung written in various type language defined by Chomsky hierarchy. Machine will stop when it didn't have proper input or a machine state which is stable but not final and show halting state of machine. In this paper we are trying to describe using a example that a Turing machine can accept a language or string defined by context sensitive, context free or regular language till it find suitable input, but it show the halting state of the machine when it does not reach to a state closure to the final state along with the complexity of the process to process of that language or string.

Abstract: We discuss the following family of problems, parameterized by integers C ≥ 2 and D ≥ 1 : Does a given one-tape q-state Turing machine make at most C n + D steps on all computations on all inputs of length n, for all n? Assuming a fixed tape and input alphabet, we show that these problems are co-NP-complete and we provide good lower bounds. Specifically, these problems cannot be solved in o ( q ( C − 1 ) / 4 ) nondeterministic time by multi-tape Turing machines. We also show that the complements of these problems can be solved in O ( q C + 2 ) nondeterministic time and cannot be solved in o ( q ( C − 1 ) / 4 ) nondeterministic time by multi-tape Turing machines.

Abstract: This paper focuses on the Turing Machine usage as a basic calculator. The basic calculator here refers the four primary arithmetic operations. Those are namely 1.Addition, 2.Subtraction, 3.Multiplication, and 4. Division. The Turing Machine has the favorable circumstances as compared with the other abstract machines the Finite Automata and the Pushdown Automata. The Turing Machine has the vital feature called bi-directional movement. This aspect makes the Turing Machine to support the construction of the basic calculator. The transition diagram and transition table also derived for each calculation. The processing of input for each operation also given.

Abstract: A rule driven system is described, that it is calculated by a restricted turing machine. Gives semi-combinatorial descriptions to turing machine.

Abstract: The early 1930s were bad years for the worldwide economy, but great years for what would eventually be called Computer Science. In 1932, Alonzo Church at Princeton described his λ-calculus as a formal system for mathematical logic,and in 1935 argued that any function on the natural numbers that can be effectively computed, can be computed with his calculus [4]. Independently in 1935, as a master’s student at Cambridge, Alan Turing was developing his machine model of computation. In 1936 he too argued that his model could compute all computable functions on the natural numbers, and showed that his machine and the λ-calculus are equivalent [6]. The fact that two such different models of computation calculate the same functions was solid evidence that they both represented an inherent class of computable functions. From this arose the so-called Church-Turing thesis, which states, roughly, that any function on the natural numbers can be effectively computed if and only if it can be computed with the λ-calculus, or equivalently, the Turing machine. Although the Church-Turing thesis by itself is one of the most important ideas in Computer Science, the influence of Church and Turing’s models go far beyond the thesis itself. The Turing machine has become the core of all complexity theory [5]. In the early 1960’s Juris Hartmanis and Richard Stearns initiated the study of the complexity of computation. In 1967 Manuel Blum developed his axiomatic theory of complexity, which although independent of any particular machine, was considered in the context of a spectrum of Turing machines (e.g., one tape, two tape, or two stack). This was followed in 1971 by Stephen Cooke and Leonid Levin who showed a complete problem for NP (satisfiability) and introducing the question of whether P = NP. Since then a huge number of complexity classes have been isolated and related, including PCP, the class of Probabilistically Checkable Proofs. All this work has been defined in terms of minor variants of the Turing machine. In the field of Algorithms, where analysis needs to be more precise, other models have been developed, such at the Random Access Machine (RAM), that are closer to physical computers while remaining relatively abstract and hence widely applicable. However even these machines were heavily influenced by the Turing machine, and most of the complexity classes defined for the Turing machine carry over to the RAM. Whereas the machine models became the foundation of complexity and algorithmic theory, it was Church’s λ-calculus that became the foundation for the theory of programming languages [3]. This came about largely under the influence of Dana Scott and Christopher Strachey who, at the suggestion of Roger Penrose, developed denotational semantics, a topologically influenced account of higher-order computations acting on infinite data objects such as functions and streams, that are inherent in Church’s formalism. Scott and Strachey’s work meshed with Church’s work on classical type theory as a foundation for mathematics, with L.E.J. Brouwer’s program of constructive foundations that led to Per Martin-Lof’s development of Intuitionistic Type Theory, and with N.G. de Bruijn’s AUTOMATH language for expressing machine-checked proof, all of which were similarly founded on the λ-calculus. The result is an integrated theory of computation and deduction, known as the Propositions as Types principle, that consolidates programs with proofs, and types with propositions. This principle lies at the heart of most programming language theory and many program verification and proof checking systems. Robin Milner’s work on the LCF prover in the 1970’s led to the emergence of functional programming, based directly on the λ-calculus, and interactive proof development, based on Scott’s logic of computable functions arising from his program of denota-

Abstract: This paper describe about polynomial time injective reduction and lack of such injection between NP and coNP. SAT can emulate NTM computation history. That is to say, there are some polynomial time injective reduction from all NP problems to SAT. And there also exist such reduction from all coNP problems to TAUT. Therefore, if NP is coNP, there is some polynomial time injective reduction between these problems (like SAT, TAUT). The other hand, We can make special problems family “ANTI” that are subset of coNP that have injective reduction from all coNP problems. If NP is coNP, each ANTI problems have polynomial time injective reduction from complement problems, but ANTI’s cardinality is amount exponential and ANTI problem’s cardinality is at most polynomial. So there are not exist polynomial time injective reduction. Therefore NP is not coNP, and also P is not NP. 1. Polynomial time injective reduction First, we describe that there exists some polynomial time injective reduction to SAT and TAUT. Definition 1.1. In this paper, we limit problems as decision problems. We use term as following; C : Complexity class that named C. Cc : C-Complete class. r : A ↪→ B : Injection from A to B. ∀a, b ∈ A (a 6= b → r (a) 6= r (b)) Arity (f) : Arity of function (or function family) f . |x| : Size of Input x. p : Complement of problem p. [f ] : Representative of equivalence formula that length is shortest. 〈f〉 : Encoding of formula f . SAT : Boolean satisfiability problems. TAUT : Tautology problems. FP : Polymonial time functions. WFF : Well formed formula of propositional logic. TM : Set of Turing Machine. ATM : Set of Alternating Turing Machine. NTM : Set of Nondeterministic Turing Machine (subset of ATM that have only existential states). NTM : Set of Negetion Nondeterministic Turing Machine (subset of ATM that have only universal states). To make polynomial time injective reduction, we define special complete class.

Abstract: This project proves universal computation in the Game of Life cellular automaton by using a Turing machine construction. Existing proofs of universality in the Game of Life rely on a counter machine. These machines require complex encoding and decoding of the input and output and the proof of universality for these machines by the Church Turing thesis is that they can perform the equivalent of a Turing machine. A proof based directly on a Turing machine is much more accessible. The computational power available today allows powerful algorithms such as HashLife to calculate the evolution of cellular automata patterns sufficiently fast that an efficient universal Turing machine can be demonstrated in a conveniently short period of time. Such a universal Turing machine is presented here. It is a direct simulation of a Turing machine and the input and output are easily interpreted. In order to achieve full universal behaviour an infinite storage media is required. The storage media used to represent the Turing machine tape is a pair of stacks. One stack representing the Turing tape to the left of the read/write head and one for the Turing tape to the right. Collision based construction techniques have been used to add stack cells to the ends of the stacks continuously. The continuous construction of the stacks is equivalent to the formatting of blank media. This project demonstrates that large areas of a cellular automata can be formatted in real time to perform complex functions.

Abstract: We consider three problems related to dynamics of one-tape Turing machines: Existence of blocking configurations, surjectivity in the trace, and entropy positiveness. In order to address them, a reversible two-counter machine is simulated by a reversible Turing machine on the right side of its tape. By completing the machine in different ways, we prove that none of the former problems is decidable. In particular, the problems about blocking configurations and entropy are shown to be undecidable for the class of reversible Turing machines.