Abstract: Consider the problem of designing a machine to solve well-defined intellectual problems. We call a problem well-defined if there is a test which can be applied to a proposed solution. In case the proposed solution is a solution, the test must confirm this in a finite number of steps. If the proposed solution is not correct, we may either require that the test indicate this in a finite number of steps or else allow it to go on indefinitely. Since any test may be regarded as being performed by a Turing machine, this means that welldefined intellectual problems may be regarded as those of inverting functions and partial functions defined by Turing machines. Let fm(n) be the partial function computed by the m th Turing machine. It is not defined for a given value of n if the computation does not come to an end. This paper deals with the problem of designing a Turing machine which, when confronted by the number pair (m, r), computes as efficiently as possible a function g(m, r) such that fm(g(m, r)) = r. Again, for particular values of m and r no g(m, r) need exist. In fact, it has been shown that the existence of g(m, r) is an undecidable question in that there does not exist a Turing machine which will eventually come to a stop and print a 1 if g(m, r) does not exist. In spite of this, it is easy to show that a Turing machine exists which will

Abstract: The algorithm, a building block of computer science, is defined from an intuitive and pragmatic point of view, through a methodological lens of philosophy rather than that of formal computation. The treatment extracts properties of abstraction, control, structure, finiteness, effective mechanism, and imperativity, and intentional aspects of goal and preconditions. The focus on the algorithm as a robust conceptual object obviates issues of correctness and minimality. Neither the articulation of an algorithm nor the dynamic process constitute the algorithm itself. Analysis for implications in computer science and philosophy reveals unexpected results, new questions, and new perspectives on current questions, including the relationship between our informally construed algorithms and Turing machines. Exploration in terms of current computational and philosophical thinking invites further developments.

Abstract: Turing's famous 1936 paper introduced a formal definition of a computing machine, a Turing machine. This model led to both the development of actual computers and to computability theory, the study of what machines can and cannot compute. This book presents classical computability theory from Turing and Post to current results and methods, and their use in studying the information content of algebraic structures, models, and their relation to Peano arithmetic. The author presents the subject as an art to be practiced, and an art in the aesthetic sense of inherent beauty which all mathematicians recognize in their subject. Part I gives a thorough development of the foundations of computability, from the definition of Turing machines up to finite injury priority arguments. Key topics include relative computability, and computably enumerable sets, those which can be effectively listed but not necessarily effectively decided, such as the theorems of Peano arithmetic. Part II includes the study of computably open and closed sets of reals and basis and nonbasis theorems for effectively closed sets. Part III covers minimal Turing degrees. Part IV is an introduction to games and their use in proving theorems. Finally, Part V offers a short history of computability theory. The author has honed the content over decades according to feedback from students, lecturers, and researchers around the world. Most chapters include exercises, and the material is carefully structured according to importance and difficulty. The book is suitable for advanced undergraduate and graduate students in computer science and mathematics and researchers engaged with computability and mathematical logic.

Abstract: In this work, we construct an adaptively secure functional encryption for Turing machines scheme, based on indistinguishability obfuscation for circuits. Our work places no restrictions on the types of Turing machines that can be associated with each secret key, in the sense that the Turing machines can accept inputs of unbounded length, and there is no limit to the description size or the space complexity of the Turing machines. Prior to our work, only special cases of this result were known, or stronger assumptions were required. More specifically, previous work implicitly achieved selectively secure FE for Turing machines with a-priori bounded input based on indistinguishability obfuscation STOC 2015, or achieved FE for general Turing machines only based on knowledge-type assumptions such as public-coin differing-inputs obfuscation TCC 2015. A consequence of our result is the first constructions of succinct adaptively secure garbling schemes even for circuits in the standard model. Prior succinct garbling schemes even for circuits were only known to be adaptively secure in the random oracle model.

Abstract: In this article we study how a subshift can simulate another one, where the notion of simulation is given by operations on subshifts inspired by the dynamical systems theory (factor, projective subaction...). There exists a correspondence between the notion of simulation and the set of forbidden patterns. The main result of this paper states that any effective subshift of dimension d -- that is a subshift whose set of forbidden patterns can be generated by a Turing machine -- can be obtained by applying dynamical operations on a subshift of finite type of dimension d + 1 -- a subshift that can be defined by a finite set of forbidden patterns. This result improves Hochman's [Hoc09].

Abstract: In this paper, we investigate multi-dimensional computational model, k-neighborhood template A-type three-dimensional bounded cellular acceptor on four-dimensional tapes, and discuss some basic properties. This model consists of a pair of a converter and a configuration-reader. The former converts the given four-dimensional tape to three-dimensional configuration. The latter determines whether or not the derived three-dimensional configuration is accepted, and concludes the acceptance or non-acceptance of given four-dimensional tape. We mainly investigate some open problems about k-neighborhood template A-type three-dimensional bounded cellular acceptor on four-dimensional tapes whose configuration-readers are L(m) space-bounded deterministic (nondeterministic) three-dimensional Turing machines.

Abstract: In this paper, we present a novel computing model, called probe machine (PM). Unlike the turing machine (TM), PM is a fully parallel computing model in the sense that it can simultaneously process multiple pairs of data, rather than sequentially process every pair of linearly adjacent data. We establish the mathematical model of PM as a nine-tuple consisting of data library, probe library, data controller, probe controller, probe operation, computing platform, detector, true solution storage, and residue collector. We analyze the computation capability of the PM model, and in particular, we show that TM is a special case of PM. We revisit two NP-complete problems, i.e., the graph coloring and Hamilton cycle problems, and devise two algorithms on basis of the established PM model, which can enumerate all solutions to each of these problems by only one probe operation. Furthermore, we show that PM can be implemented by leveraging the nano-DNA probe technologies. The computational power of an electronic computer based on TM is known far more than that of the human brain. A question naturally arises: will a future computer based on PM outperform the human brain in more ways beyond the computational power?

Abstract: In 1936, when he was just twenty-four years old, Alan Turing wrote a remarkable paper in which he outlined the theory of computation, laying out the ideas that underlie all modern computers. This groundbreaking and powerful theory now forms the basis of computer science. In Turing's Vision, Chris Bernhardt explains the theory, Turing's most important contribution, for the general reader. Bernhardt argues that the strength of Turing's theory is its simplicity, and that, explained in a straightforward manner, it is eminently understandable by the nonspecialist. As Marvin Minsky writes, "The sheer simplicity of the theory's foundation and extraordinary short path from this foundation to its logical and surprising conclusions give the theory a mathematical beauty that alone guarantees it a permanent place in computer theory." Bernhardt begins with the foundation and systematically builds to the surprising conclusions. He also views Turing's theory in the context of mathematical history, other views of computation (including those of Alonzo Church), Turing's later work, and the birth of the modern computer. In the paper, "On Computable Numbers, with an Application to the Entscheidungsproblem," Turing thinks carefully about how humans perform computation, breaking it down into a sequence of steps, and then constructs theoretical machines capable of performing each step. Turing wanted to show that there were problems that were beyond any computer's ability to solve; in particular, he wanted to find a decision problem that he could prove was undecidable. To explain Turing's ideas, Bernhardt examines three well-known decision problems to explore the concept of undecidability; investigates theoretical computing machines, including Turing machines; explains universal machines; and proves that certain problems are undecidable, including Turing's problem concerning computable numbers.

Abstract: An unsolved problem in neuroevolution (NE) is to evolve artificial neural networks (ANN) that can store and use information to change their behavior online. While plastic neural networks have shown promise in this context, they have difficulties retaining information over longer periods of time and integrating new information without losing previously acquired skills. Here we build on recent work by Graves et al. [5] who extended the capabilities of an ANN by combining it with an external memory bank trained through gradient descent. In this paper, we introduce an evolvable version of their Neural Turing Machine (NTM) and show that such an approach greatly simplifies the neural model, generalizes better, and does not require accessing the entire memory content at each time-step. The Evolvable Neural Turing Machine (ENTM) is able to solve a simple copy tasks and for the first time, the continuous version of the double T-Maze, a complex reinforcement-like learning problem. In the T-Maze learning task the agent uses the memory bank to display adaptive behavior that normally requires a plastic ANN, thereby suggesting a complementary and effective mechanism for adaptive behavior in NE.

Abstract: Slot and van Emde Boas' weak invariance thesis states that reasonable machines can simulate each other within a polynomial overhead in time. Is λ-calculus a reasonable machine? Is there a way to measure the computational complexity of a λ-term? This paper presents the first complete positive answer to this long-standing problem. Moreover , our answer is completely machine-independent and based on a standard notion in the theory of λ-calculus: the length of a leftmost-outermost derivation to normal form is an invariant , i.e. reasonable, cost model. Such a theorem cannot be proved by directly relating λ-calculus with Turing machines or random access machines, because of the size-explosion problem: there are terms that in a linear number of steps produce an exponentially large output. The first step towards the solution is to shift to a notion of evaluation for which the length and the size of the output are linearly related. This is done by adopting the linear substitution calculus (LSC), a calculus of explicit substitutions modeled after linear logic proof nets and admitting a decomposition of leftmost-outermost derivations with the desired property. Thus, the LSC is invariant with respect to, say, random access machines. The second step is to show that the LSC is invariant with respect to the λ-calculus. The size explosion problem seems to imply that this is not possible: having the same notions of normal form, evaluation in the LSC is exponentially longer than in the λ-calculus. We solve such an impasse by introducing a new form of shared normal form and shared reduction , called useful. Useful evaluation produces a compact, shared representation of the normal form, by avoiding those steps that only unshare the output without contributing to β-redexes, i.e. the steps that cause the blow-up in size. The main technical contribution of the paper is indeed the definition of useful reductions and the thorough analysis of their properties.

Abstract: The capacity of recurrent neural networks to learn complex sequential patterns is improving. Recent developments such as Clockwork RNN, Stack RNN, Memory networks and Neural Turing Machine all aim to increase long-term memory capacity of recurrent neural networks. In this study, we investigate properties of Neural Turing Machine, compare it with ensembles of Stack RNN on artificial benchmarks and applied it to learn human mobility patterns. We show, that Neural Turing Machine based predictor outperformed not only n-gram based prediction, but also neighborhood based predictor, that was designed to solve this particular problem. Our models will be deployed in anti-drug police department to predict mobility of suspects.

Abstract: The investigation of dynamical systems has revealed a deep-rooted difference between waves and objects regarding temporal reversibility and particlelike objects. In nondissipative chaos, the dynamic of waves always remains time reversible, unlike that of particles. Here, we explore the dynamics of a wave-particle entity. It consists in a drop bouncing on a vibrated liquid bath, self-propelled and piloted by the surface waves it generates. This walker, in which there is an information exchange between the particle and the wave, can be analyzed in terms of a Turing machine with waves as the information repository. The experiments reveal that in this system, the drop can read information backwards while erasing it. The drop can thus backtrack on its previous trajectory. A transient temporal reversibility, restricted to the drop motion, is obtained in spite of the system being both dissipative and chaotic.

Abstract: Since the definition of the Busy Beaver function by Rado in 1962, an interesting open question has been what the smallest value of n for which BB(n) is independent of ZFC set theory. Is this n approximately 10, or closer to 1,000,000, or is it even larger? In this paper, we show that it is at most 7,918 by presenting an explicit description of a 7,918-state Turing machine Z with 1 tape and a 2-symbol alphabet that cannot be proved to run forever in ZFC (even though it presumably does), assuming ZFC is consistent. The machine is based on work of Harvey Friedman on independent statements involving order-invariant graphs. In doing so, we give the first known upper bound on the highest provable Busy Beaver number in ZFC. We also present a 4,888-state Turing machine G that halts if and only if there is a counterexample to Goldbach’s conjecture, and a 5,372-state Turing machine R that halts if and only if the Riemann hypothesis is false. To create G, R, and Z, we develop and use a higher-level language, Laconic, which is much more convenient than direct state manipulation.

Abstract: We propose a new computing model called chemical reaction automata (CRAs) as a simplified variant of reaction automata (RAs) studied in recent literature (Okubo in RAIRO Theor Inform Appl 48:23---38 2014; Okubo et al. in Theor Comput Sci 429:247---257 2012a, Theor Comput Sci 454:206---221 2012b). We show that CRAs in maximally parallel manner are computationally equivalent to Turing machines, while the computational power of CRAs in sequential manner coincides with that of the class of Petri nets, which is in marked contrast to the result that RAs (in both maximally parallel and sequential manners) have the computing power of Turing universality (Okubo 2014; Okubo et al. 2012a). Intuitively, CRAs are defined as RAs without inhibitor functioning in each reaction, providing an offline model of computing by chemical reaction networks (CRNs). Thus, the main results in this paper not only strengthen the previous result on Turing computability of RAs but also clarify the computing powers of inhibitors in RA computation.

Abstract: It has been a long-standing open problem whether the strong $$\lambda $$-calculus is a reasonable computational model, i.e. whether it can be implemented within a polynomial overhead with respect to the number of $$\beta $$-steps on models like Turing machines or RAM. Recently, Accattoli and Dal Lago solved the problem by means of a new form of sharing, called useful sharing, and realised via a calculus with explicit substitutions. This paper presents a new abstract machine for the strong $$\lambda $$-calculus based on useful sharing, the Useful Milner Abstract Machine, and proves that it reasonably implements leftmost-outermost evaluation. It provides both an alternative proof that the $$\lambda $$-calculus is reasonable and an improvement on the technology for implementing strong evaluation.

Abstract: Many important lines of argumentation have been presented during the last decades claiming that machines cannot think like people. Yet, it has been possible to construct devices and information systems, which replace people in tasks which have previously been occupied by people as the tasks require intelligence. The long and versatile discourse over, what machine intelligence is, suggests that there is something unclear in the foundations of the discourse itself. Therefore, we critically studied the foundations of used theory languages. By looking critically some of the main arguments of machine thinking, one can find unifying factors. Most of them are based on the fact that computers cannot perform sense-making selections without human support and supervision. This calls attention to mathematics and computation itself as a representational constructing language and as a theory language in analysing human mentality. It is possible to notice that selections must be based on relevance, i.e., on why some elements of sets belong to one class and others do not. Since there is no mathematical justification to such selection, it is possible to say that relevance and related concepts are beyond the power of expression of mathematics and computation. Consequently, Turing erroneously assumed that mathematics and formal language is equivalent with natural languages. He missed the fact that mathematics cannot express relevance, and for this reason, mathematical representations cannot be complete models of the human mind.

Abstract: Following the recent trend in explicit neural memory structures, we present a new design of an external memory, wherein memories are stored in an Euclidean key space $\mathbb R^n$. An LSTM controller performs read and write via specialized read and write heads. It can move a head by either providing a new address in the key space (aka random access) or moving from its previous position via a Lie group action (aka Lie access). In this way, the "L" and "R" instructions of a traditional Turing Machine are generalized to arbitrary elements of a fixed Lie group action. For this reason, we name this new model the Lie Access Neural Turing Machine, or LANTM. We tested two different configurations of LANTM against an LSTM baseline in several basic experiments. We found the right configuration of LANTM to outperform the baseline in all of our experiments. In particular, we trained LANTM on addition of $k$-digit numbers for $2 \le k \le 16$, but it was able to generalize almost perfectly to $17 \le k \le 32$, all with the number of parameters 2 orders of magnitude below the LSTM baseline.

Abstract: In this article, we will show that uncomputability is a relative property not only of oracle Turing machines, but also of subrecursive classes. We will define the concept of a Turing submachine, and a recursive relative version for the Busy Beaver function which we will call Busy Beaver Plus function. Therefore, we will prove that the computable Busy Beaver Plus function defined on any Turing submachine is not computable by any program running on this submachine. We will thereby demonstrate the existence of a "paradox" of computability a la Skolem: a function is computable when "seen from the outside" the subsystem, but uncomputable when "seen from within" the same subsystem. Finally, we will raise the possibility of defining universal submachines, and a hierarchy of negative Turing degrees.

Abstract: The nonuniversality in computation theorem NCT states that no computer capable of a finite and fixed number of basic operations per time unit can be universal. This result, obtained in 2005, disproves major prevailing dogmas in computer science, on the strength of several counterexamples that differ significantly from one another. Thus, the NCT shows that the Church–Turing thesis CTT is false: It is not true that the Turing Machine can execute any computation possible on any other computer. It also disproves the inflated CTT, whereby there exists a universal computer, that is, a physical device capable of carrying out any computation conceivable. At the heart of the NCT is a refutation of the simulation principle, which states that any computation possible on a general-purpose computer can be simulated, more or less efficiently, on any other general-purpose computer . While more than 10 years have now elapsed since the NCT was established, the result is still widely misunderstood. This state of affairs is due to a number of misconceptions about the nature of the counterexamples and the significance of the NCT. The purpose of this paper is to dispel these misconceptions and convey the simple, yet important idea that universality in computation cannot be achieved, except by a computer capable, each time unit, of an infinite number of basic operations executed in parallel.

Abstract: The theory of computer science is based around Universal Turing Machines (UTMs): abstract machines able to execute all possible algorithms. Modern digital computers are physical embodiments of UTMs. The nondeterministic polynomial (NP) time complexity class of problems is the most significant in computer science, and an efficient (i.e. polynomial P) way to solve such problems would be of profound economic and social importance. By definition nondeterministic UTMs (NUTMs) solve NP complete problems in P time. However, NUTMs have previously been believed to be physically impossible to construct. Thue string rewriting systems are computationally equivalent to UTMs, and are naturally nondeterministic. Here we describe the physical design for a NUTM that implements a universal Thue system. The design exploits the ability of DNA to replicate to execute an exponential number of computational paths in P time. Each Thue rewriting step is embodied in a DNA edit implemented using a novel combination of polymerase chain reactions and site-directed mutagenesis. We demonstrate that this design works using both computational modelling and in vitro molecular biology experimentation. The current design has limitations, such as restricted error-correction. However, it opens up the prospect of engineering NUTM based computers able to outperform all standard computers on important practical problems.

Abstract: We look at small Turing machines (TMs) that work with just two colors (alphabet symbols) and either two or three states. For any particular such machine and any particular input , we consider what we call the space-time diagram which is basically the collection of consecutive tape configurations of the computation . In our setting, it makes sense to define a fractal dimension for a Turing machine as the limiting fractal dimension for the corresponding space-time diagrams. It turns out that there is a very strong relation between the fractal dimension of a Turing machine of the above-specified type and its runtime complexity. In particular, a TM with three states and two colors runs in at most linear time, if and only if its dimension is 2, and its dimension is 1, if and only if it runs in superpolynomial time and it uses polynomial space. If a TM runs in time , we have empirically verified that the corresponding dimension is , a result that we can only partially prove. We find the results presented here remarkable because they relate two completely different complexity measures: the geometrical fractal dimension on one side versus the time complexity of a computation on the other side.

Abstract: A constrained pseudo random function (PRF) behaves like a standard PRF, but with the added feature that the (master) secret key holder, having secret key K, can produce a constrained key, K{f}, that allows for the evaluation of the PRF on all inputs satisfied by the constraint f . Most existing constrained PRF constructions can handle only bounded length inputs. In a recent work, Abusalah et al. [AFP14] constructed a constrained PRF scheme where constraints can be represented as Turing machines with unbounded inputs. Their proof of security, however, requires risky “knowledge type” assumptions such as (public coins) differing inputs obfuscation for circuits and SNARKs. In this work, we construct a constrained PRF scheme for Turing machines with unbounded inputs under weaker assumptions, namely, the existence of indistinguishability obfuscation for circuits (and DDH). ∗This work was done while the author was visiting the Simons Institute for the Theory of Computing, supported by the Simons Foundation and by the DIMACS/Simons Collaboration in Cryptography through NSF grant #CNS-1523467. †Supported by NSF CNS-0952692, CNS-1228599 and CNS-1414082. DARPA through the U.S. Office of Naval Research under Contract N00014-11-1-0382, Google Faculty Research award, the Alfred P. Sloan Fellowship, Microsoft Faculty Fellowship, and Packard Foundation Fellowship.

Abstract: Alan Turing (1912--1954) has been increasingly recognised as an important mathematician who, despite his short life, developed mathematical ideas that today have led to foundational aspects of computer science, especially with respect to computability, artificial intelligence and morphogenesis (the growth of biological patterns, important in mathematical biology). Some of Turing's mathematics and related ideas can be visualised in interesting and even artistic ways, aided using software. In addition, a significant corpus of the historical documentation on Turing can now be accessed online as a number of major archives have digitised material related to Turing. This paper introduces some of the major scientific work of Turing and ways in which it can be visualised, often artistically. Turing's fame has, especially since his centenary in 2012, reached a level where he has had a cultural influence on the arts in general. Although the story of Turing can be seen as one of tragedy, with his life cut short, from a historical viewpoint Turing's contribution to humankind has been triumphant.

Abstract: The oritatami system (OS) is a model of computation by cotranscriptional folding, being inspired by the recent experimental succeess of RNA origami to self-assemble an RNA tile cotranscriptionally. The OSs implemented so far, including binary counter and Turing machine simulator, are deterministic, that is, uniquely fold into one conformation, while nondeterminism is intrinsic in biomolecular folding. We introduce nondeterminism to OS (NOS) and propose an NOS that chooses an assignment of Boolean values nondeterministically and evaluates a logical formula on the assignment. This NOS is seedless in the sense that it does not require any initial conformation to begin with like the RNA origami. The NOS allows to prove the co-NP hardness of deciding, given two NOSs, if there exists no conformation that one of them folds into but the other does not.

Abstract: We adapt an algorithmic approach to the problem of local realism in a bipartite scenario. We assume that local outcomes are simulated by spatially separated universal Turing machines. The outcomes are calculated from inputs encoding information about a local measurement setting and a description of the bipartite system sent to both parties. In general, such a description can encode some additional information not available in quantum theory, i.e., local hidden variables. Using the Kolmogorov complexity of local outcomes we derive an inequality that must be obeyed by any local realistic theory. Since the Kolmogorov complexity is in general uncomputable, we show that this inequality can be expressed in terms of lossless compression of the data generated in such experiments and that quantum mechanics violates it. Finally, we confirm experimentally our findings using pairs of polarisation-entangled photons and readily available compression software. We argue that our approach relaxes the independent and identically distributed (i.i.d.) assumption, namely that individual bits in the outcome bit-strings do not have to be i.i.d.

Abstract: We compare the computational power of different classes of computational systems and relate it to whether they contain closed loops. Adding closed loops to the architecture of computational systems increases their computational power. Different computational models are apt for capturing the computational power of different classes of neural systems. We argue that while ordinary Turing machines (TMs) are a poor model for a kind of feedback that the closed-loop approach to neuroscience highlights, suitably modified TMs are a better fit.