Abstract: The universal composability paradigm allows for the modular design and analysis of cryptographic protocols. It has been widely and successfully used in cryptography. However, devising a coherent yet simple and expressive model for universal composability is, as the history of such models shows, highly non-trivial. For example, several partly severe problems have been pointed out in the literature for the UC model. In this work, we propose a coherent model for universal composability, called the IITM model (“Inexhaustible Interactive Turing Machine”). A main feature of the model is that it is stated without a priori fixing irrelevant details, such as a specific way of addressing of machines by session and party identifiers, a specific modeling of corruption, or a specific protocol hierarchy. In addition, we employ a very general notion of runtime. All reasonable protocols and ideal functionalities should be expressible based on this notion in a direct and natural way, and without tweaks, such as (artificial) padding of messages or (artificially) adding extra messages. Not least because of these features, the model is simple and expressive. Also the general results that we prove, such as composition theorems, hold independently of how such details are fixed for concrete applications. Being inspired by other models for universal composability, in particular the UC model and because of the flexibility and expressivity of the IITM model, conceptually, results formulated in these models directly carry over to the IITM model.

Abstract: Spectral factorization is an operation which appears in many different engineering applications. This paper studies whether spectral factorization can be algorithmically computed on an abstract machine (a Turing machine). It is shown that there exist computable spectral densities with very good analytic properties (i.e. smooth with finite energy) such that the corresponding spectral factor cannot be determined on a Turing machine. Further, it will be proved that it is impossible to decide algorithmically whether or not a given computable density possesses a computable spectral factor. This negative result has consequences for applications of spectral factorization in computer-aided design, because there it is necessary that this problem be decidable. Conversely, this paper will show that if the logarithm of a computable spectral density belongs to certain Sobolev space of sufficiently smooth functions, then the spectral factor is always computable. As an application, the paper discusses the possibility of calculating the optimal causal Wiener filter on an abstract machine.

Abstract: We present a framework for the verified programming of multi-tape Turing machines in Coq. Improving on prior work by Asperti and Ricciotti in Matita, we implement multiple layers of abstraction. The highest layer allows a user to implement nontrivial algorithms as Turing machines and verify their correctness, as well as time and space complexity compositionally. The user can do so without ever mentioning states, symbols on tapes or transition functions: They write programs in an imperative language with registers containing values of encodable data types, and our framework constructs corresponding Turing machines. As case studies, we verify a translation from multi-tape to single-tape machines as well as a universal Turing machine, both with polynomial time overhead and constant factor space overhead.

Abstract: Modern communication systems need to satisfy pre-specified requirements on spectral efficiency and security. Physical layer security is a concept that unifies both and connects them with entropic quantities. In this paper, a framework based on Turing machines is developed to address the question of deciding whether or not a communication system meets these requirements. It is known that the class of Turing solvable problems coincides with the class of algorithmically solvable problems so that this framework provides the theoretical basis for effective verification of such performance requirements. A key issue here is to decide whether or not the performance functions, i.e., capacities, of relevant communication scenarios, particularly those with secrecy requirements and active adversaries, are Turing computable. This is a necessary condition for the corresponding communication protocols to be effectively verifiable. Within this framework, it is then shown that for certain scenarios including the wiretap channel the corresponding capacities are Turing computable. Next, a general necessary condition on the performance function for Turing computability is established. With this, it is shown that for certain scenarios, including the wiretap channel with an active jammer, the performance functions are not computable when deterministic codes are used. As a consequence, such performance functions are also not computable on all other computer architectures such as the von Neumann-architecture or the register machines.

Abstract: The quest to understand consciousness, once the purview of philosophers and theologians, is now actively pursued by scientists of many stripes. This paper studies consciousness from the perspective of theoretical computer science. It formalizes the Global Workspace Theory (GWT) originated by cognitive neuroscientist Bernard Baars and further developed by him, Stanislas Dehaene, and others. Our major contribution lies in the precise formal definition of a Conscious Turing Machine (CTM), also called a Conscious AI. We define the CTM in the spirit of Alan Turing's simple yet powerful definition of a computer, the Turing Machine (TM). We are not looking for a complex model of the brain nor of cognition but for a simple model of (the admittedly complex concept of) consciousness. After formally defining CTM, we give a formal definition of consciousness in CTM. We then suggest why the CTM has the feeling of consciousness. The reasonableness of the definitions and explanations can be judged by how well they agree with commonly accepted intuitive concepts of human consciousness, the breadth of related concepts that the model explains easily and naturally, and the extent of its agreement with scientific evidence.

Abstract: We construct an automaton group with a PSPACE-complete word problem, proving a conjecture due to Steinberg. Additionally, the constructed group has a provably more difficult, namely EXPSPACE-complete, compressed word problem. Our construction directly simulates the computation of a Turing machine in an automaton group and, therefore, seems to be quite versatile. It combines two ideas: the first one is a construction used by D'Angeli, Rodaro and the first author to obtain an inverse automaton semigroup with a PSPACE-complete word problem and the second one is to utilize a construction used by Barrington to simulate circuits of bounded degree and logarithmic depth in the group of even permutations over five elements.

Abstract: Solving continuous problems on digital computers gives generally only approximations of the continuous solutions. It is therefore crucial that the error between the continuous solution and the digital approximation can effectively be controlled. This paper investigates the possibility of simulating linear, time-invariant (LTI) systems on Turing machines. It is shown that there exist elementary LTI systems for which an admissible and computable input signal results in a non-computable output signal. For these LTI systems, the paper gives sharp characterizations of input spaces such that the output is guaranteed to be computable. The second part of the paper discusses the computability of the impulse and step response of LTI systems. It is shown that the computability of the step response implies not the computability of the impulse response. Moreover, there exist impulse responses which cannot be computed from the transfer function using the inverse Laplace transform. Finally, the paper gives a stronger version of a classical non-computability result, showing that there exist admissible and computable initial values for the wave equation so that the solution cannot be computed at certain points in space and time.

Abstract: One of the significant objectives of artificial intelligence is to design learning algorithms that are executed on general-purpose computational machines inspired by the human brain. Neural Turing Machine (NTM) is a step towards realizing such a computational machine. In the literature, a variety of approaches have been presented for the NTM; however, there is no existing comprehensive survey and taxonomy for NTM methods. This article presents an overview of taxonomies characterizing the critical concepts of the NTM through a comprehensive survey on the related research activities. This in-depth analysis of taxonomies can provide researchers, designers, and application developers with a clear guideline to compare NTM methods. The taxonomy of machine learning, neural networks, and the Turing machine is introduced. The NTM is also inspected in terms of concepts, structure, implemented tasks, and related works. The article further presents research discussions and future challenges in this area.

Abstract: Against a remarkable hardware oblivion in discussions of algorithmic intelligence, this article insists that algorithmic thought, or abstract computation, cannot be separated from its technological...

Abstract: Spiking neural P (SN P) systems are models of computation inspired by spiking neurons and part of the third generation of neuron models. SN P systems are equivalent to Turing machines and are able to solve computationally hard problems using a space-time trade-off. Research in SN P systems theory is especially active, more so in recent years as more efforts are directed towards their real-world applications. Usually, SN P systems are represented visually as a directed graph and simulated through mainly text-based simulations or tables. Thus, there is a need for tools that can simulate and create SN P Systems in a visual and easy-to-use manner. Snapse is such a tool which aims to hasten the speed and ease at which researchers may create and experiment with SN P systems. Furthermore, visual tools such as Snapse can help further bring SN P systems outside of theoretical computer science.

Abstract: The machinery of the human brain -- analog, probabilistic, embodied -- can be characterized computationally, but what machinery confers what computational powers? Any such system can be abstractly cast in terms of two computational components: a finite state machine carrying out computational steps, whether via currents, chemistry, or mechanics; plus a set of allowable memory operations, typically formulated in terms of an information store that can be read from and written to, whether via synaptic change, state transition, or recurrent activity. Probing these mechanisms for their information content, we can capture the difference in computational power that various systems are capable of. Most human cognitive abilities, from perception to action to memory, are shared with other species; we seek to characterize those (few) capabilities that are ubiquitously present among humans and absent from other species. Three realms of formidable constraints -- a) measurable human cognitive abilities, b) measurable allometric anatomic brain characteristics, and c) measurable features of specific automata and formal grammars -- illustrate remarkably sharp restrictions on human abilities, unexpectedly confining human cognition to a specific class of automata ("nested stack"), which are markedly below Turing machines.

Abstract: Living systems process information using chemistry. Computations can be viewed as language recognition problems where both languages and automata recognizing them form an inclusive hierarchy. Chemical realizations, without using biochemistry, of the main classes of computing automata, Finite Automata (FA), 1-stack Push Down Automata (1-PDA) and Turing Machine (TM) have recently been presented. These use chemistry for the representation of input information, its processing and output information. The Turing machine uses the Belousov-Zhabotinsky (BZ) oscillatory reaction to recognize a representative Context-Sensitive Language (CSL), the 1-PDA uses a pH network to recognize a Context Free Language (CFL) and a FA for a Regular Language (RL) uses a precipitation reaction. By chemically reconfiguring them to recognize representative languages in the lower classes of the Chomsky hierarchy we illustrate the inclusiveness of the hierarchy of native chemical automata. These examples open the door for chemical programming without biochemistry. Furthermore, the thermodynamic metric originally introduced to identify the accept/reject state of the chemical output for the CSL, can equally be used for recognizing CFL and RL by the automata. Finally, we point out how the chemical and thermodynamic duality of accept/reject criteria can be used in the optimization of the energetics and efficiency of computations.

Abstract: Deep learning based knowledge tracing approaches achieve high accuracy in mastery prediction with pattern extraction on a large learning behavior data set. However, when there is little training data available, these approaches either fail to extract the key patterns or result in over fitting. Ideally, we aim to provide a similar learning experience to both the first group of learners, who interact with a new course or a new activity with little learning behavior data to provide personalized guidance, and the learners who interact with the course later. We propose a novel architecture, Attentive Neural Turing Machine (ANTM), to solve the cold start knowledge tracing problem. The proposed ANTM comprises an attentive controller module and differential reading and writing processes with extra memory bank. Accuracy (ACC) and Area Under Curve (AUC) measures are used for model performance comparison. Results show the proposed approach can learn fast and generalize well to unseen data. It achieves around 95% ACC trained with only 3 learners, while conventional deep learning based approaches achieve only 65% ACC with over prediction issues.

Abstract: (This is known as “system 0” in the proof below.) The proof I intend to give demonstrates that this Turing machine can emulate any twocolor cyclic tag system for an infinite number of steps; in order to do this, I first show that this Turing machine can emulate any two-color cyclic tag system for an arbitrary finite number of steps (with the number of steps encoded in the initial condition), and then use this result as a basis for proving the more general result that this Turing machine can carry on such an emulation indefinitely. One main problem is to ensure that it is the system itself that is doing the calculation, and not the case that the universal part of the calculation is complete before the initial condition is even constructed. The construction I will give to show that system 0 can emulate any two-color cyclic tag system for an arbitrary number of steps is quite complex, and it is not immediately obvious that it does not itself do the calculation rather than leaving system 0 to do the calculation. However, the proofs below leave many options open in the initial conditions (there is more than one initial condition that emulates a cyclic tag system), and after the rest of the proof I show that at least one of these initial conditions can be constructed by a process that is clearly not universal itself.

Abstract: Universal Turing Machines are well known in computer science but they are about manual programming for general purposes. Although human children perform conscious learning (learning while being conscious) from infancy, it is unknown that Universal Turing Machines can facilitate not only our understanding of Autonomous Programming For General Purposes (APFGP) by machines, but also enable early-age conscious learning. This work reports a new kind of AI-conscious learning AI from a machine's “baby” time. Instead of arguing what static tasks a conscious machine should be able to do during its “adulthood”, this work suggests that APFGP is a computationally clearer and necessary criterion for us to judge whether a machine is capable of conscious learning so that it can autonomously acquire skills along its “career path”. The results here report new concepts and experimental studies for early vision, audition, natural language understanding, and emotion, with conscious learning capabilities that are absent from traditional AI systems.

Abstract: We introduce a realisability semantics for infinitary intuitionistic set theory that is based on Ordinal Turing Machines (OTMs). We show that our notion of OTM-realisability is sound with respect to certain systems of infinitary intuitionistic logic, and that all axioms of infinitary Kripke-Platek set theory are realised. As an application of our technique, we show that the propositional admissible rules of (finitary) intuitionistic Kripke-Platek set theory are exactly the admissible rules of intuitionistic propositional logic.

Abstract: It is known that the polarizationless P systems of the kind involved in the definition of the P conjecture are able to solve problems in the complexity class $$\textsf {P}$$ by leveraging their uniformity condition. Here, we show that they are indeed able to simulate a deterministic Turing machine working in polynomial time with a weaker uniformity condition and using only one level of membrane nesting. This allows us to embed this construction into more complex membrane structures, possibly showing that constructions similar to the one performed for P systems with charges can be carried on.

Abstract: We consider notions of space complexity for Infinite Time Turing Machines (ITTMs) that were introduced by B. Lowe and studied further by J. Winter. We answer several open questions about these notions, among them whether low space complexity implies low time complexity (it does not) and whether one of the equalities P=PSPACE, P$_{+}=$PSPACE$_{+}$ and P$_{++}=$PSPACE$_{++}$ holds for ITTMs (all three are false). We also show various separation results between space complexity classes for ITTMs. This considerably expands our earlier observations on the topic in section 7.2.2 of \cite{Ca2}, which appear here as Lemma $6$ up to Corollary $9$.

Abstract: Making an artificial brain is not a part of artificial intelligence. It will be a revolutionary journey of mankind exploring a science where one cannot write an equation, a material will vibrate like geometric shape, and then those shapes will change to make decisions. Geometry of silence plays like a musical instrument to mimic a human brain; our thoughts, imagination, everything would be a 3D shape playing as music; composing music would be the brain’s singular job. For a century, the Turing machine ruled human civilization; it was believed that irrespective of complexity all events add up linearly. This book is a thesis to explore the science of decision-making where events are 3D-geometric shapes, events grow within and above, never side by side. ​ The book documents inventions and discoveries in neuroscience, computer science, materials science, mathematics and chemistry that explore the possibility of brain or universe as a time crystal. The philosophy of Turing, the philosophy of membrane-based neuroscience and the philosophy of linear, sequential thought process are challenged here by considering that a nested time crystal encompasses the entire conscious universe. Instead of an algorithm, the pattern of maximum free will is generated mathematically and that very pattern is encoded in materials such that its natural vibration integrates random events exactly similar to the way nature does it in every remote corner of our universe. Find how an artificial brain avoids any necessity for algorithm or programming using the pattern of free will.

Abstract: Although natural and bioinspired computing has developed significantly, the relationship between the computational universality and efficiency beyond the Turing machine has not been studied in detail. Here, we investigate how asynchronous updating can contribute to the universal and efficient computation in cellular automata (CA). First, we define the computational universality and efficiency in CA and show that there is a trade-off relation between the universality and efficiency in CA implemented in synchronous updating. Second, we introduce asynchronous updating in CA and show that asynchronous updating can break the trade-off found in synchronous updating. Our finding spells out the significance of asynchronous updating or the timing of computation in robust and efficient computation.

Abstract: The entailment between separation logic formulae with inductive predicates, also known as symbolic heaps, has been shown to be decidable for a large class of inductive definitions. Recently, a 2-EXPTIME algorithm was proposed and an EXPTIME-hard bound was established; however no precise lower bound is known. In this paper, we show that deciding entailment between predicate atoms is 2-EXPTIME-hard. The proof is based on a reduction from the membership problem for exponential-space bounded alternating Turing machines.

Abstract: The zero-error capacity of a discrete classical channel was first defined by Shannon as the least upper bound of rates for which one transmits information with zero probability of error. The problem of finding the zero-error capacity C 0 , which assigns a capacity to each channel as a function, was reformulated in terms of graph theory as a function Θ, which assigns a value to each simple graph. This paper studies the computability of the zero-error capacity. For the computability, the concept of a Turing machine and a Kolmogorov oracle is used. It is unknown if the zero-error capacity is computable in general. We show that in general the zero-error capacity is semi-computable with the help of a Kolmogorov Oracle. Furthermore, we show that C 0 and Θ are computable functions if and only if there is a computable sequence of computable functions of upper bounds, i.e. the converse exist in the sense of information theory, which point-wise converges to C 0 or Θ. Finally, we examine Zuiddam’s characterization of C 0 and Θ in terms of algorithmic computability.

Abstract: 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.