Abstract: The Enigma was a cryptographic (enciphering) machine used by the German military during WWII. The German navy changed part of the Enigma keys every other day. One of the important cryptanalytic attacks against the naval usage was called Banburismus, a sequentiai Bayesian procedure (anticipating sequential analysis) which was used from the sorine of 1941 until the middle of 1943. It was invented mainlv bv A. M. Turina and was perhaps the first important sequential Bayesian IE is unnecessab to describe it here. Before Banburismus could be started on a given day it was necessary to identifv which of nine ‘biaram’ (or ‘diaraph’) tables was in use on that day. In Turing’s approach to this identification hk had io istimate the probabilities of certain ‘trigraphs’. rrhese trigraphs were used. as described below. for determinine the initial wheel settings of messages). For estimatidg the probabilities, Turing inventedin important special case o the nonparametric (nonhypermetric) Empirid Bayes method independently...

Abstract: We give the first nontrivial model-independent time?space tradeoffs for satisfiability. Namely, we show that SAT cannot be solved in n1+o(1) time and n1?? space for any ?>0 general random-access nondeterministic Turing machines. In particular, SAT cannot be solved deterministically by a Turing machine using quasilinear time and n space. We also give lower bounds for log-space uniform NC1 circuits and branching programs. Our proof uses two basic ideas. First we show that if SAT can be solved nondeterministically with a small amount of time then we can collapse a nonconstant number of levels of the polynomial-time hierarchy. We combine this work with a result of Nepomnja??ii that shows that a nondeterministic computation of superlinear time and sublinear space can be simulated in alternating linear time. A simple diagonalization yields our main result. We discuss how these bounds lead to a new approach to separating the complexity classes NL and NP. We give some possibilities and limitations of this approach.

Abstract: Persistent Turing Machines (PTMs) are multitape machines with a persistent worktape preserved between interactions, whose inputs and outputs are dynamically generated streams of tokens (strings). They are a minimal extension of Turing Machines (TMs) that express interactive behavior. They provide a natural model for sequential interactive computation such as single-user databases and intelligent agents. PTM behavior is characterized observationally, by input-output streams; the notions of equivalence and expressiveness for PTMs are defined relative to its behavior. Four different models of PTM behavior are examined: language-based, automaton-based, function-based, and environment-based. A number of special subclasses of PTMs are identified; several expressiveness results are obtained, both for the general class of all PTMs and for the special subclasses, proving the conjecture in [We2] that interactive computing devices are more expressive than TMs. The methods and tools for formalizing PTM computation developed in this paper can serve as a basis for a more comprehensive theory of interactive computation.

Abstract: We characterise explicitly the decidable predicates on integers of Infinite Time Turing machines. in terms of admissibility theory and the constructible hierarchy. We do this by pinning down I. the least ordinal not the length of any eventual output of an Infinite Time Turing machine (halting or otherwise): using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy, further that the natural ordinals associated with the jump operator satisfy a Spector criterion, and correspond to the LU -stables. It also implies that the machines devised are "E) Complete" amongst all such other possible machines. It is shown that least upper bounds of an "eventual jump" hierarchy exist on an initial segment. ?

Abstract: This paper investigates computation by linear assemblies of complex DNA tiles, which we call string tiles. By keeping track of the strands as they weave back and forth through the assembly, we show that surprisingly sophisticated calculations can be performed using linear self-assembly. Examples range from generating an addition table to providing O(1) solutions to CNF-SAT and DHPP. We classify the families of languages that can be generated by various types of DNA molecules, and establish a correspondence to the existing classes ET0Lml and ET0Lfin. Thus, linear self-assembly of string tiles can generate the output languages of finite-visit Turing Machines.

Abstract: Quantum finite automata, as well as quantum pushdown automata were first introduced by C. Moore, J. P. Crutchfield [13]. In this paper we introduce the notion of quantum pushdown automata (QPA) in a non-equivalent way, including unitarity criteria, by using the definition of quantum finite automata of [11]. It is established that the unitarity criteria of QPA are not equivalent to the corresponding unitarity criteria of quantum Turing machines [4]. We show that QPA can recognize every regular language. Finally we present some simple languages recognized by QPA, two of them are not recognizable by deterministic pushdown automata and one seems to be not recognizable by probabilistic pushdown automata as well.

Abstract: Permanents and determinants are special cases of immanants. The latter are polynomial matrix functions defined in terms of characters of symmetric groups and corresponding to Young diagrams. Valiant has proved that the evaluation of permanents is a complete problem in both the Turing machine model (#P-completeness) as well as in his algebraic model (VNP-completeness). We show that the evaluation of immanants corresponding to hook diagrams or rectangular diagrams of polynomially growing width is both #P-complete and VNP-complete.

Abstract: A set is autoreducible if it can be reduced to itself by a Turing machine that does not ask its own input to the oracle. We use autoreducibility to separate the polynomial-time hierarchy from exponential space by showing that all Turing complete sets for certain levels of the exponential-time hierarchy are autoreducible but there exists some Turing complete set for doubly exponential space that is not. Although we already knew how to separate these classes using diagonalization, our proofs separate classes solely by showing they have different structural properties, thus applying Post's program to complexity theory. We feel such techniques may prove unknown separations in the future. In particular, if we could settle the question as to whether all Turing complete sets for doubly exponential time are autoreducible, we would separate either polynomial time from polynomial space, and nondeterministic logarithmic space from nondeterministic polynomial time, or else the polynomial-time hierarchy from exponential time. We also look at the autoreducibility of complete sets under nonadaptive, bounded query, probabilistic, and nonuniform reductions. We show how settling some of these autoreducibility questions will also lead to new complexity class separations.

Abstract: We start with an abstract definition of interior-point methods. These are the methods of solving convex optimization problems by generating a sequence of points which lies in the relative interior of a convex set defined by the difficult constraints (see [797]). In this definition, we are thinking about a formulation of the underlying problem in which one has a maximal set of linear equality constraints and some convex set constraint. The fundamental idea of interior-point algorithms goes back at least to Frisch [257] and to Fiacco and McCormick [230]. However, almost all of the new interior-point algorithms have their foundations in Karmarkar’s seminal work [404]. Karrnarkar’s work introduced many ingredients of the interior-point algorithms with polynomial iteration complexity that have been studied extensively since 1984. These algorithms possess polynomial bounds on the number of arithmetic operations required to solve a linear programming (LP) problem. In this chapter, we will be content with the bounds on the number of iterations. To date, it is not known whether even the SDP feasibility problem belongs to the class of decision problems solvable in polynomial time (assuming rational data and using the “bit model” on a Turing machine).

Abstract: Many IT-systems behave very differently from classical machine models: they interact with an unpredictable environment, they never terminate, and their behavior changes over time. Wegner [25,26] (see also [28]) recently argued that the power of interaction goes beyond the Church-Turing thesis. To explore interaction from a computational viewpoint, we describe a generic model of an 'interactive machine' which interacts with the environment using single streams of input and output signals over a simple alphabet. The model uses ingredients from the theory of ω-automata. Viewing the interactive machines as transducers of infinite streams of signals, we show that their interactive recognition and generation capabilities are identical. It is also shown that, in the given model, all interactively computable functions are limit-continuous.

Abstract: This paper examines certain claims of 'cognitive significance' which (wisely or not) have been based upon the theoretical powers of three distinct classes of connectionist networks, namely, the 'universal function approximators', recurrent finite-state simulation networks and Turing equivalent networks. Each class will be considered with respect to its potential in the realm of cognitive modelling. Regarding the first class, I argue that, contrary to the claims of some influential connectionists, feed-forward networks do not possess the theoretical capacity to approximate all functions of interest to cognitive scientists. For example, they cannot approximate many important, recursive (halting) functions which map symbolic strings onto other symbolic strings. By contrast, I argue that a certain class of recurrent networks (i.e. those which closely approximate deterministic finite automata (DFA)) shows considerably greater promise in some domains. However, from a cognitive standpoint, difficulties arise whe...

Abstract: The paper is about the comparison between (apparently) different cartesian closed extensions of the category of topological spaces. Since topological spaces do not in general allow formation of function spaces, the problem of determining suitable categories with such a property—and nicely related to that of topological spaces—was studied from many different perspectives: general topology, functional analysis, measure theory, computability. From a computational perspective, the interest about topological properties of the function spaces arose after the discovery of topological models of the λ-calculus by D.S. Scott, see [22], and the work of Ersov on the partial continuous functionals, see [12]. At the same time, work on cartesian closed extensions of the category of topological spaces in general topology produced interesting quasitoposes of filter spaces where a notion of convergence replaced that of neighbourhood system, see [23] for a complete review. The semantical analysis of computer behaviour requires that the mechanical black box which is the computer is simply a rule-executing grinder which determines values/outputs from given arguments/inputs. Although the mathematical concept of a Turing machine explains very precisely how the grinder operates, it is extremely useful to be able to have a more conceptual intuition about what a machine does. One of the most useful approximations to this intuition produced so far is the following: a computer executing a program evaluates a partial function. It follows that one of the most important things to know is whether, given a certain collection of inputs, the function evaluates on them all. Since programs can be stored in the machine, and operated upon by the grinder, a direct consequence is that extensional models of programs are to be cartesian closed categories. By an extensional model, here we mean a mathematical structure where two maps are distinguished by their values on global elements, e.g. a category where the terminal object generates. This does not necessarily mean that a general notion of model of computation

Abstract: We examine the selection and competition of patterns in the Brusselator model, one of the simplest reaction-diffusion systems giving rise to Turing instabilities. Simulations of this model show a significant change in the wave number of stable patterns as the control parameter is increased. A weakly nonlinear analysis makes it possible to obtain the amplitude equations for the concentration fields near the instability threshold. Together with the linear diffusive terms, these equations also contain nonvariational spatial terms. When these terms are included, the stability diagrams and the thresholds for secondary instabilities are heavily modified with respect to the usual diffusive case. The results obtained from the numerical simulations fit very well into the calculated stability regions.

Abstract: In this first of two papers, strong limits on the accuracy of physical computation are established. First it is proven that there cannot be a physical computer C to which one can pose any and all computational tasks concerning the physical universe. Next it is proven that no physical computer C can correctly carry out any computational task in the subset of such tasks that can be posed to C. This result holds whether the computational tasks concern a system that is physically isolated from C, or instead concern a system that is coupled to C. As a particular example, this result means that there cannot be a physical computer that can, for any physical system external to that computer, take the specification of that external system's state as input and then correctly predict its future state before that future state actually occurs; one cannot build a physical computer that can be assured of correctly 'processing information faster than the universe does'. The results also mean that there cannot exist an infallible, general-purpose observation apparatus, and that there cannot be an infallible, general-purpose control apparatus. These results do not rely on systems that are infinite, and/or non-classical, and/or obey chaotic dynamics. They also hold even if one uses an infinitely fast, infinitely dense computer, with computational powers greater than that of a Turing Machine. This generality is a direct consequence of the fact that a novel definition of computation - a definition of 'physical computation' - is needed to address the issues considered in these papers. While this definition does not fit into the traditional Chomsky hierarchy, the mathematical structure and impossibility results associated with it have parallels in the mathematics of the Chomsky hierarchy. The second in this pair of papers presents a preliminary exploration of some of this mathematical structure, including in particular that of prediction complexity, which is a 'physical computation analogue' of algorithmic information complexity. It is proven in that second paper that either the Hamiltonian of our universe proscribes a certain type of computation, or prediction complexity is unique (unlike algorithmic information complexity), in that there is one and only version of it that can be applicable throughout our universe.

Abstract: Alan Turing anticipated many areas of current research in computer and cognitive science. This article outlines his contributions to Artificial Intelligence, connectionism, hypercomputation, and Artificial Life, and also describes Turing's pioneering role in the development of electronic stored-program digital computers. It locates the origins of Artificial Intelligence in postwar Britain. It examines the intellectual connections between the work of Turing and of Wittgenstein in respect of their views on cognition, on machine intelligence, and on the relation between provability and truth. We criticise widespread and influential misunderstandings of the Church–Turing thesis and of the halting theorem. We also explore the idea of hypercomputation, outlining a number of notional machines that “compute the uncomputable.”

Abstract: We investigate the computational properties of finite binary- and analog-state discrete-time symmetric Hopfield nets. For binary networks, we obtain a simulation of convergent asymmetric networks by symmetric networks with only a linear increase in network size and computation time. Then we analyze the convergence time of Hopfield nets in terms of the length of their bit representations. Here we construct an analog symmetric network whose convergence time exceeds the convergence time of any binary Hopfield net with the same representation length. Further, we prove that the MIN ENERGY problem for analog Hopfield nets is NP-hard and provide a polynomial time approximation algorithm for this problem in the case of binary nets. Finally, we show that symmetric analog nets with an external clock are computationally Turing universal.

Abstract: I would like to talk about some crazy stuff. The general idea is that sometimes ideas are very powerful. I’d like to talk about theory, about the computer as a concept, a philosophical concept.

Abstract: The Turing Test (TT), as originally specified, centres on the ability to perform a social role. The TT can be seen as a test of an ability to enter into normal human social dynamics. In this light it seems unlikely that such an entity can be wholly designed in an “off-line” mode; rather a considerable period of training in situ would be required. The argument that since we can pass the TT, and our cognitive processes might be implemented as a Turing Machine (TM), that consequently a TM that could pass the TT could be built, is attacked on the grounds that not all TMs are constructible in a planned way. This observation points towards the importance of developmental processes that use random elements (e.g., evolution), but in these cases it becomes problematic to call the result artificial. This has implications for the means by which intelligent agents could be developed.

Abstract: We propose a systematic investigation of the (semi-) automatic verifiability of ASMs. As a first step, we put forward two verification problems concerning the correctness of ASMs and investigate the decidability and complexity of both problems.

Abstract: This paper develops a database query language called Transducer Datalog motivated by the needs of a new and emerging class of database applications. In these applications, such as text databases and genome databases, the storage and manipulation of long character sequences is a crucial feature. The issues involved in managing this kind of data are not addressed by traditional database systems, either in theory or in practice. To address these issues, we recently introduced a new machine model called a generalized sequence transducer. These generalized transducers extend ordinary transducers by allowing them to invoke other transducers as “subroutines.” This paper establishes the computational properties of Transducer Datalog, a query language based on this new machine model. In the process, we develop a hierarchy of time-complexity classes based on the Ackermann function. The lower levels of this hierarchy correspond to well-known complexity classes, such as polynomial time and hyper-exponential time. We establish a tight relationship between levels in this hierarchy and the depth of subroutine calls within Transducer Datalog programs. Finally, we show that Transducer Datalog programs of arbitrary depth express exactly the sequence functions computable in primitive-recursive time.

Abstract: We are interested in the expressiveness of constraints represented by general first order formulae, with equality as unique relational symbol and functional symbols taken from an infinite set F. The chosen domain is the set of trees whose nodes, in possibly infinite number, are labeled by elements of F. The operation linked to each element f of F is the mapping (a1, . . . , an) → b, where b is the tree whose initial node is labeled f and whose sequence of daughters is a1, . . . , an. We first consider constraints involving long alternated sequences of quantifiers ∃∀∃∀ ... We show how to express winning positions of two-person games with such constraints and apply our results to two examples. We then construct a family of strongly expressive constraints, inspired by a constructive proof of a complexity result by Pawel Mielniczuk. This family involves the huge number α(k), obtained by evaluating top down a power tower of 2's, of height k. With elements of this family, of sizes at most proportional to k, we define a finite tree having α(k) nodes, and we express the result of a Prolog machine executing at most α(k) instructions. By replacing the Prolog machine by a Turing machine we rediscover the following result of Sergei Vorobyov: the complexity of an algorithm, deciding whether a constraint without free variables is true, cannot be bounded above by a function obtained by finite composition of elementary functions including exponentiation. Finally, taking advantage of the fact that we have at our disposal an algorithm for solving such constraints in all their generality, we produce a set of benchmarks for separating feasible examples from purely speculative ones. Among others we solve constraints involving alternated sequences of more than 160 quantifiers.

Abstract: Our model of computation (theoretical machine) was designed for the analysis of analog Fourier optical processors, its basic data unit being a continuous image of unbounded resolution. In this paper, we demonstrate the universality of our machine by presenting a framework for arbitrary Turing machine simulation. Computational complexity benefits are also demonstrated by providing a O(log 2 n) algorithm for a search problem that has a lower bound of n - 1 on a Turing machine.

Abstract: In a number of recent studies the question has arisen whether the familiar Church-Turing thesis is still adequate to capture the powers and limitations of modern computational systems. In this presentation we review two developments that may lead to an extension of the classical Turing machine paradigm: interactiveness, and global computing.

Abstract: $-calculus is a higher-order polyadic process algebra for resource bounded computation. It has been designed to handle autonomous agents, evolutionary computing, neural nets, expert systems, machine learning, and distributed interactive AI systems, in general. $-calculus has built-in cost-optimization mechanism allowing to deal with nondeterminism, incomplete and uncertain information. In this paper, we investigate expressiveness of $-calculus. We show that due to infinitary means, it allows to express models having richer behavior than Turing machine, including cellular automata, interaction machines, neural networks, and random automata networks. We also investigate the importance of synchronization, representation of continuity, and higher-order.

Abstract: We assume the multitape real-time Turing machine as a formal model for parallel real-time computation. Then, we show that, for any positive integer k, there is at least one language Lk which is accepted by a k-tape real-Turing machine, but cannot be accepted by a (k - 1)-tape real-time Turing machine. It follows therefore that the languages accepted by real-time Turing machines form an infinite hierarchy with respect to the number of tapes used. Although this result was previously obtained elsewhere, our proof is considerably shorter, and explicitly builds the languages Lk. The ability of the real-time Turing machine to model practical real-time and/or parallel computations is open to debate. Nevertheless, our result shows how a complexity theory based on a formal model can draw interesting results that are of more general nature than those derived from examples. Thus, we hope to offer a motivation for looking into realistic parallel real-time models of computation.