Abstract: We consider the question of whether P ne NP implies that there exists some concept class that is efficientlyrepresentable but is still hard to learn in the PAC model of Valiant (CACM '84), where the learner is allowed to output any efficient hypothesis approximating the concept, including an "improper" hypothesis that is not itself in the concept class. We show that unless the polynomial hierarchy collapses, such a statement cannot be proven via a large class of reductions including Karp reductions, truth-table reductions, and a restricted form of non-adaptive Turing reductions. Also, a proof that uses a Turing reduction of constant levels of adaptivity would imply an important consequence in cryptography as it yields a transformation from any average-case hard problem in NP to a one-way function. Our results hold even in the stronger model of agnostic learning. These results are obtained by showing that lower bounds for improper learning are intimately related to the complexity of zero-knowledge arguments and to the existence of weak cryptographic primitives. In particular, we prove that if alanguage L reduces to the task of improper learning of circuits, then, depending on the type of the reduction in use, either (1) L has a statistical zero-knowledge argument system, or (2) the worst-case hardness of L implies the existence of a weak variant of one-way functions defined by Ostrovsky-Wigderson (ISTCS '93). Interestingly, we observe that the converse implication also holds. Namely, if (1) or (2) hold then the intractability of L implies that improper learning is hard.

Abstract: Church's and Turing's theses dogmatically assert that an informal notion of computability is captured by a particular mathematical concept. I present an analysis of computability that leads to precise concepts, but dispenses with theses. To investigate computability is to analyze processes that can in principle be carried out by calculators. Drawing on this lesson we owe to Turing and recasting work of Gandy, I formulate finiteness and locality conditions for two types of calculators, human computing agents and mechanical computing devices; the distinctive feature of the latter is that they can operate in parallel. The analysis leads to axioms for discrete dynamical systems (representing human and machine computations) and allows the reduction of models of these axioms to Turing machines. Cellular automata and a variety of artificial neural nets can be shown to satisfy the axioms for machine computations.

Abstract: We propose a characterization of PSPACE by means of atype assignment for an extension of lambda calculus with a conditional construction. The type assignment STAB is an extension of STA, a type assignment for lambda-calculus inspired by Lafont's Soft Linear Logic.We extend STA by means of a ground type and terms for booleans. The key point is that the elimination rule for booleans is managed in an additive way. Thus, we are able to program polynomial time Alternating Turing Machines. Conversely, we introduce a call-by-name evaluation machine in order tocompute programs in polynomial space. As far as we know, this is the first characterization of PSPACE which is based on lambda calculusand light logics.

Abstract: This is Turing's classical paper with every passage quote/commented to highlight what Turing said, might have meant, or should have meant. The paper was equivocal about whether the full robotic test was intended, or only the email/penpal test, whether all candidates are eligible, or only computers, and whether the criterion for passing is really total, liefelong equavalence and indistinguishability or merely fooling enough people enough of the time. Once these uncertainties are resolved, Turing's Test remains cognitive science's rightful (and sole) empirical criterion today.

Abstract: We discuss combining physical experiments with machine computations and introduce a form of analogue–digital (AD) Turing machine. We examine in detail a case study where an experimental procedure based on Newtonian kinematics is combined with a class of Turing machines. Three forms of AD machine are studied, in which physical parameters can be set exactly and approximately. Using non-uniform complexity theory, and some probability, we prove theorems that show that these machines can compute more than classical Turing machines.

Abstract: This paper explores the impact of geometry on computability and complexity in Winfree's model of nanoscale self-assembly. We work in the two-dimensional tile assembly model, i.e., in the discrete Euclidean plane ? ×?. Our first main theorem says that there is a roughly quadratic function fsuch that a set A? ?+is computably enumerable if and only if the set X A = { (f(n), 0) | n? A} --- a simple representation of Aas a set of points on the x-axis --- self-assembles in Winfree's sense. In contrast, our second main theorem says that there are decidable sets D? ? ×? that do notself-assemble in Winfree's sense. Our first main theorem is established by an explicit translation of an arbitrary Turing machine Mto a modular tile assembly system $\mathcal{T}_M$, together with a proof that $\mathcal{T}_M$ carries out concurrent simulations of Mon all positive integer inputs.

Abstract: Determining the birth date of computer science is a very complicated task and certainly reliant to the standpoint chosen. Some may point out the work of Kurt Godel [18], Alan Turing [31], and Alonso Church [11], thus locating the appearance of computer science to 1930's. Some want to mention Charles Babbage's engines, some Gottfried Leibniz' Calculus Ratiocinator, and some refer back to the Euclidean algorithm.

Abstract: Infinite time register machines(ITRMs) are register machines which act on natural numbers and which are allowed to run for arbitrarily many ordinal steps Successor steps are determined by standard register machine commands At limit times a register content is defined as a $\liminf$ of previous register contents, if that limit is finite; otherwise the register is resetto 0 (A previous weaker version of infinitary register machines, in [6], would halt without a result in case of such an overflow) The theory of infinite time register machines has similarities to the infinite time Turing machines (ITTMs) of Hamkins and Lewis Indeed ITRMs can decide all $\Pi^1_1$ sets, yet they are strictly weaker than ITTMs

Abstract: We introduce a simple game family, called constraint logic, where players reverse edges in a directed graph while satisfying vertex in-flow constraints. This game family can be interpreted in many different game-theoretic settings, ranging from zero-player automata to a more economic setting of team multiplayer games with hidden information. Each setting gives rise to a model of computation that we show corresponds to a classic complexity class. In this way we obtain a uniform framework for modeling various complexities of computation as games. Most surprising among our results is that a game with three players and a bounded amount of state can simulate any (infinite) Turing computation, making the game undecidable. Our framework also provides a more graphical, less formulaic viewpoint of computation. This graph model has been shown to be particularly appropriate for reducing to many existing combinatorial games and puzzles - such as Sokoban, rush hour, river crossing, tipover, the warehouseman's problem, pushing blocks, hinged-dissection reconfiguration, Amazons, and Konane (hawaiian checkers) - which have an intrinsically planar structure. Our framework makes it substantially easier to prove completeness of such games in their appropriate complexity classes.

Abstract: We study lower bounds on space and input head reversals for deterministic, nondeterministic, and alternating Turing machines accepting nonregular languages. Three notions of space, namely strong, middle, weak are considered, and another notion, called accept, is introduced. In all cases, we obtain tight lower bounds. Moreover, we show that, while for determinism and nondeterminism such lower bounds are optimal even with respect to unary languages, for alternation optimal lower bounds for unary languages turn out to be strictly higher than those for languages over alphabets with two or more symbols.

Abstract: Classical models of computation no longer fully correspond to the current notions of computing in modern systems. Even in the sciences, many natural systems are now viewed as systems that compute. Can one devise models of computation that capture the notion of computing as seen today and that could play the same role as Turing machines did for the classical case? We propose two models inspired from key mechanisms of current systems in both artificial and natural environments: evolving automata and interactive Turing machines with advice. The two models represent relevant adjustments in our apprehension of computing: the shift to potentially non-terminating interactive computations, the shift towards systems whose hardware and/or software can change over time, and the shift to computing systems that evolve in an unpredictable, non-uniform way. The two models are shown to be equivalent and both are provably computationally more powerful than the models covered by the old computing paradigm. The models also motivate the extension of classical complexity theory by non-uniform classes, using the computational resources that are natural to these models. Of course, the additional computational power of the models cannot in general be meaningfully exploited in concrete goal-oriented computations.

Abstract: Numerous results for simple computationally universal systems are presented, with a particular focus on small universal Turing machines. These results are towards finding the simplest universal systems. We add a new aspect to this area by examining trade-offs between the simplicity of universal systems and their time/space computational complexity. Improving on the earliest results we give the smallest known universal Turing machines that simulate Turing machines in O(t2) time. They are also the smallest known machines where direct simulation of Turing machines is the technique used to establish their universality. This result gives a new algorithm for small universal Turing machines. We show that the problem of predicting t steps of the 1D cellular automaton Rule 110 is P-complete. As a corollary we find that the small weakly universal Turing machines of Cook and others run in polynomial time, an exponential improvement on their previously known simulation time overhead. These results are achieved by improving the cyclic tag system simulation time of Turing machines from exponential to polynomial. A new form of tag system which we call a bi-tag system is introduced. We prove that bi-tag systems are universal by showing they efficiently simulate Turing machines. We also show that 2-tag systems efficiently simulate Turing machines in polynomial time. As a corollary we find that the small universal Turing machines of Rogozhin, Minsky and others simulate Turing machines in polynomial time. This is an exponential improvement on the previously known simulation time overhead and improves on a forty-year old result. We present new small polynomial time universal Turing machines with state-symbol pairs of (5, 5), (6, 4), (9, 3) and (15, 2). These machines simulate bi-tag systems and are the smallest known universal Turing machines with 5, 4, 3 and 2-symbols, respectively. The 5-symbol machine uses the same number of instructions (22) as the current smallest known universal Turing machine (Rogozhin’s 6-symbol machine). We give the smallest known weakly universal Turing machines. These machines have state-symbol pairs of (6, 2), (3, 3) and (2, 4). The 3-state and 2-state machines are very close to the minimum possible size for weakly universal machines with 3 and 2 states, respectively.

Abstract: State Machines (hencefort referred to as just ASM) were introduced as “a computation model that is more powerful and more universal than standard computation models” by Yuri Gurevich in 1985.

Abstract: It is shown that there is no standard spiking neural P system that simulates Turing machines with less than exponential time and space overheads. The spiking neural P systems considered here have a constant number of neurons that is independent of the input length. Following this we construct a universal spiking neural P system with exhaustive use of rules that simulates Turing machines in polynomial time and has only 18 neurons.

Abstract: In this paper, we show how to construct secure obfuscation for Deterministic Finite Automata, assuming non-uniformly strong one-way functions exist. We revisit the software protection approaches originally proposed by [5, 10, 12, 17] and revise them to the current obfuscation setting of Barak et al. [2]. Under this model, we introduce an efficient oracle that retains some 'small' secret about the original program. Using this secret, we can construct an obfuscator and two-party protocol that securely obfuscates Deterministic Finite Automata against malicious adversaries. The security of this model retains the strong 'virtual black box' property originally proposed in [2] while incorporating the stronger condition of dependent auxiliary inputs in [15]. Additionally, we show that our techniques remain secure under concurrent self-composition with adaptive inputs and that Turing machines are obfuscatable under this model.

Abstract: We aim for a more rigorous discussion of "complexity" for Artificial Embryogeny. Initially, we review several existing measures from Biology and Mathematics. We argue that measures which rank complexity through a Turing machine, or measures of information contained in a genome about an environment, are not desireable here; Instead, we argue for measures which provide the environment "for free", allowing us to quantify the capacity for a genome to exploit a provided area of growth. This leads to our definition of Environmental Kolmogorov Complexity and Logical Depth, along with our introduction of novel measures of functional complexity. Next, we attempt at defining an exceptionally simple model of embryogenesis, the Terminating Cellular Automata. The described measures are computed in this context, and contrasted.

Abstract: The Exact Geometric Computation (EGC) mode of computation has been developed over the last decade in response to the widespread problem of numerical non-robustness in geometric algorithms Its technology has been encoded in libraries such as LEDA, CGAL and Core Library The key feature of EGC is the necessity to decide zero in its computation This paper addresses the problem of providing a foundation for the EGC mode of computation This requires a theory of real computation that properly addresses the Zero Problem The two current approaches to real computation are represented by the analytic school and algebraic school We propose a variant of the analytic approach based on real approximation To capture the issues of representation, we begin with a reworking of van der Waerden's idea of explicit rings and fields We introduce explicit sets and explicit algebraic structures Explicit rings serve as the foundation for real approximation: our starting point here is not ?, but $\mathbb{F}\subseteq \mathbb{R}$, an explicit ordered ring extension of ? that is dense in ? We develop the approximability of real functions within standard Turing machine computability, and show its connection to the analytic approach Current discussions of real computation fail to address issues at the intersection of continuous and discrete computation An appropriate computational model for this purpose is obtained by extending Schonhage's pointer machines to support both algebraic and numerical computation Finally, we propose a synthesis wherein both the algebraic and the analytic models coexist to play complementary roles Many fundamental questions can now be posed in this setting, including transfer theorems connecting algebraic computability with approximability

Abstract: Interesting subclasses of CHR are still Turing-complete: CHR with only one kind of rule, with only one rule, and propositional refined CHR. This is shown by programming a simulator of Turing machines (or Minsky machines) within those subclasses. Single-headed CHR without host language and propositional abstract CHR are not Turing-complete.

Abstract: This paper gives an implicit characterization of the class of functions computable in polynomial space by deterministic Turing machines – PSPACE. It gives an inductive characterization of PSPACE with no ad-hoc initial functions and with only one recursion scheme. The main novelty of this characterization is the use of pointers (also called path information) to reach PSPACE. The presence of the pointers in the recursion on notation scheme is the main difference between this characterization of PSPACE and the well-known Bellantoni-Cook characterization of the polytime functions – PTIME. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Abstract: This paper presents an overview of accelerating machines. We begin by exploring the history of the accelerating machine model and the potential power that it provides. We look at a number of computations that could be performed with an accelerating machine, and review various possible implementation methods that have been proposed. Finally, we expose the limitations of accelerating machines and conclude by posing some problems for further research.

Abstract: Several semantic-based malware analyzers have recently been put forward, each one defining its own model to capture the code behavior. All these semantic models, and abstract virology models likewise, fundamentally rely on formalisms equivalent to Turing Machines. However, as stated by recent advances in computer theory, these same formalisms do not capture appropriately interactions and concurrency. Unfortunately, malware, adaptable and resilient by essence, are likely to use these mechanisms intensively. In this paper, we thus extend the malware models to the specifically designed Interaction Machines. We first introduce two formal definitions for the interactive and the distributed viruses. According to different classes of interactions, their detection complexity is strongly impacted. Based on interactive languages, we then design an operational framework to describe malicious behaviors. Descriptions for some representative behaviors are given to complete and assess this framework.

Abstract: If we measure the position of a point particle, then we will come about with an interval [an, bn] into which the point falls. We make use of a Gedankenexperiment to find better and better values of an and bn, by reducing their relative distance, in a succession of intervals [a1, b1] ⊃ [a2, b2] ⊃ ... ⊃ [an, bn]] that contain the point.We then use such a point as an oracle to perform relative computation in polynomial time, by considering the succession of approximations to the point as suitable answers to the queries in an oracle Turing machine. We prove that, no matter the precision achieved in such a Gedankenexperiment, within the limits studied, the Turing Machine, equipped with such an oracle, will be able to compute above the classical Turing limit for the polynomial time resource, either generating the class P/poly either generating the class BPP//log*, if we allow for an arbitrary precision in measurement or just a limited precision, respectively.We think that this result is astonishingly interesting for Classical Physics and its connection to the Theory of Computation, namely for the implications on the nature of space and the perception of space in Classical Physics. (Some proofs are provided, to give the flavor of the subject. Missing proofs can be found in a detailed long report at the address http://fgc.math.ist.utl.pt/papers/sm.pdf.)

Abstract: This chapter provides a self-contained introduction to a collection of topics in computer science that focusses on the abstract, logical, and mathematical aspects of computing. First, mathematical structures called graphs are described that are used to model pairwise relations between objects from a certain collection. Second, abstract machines with a finite number of states called finite state automata are detailed. Third, mathematical models of computation are studied and their relationships to formal grammars are explained. Fourth, combinatorial logic is introduced, which describes logic circuits whose output is a pure function of the present input only. Finally, the degrees of complexity to solve a problem on a computer are outlined.

Abstract: In the following discussion we are going to deal with the problem of computability of differential equations, and we will outline some of the most important results achieved in this area, mainly due to K. Weihrauch and N. Zhong. In particular, a large part of the paper will concern the debate about the computability of the wave equation.

Abstract: In this paper we will try to understand how oracles and advice functions, which are mathematical abstractions in the theory of computability and complexity, can be seen as physical measurements in Classical Physics. First, we consider how physical measurements are a natural external source of information to an algorithmic computation, using a simple and engaging case study, namely: Hoyle's algorithm for calculating eclipses at Stonehenge. Next, we argue that oracles and advice functions can help us understand how the structure of space and time has information content that can be processed by Turing machines. Using an advanced case study from Newtonian kinematics, we show that non-uniform complexity is an adequate framework for classifying feasible computations by Turing machines interacting with an oracle in Nature, and that by classifying the information content of such a natural oracle, using Kolmogorov complexity, we obtain a hierarchical structure based on measurements, advice classes and information.