Abstract: We show that. the problem of predicting t steps of the ID cellular automaton Rule 110 is P-complete. The result is found by showing that Rule 110 simulates deterministic Turing machines in polynomial time. As a corollary we find that the small universal Turing machines of Mathew Cook run in polynomial time, this is an exponential improvement on their previously known simulation time overhead.

Abstract: Recently, there has been much interest in extending models for simulation-based security in such a way that the runtime of protocols may depend on the length of their input. Finding such extensions has turned out to be a non-trivial task. In this work, we propose a simple, yet expressive general computational model for systems of interactive Turing machines (ITMs) where the runtime of the ITMs may be polynomial per activation and may depend on the length of the input received. One distinguishing feature of our model is that the systems of ITMs that we consider involve a generic mechanism for addressing dynamically generated copies of ITMs. We study properties of such systems and, in particular, show that systems satisfying a certain acyclicity condition run in polynomial time. Based on our general computational model, we state different notions of simulation-based security in a uniform and concise way, study their relationships, and prove a general composition theorem for composing a polynomial number of copies of protocols, where the polynomial is determined by the environment. The simplicity of our model is demonstrated by the fact that many of our results can be proved by mere equational reasoning based on a few equational principles on systems.

Abstract: Recently, there has been much interest in extending models for simulation-based security in such a way that the runtime of protocols may depend on the length of their input. Finding such extensions has turned out to be a non-trivial task. In this work, we propose a simple, yet expressive general computational model for systems of interactive Turing machines (ITMs) where the runtime of the ITMs may be polynomial per activation and may depend on the length of the input received. One distinguishing feature of our model is that the systems of ITMs that we consider involve a generic mechanism for addressing dynamically generated copies of ITMs. We study properties of such systems and, in particular, show that systems satisfying a certain acyclicity condition run in polynomial time. Based on our general computational model, we state different notions of simulation-based security in a uniform and concise way, study their relationships, and prove a general composition theorem for composing a polynomial number of copies of protocols, where the polynomial is determined by the environment. The simplicity of our model is demonstrated by the fact that many of our results can be proved by mere equational reasoning based on a few equational principles on systems.

Abstract: We consider parallel algorithms working in sequential global time, for example, circuits or parallel random access machines (PRAMs). Parallel abstract state machines (parallel ASMs) are such parallel algorithms, and the parallel ASM thesis asserts that every parallel algorithm is behaviorally equivalent to a parallel ASM. In an earlier article, we axiomatized parallel algorithms, proved the ASM thesis, and proved that every parallel ASM satisfies the axioms. It turned out that we were too timid in formulating the axioms; they did not allow a parallel algorithm to create components on the fly. This restriction did not hinder us from proving that the usual parallel models, like circuits or PRAMs or even alternating Turing machines, satisfy the postulates. But it resulted in an error in our attempt to prove that parallel ASMs always satisfy the postulates. To correct the error, we liberalize our axioms and allow on-the-fly creation of new parallel components. We believe that the improved axioms accurately express what parallel algorithms ought to be. We prove the parallel thesis for the new, corrected notion of parallel algorithms, and we check that parallel ASMs satisfy the new axioms.

Abstract: We study channel systems whose behaviour (sending and receiving messages via unbounded FIFO channels) must follow given timing constraints specifying the execution speeds of the local components. We propose Communicating Timed Automata (CTA) to model such systems. The goal is to study the borderline between decidable and undecidable classes of channel systems in the timed setting. Our technical results include: (1) CTA with one channel without shared states in the form (A 1 ,A 2 ,c 1,2 ) is equivalent to one-counter machine, implying that verification problems such as checking state reachability and channel boundedness are decidable, and (2) CTA with two channels without sharing states in the form (A 1 , A 2 ,A 3 ,c 1,2 ,c 2,3 ) has the power of Turing machines. Note that in the untimed setting, these systems are no more expressive than finite state machines. This shows that the capability of synchronizing on time makes it substantially more difficult to verify channel systems.

Abstract: We introduce the zeta number, natural halting probability and natural complexity of a Turing machine and we relate them to Chaitin's Omega number, halting probability, and program-size complexity. A classification of Turing machines according to their zeta numbers is proposed: divergent, convergent and tuatara. We prove the existence of universal convergent and tuatara machines. Various results on (algorithmic) randomness and partial randomness are proved. For example, we show that the zeta number of a universal tuatara machine is c.e. and random. A new type of partial randomness, asymptotic randomness, is introduced. Finally we show that in contrast to classical (algorithmic) randomness--which cannot be naturally characterised in terms of plain complexity--asymptotic randomness admits such a characterisation.

Abstract: It is shown that a set is low for weakly 1-generic iff it has neither dnr nor hyperimmune Turing degree. As this notion is more general than being recursively traceable, this answers negatively a recent question on the characterization of these sets. Furthermore, it is shown that every set which is low for weakly 1-generic is also low for Kurtz-random.

Abstract: Implications of a formal context (G,M,I) have a minimal implication basis, called Duquenne-Guigues basis or stem base. It is shown that the problem of deciding whether a set of attributes is a premise of the stem base is in coNP and determining the size of the stem base is polynomially Turing equivalent to a #P-complete problem.

Abstract: Current P systems which solve NP–complete numerical problems represent the instances of the problems in unary notation. However, in classical complexity theory, based upon Turing machines, switching from binary to unary encoded instances generally corresponds to simplify the problem. In this paper we show that, when working with P systems, we can assume without loss of generality that instances are expressed in binary notation. More precisely, we propose a simple method to encode binary numbers using multisets, and a family of P systems which transforms such multisets into the usual unary notation. Such a family could thus be composed with the unary P systems currently proposed in the literature to obtain (uniform) families of P systems which solve NP–complete numerical problems with instances encoded in binary notation. We introduce also a framework which can be used to design uniform families of P systems which solve NP–complete problems (both numerical and non-numerical) working directly on binary encoded instances, i.e., without first transforming them to unary notation. We illustrate our framework by designing a family of P systems which solves the 3-SAT problem. Next, we discuss the modifications needed to obtain a family of P systems which solves the PARTITION numerical problem.

Abstract: In this paper, we present two new results regarding ANSPs. The first one states that every recursively enumerable language can be accepted by an ANSP of size 7 out of which 6 do not depend on the given language. Then we propose a method for constructing, given an NP-language, an ANSP of size 7 accepting that language in polynomial time. Unlike the previous case, all nodes of this ANSP depend on the given language. Since each ANSP may be viewed as a problem solver as shown in [6], the later result may be interpreted as a method for solving every NP-problem in polynomial time by an ANSP of size 7.

Abstract: We compare various computational complexity classes defined within the framework of membrane systems, a distributed parallel computing device which is inspired from the functioning of the cell, with usual computational complexity classes for Turing machines. In particular, we focus our attention on the comparison among complexity classes for membrane systems with active membranes (where new membranes can be created by division of existing membranes) and the classes PSPACE, EXP, and EXPSPACE.

Abstract: We improve upon indirect diagonalization arguments for lower bounds on explicit problems within the polynomial hierarchy. Our contributions are summarized as follows. 1. We present a technique that uniformly improves upon most known nonlinear time lower bounds for nondeterminism and alternating computation, on both subpolynomial (n o(1)) space RAMs and sequential one-tape machines with random access to the input. We obtain improved lower bounds for Boolean satisfiability (SAT), as well as all NP-complete problems that have efficient reductions from SAT, and k-SAT, for constant k 2. For example, SAT cannot be solved by random access machines using time and subpolynomial space. 2. We show how indirect diagonalization leads to time-space lower bounds for computation with bounded nondeterminism. For both the random access and multitape Turing machine models, we prove that for all k 1, there is a constant c k > 1 such that linear time with n 1/k nondeterministic bits is not contained in deterministic $$n^{{c}_{k}}$$ time with subpolynomial space. This is used to prove that satisfiability of Boolean circuits with n inputs and n k size cannot be solved by deterministic multitape Turing machines running in $${n^{{k \cdot {c}}_{k}}}$$ time and subpolynomial space.

Abstract: In this paper we consider the computation of reachable, viable and invariant sets for discrete-time systems. We use the framework of type-two effectivity, in which computations are performed by Turing machines with infinite input and output tapes, with the representations of computable topology. We see that the reachable set is lower-semicomputable, and the viability and invariance kernels are upper-semicomputable. We then define an upper-semicomputable over-approximation to the reachable set, and lower-semicomputable under-approximations to the viability and invariance kernels, and show that these approximations are optimal

Abstract: This essay is a slightly edited transcription of a lecture given in 1997 in King’s College, Cambridge, where Alan Turing had been a Fellow. The lecture was part of a meeting to celebrate the 60th anniversary of the publication of Turing’s paper On computable numbers, with an application to the Entscheidungsproblem, published in the Proceedings of the London Mathematical Society in 1937.

Abstract: Models of computation in theoretical computer science very frequently consist of a device performing some type of process, like a Turing machine and its computation or a grammar and its derivation. After the process halts, only some final output is regarded as the result. In adding an observer to such a device, one can obtain a protocol of the entire process and then use this as the result of the computation. In several recent articles this approach has proved to often exceed greatly the power of the observed system. Here we apply this architecture to string-rewriting systems. They receive a string as input, and a combination of observer and decider then determines whether this string is accepted. This decision is based only on the rewriting process performed. First we determine the power of painter, context-free, and inverse context-free rewriting systems in terms of McNaughton languages. Then these are investigated as components of rewriting/observer systems, and we obtain characterizations of the classes of context-sensitive and recursively enumerable languages. Finally we investigate some limitations, i.e. characterize some systems, where observation does not increase power.

Abstract: Can general relativistic computers break the Turing barrier? – Are there final limits to human knowledge? – Limitative results versus human creativity (paradigm shifts). – Godel's logical results in comparison/combination with Godel's relativistic results. – Can Hilbert's programme be carried through after all?

Abstract: Ohya and Volovich have proposed a new quantum computation model with chaotic amplification to solve the SAT problem, which went beyond usual quantum algorithm. In this paper, we generalize quantum Turing machine, and we show in this general quantum Turing machine (GQTM) that we can treat the Ohya-Volovich (OV) SAT algorithm.

Abstract: We show that there exists a universal quantum Turing machine (UQTM) that can simulate every other QTM until the other QTM has halted and then halt itself with probability one. This extends work by Bernstein and Vazirani who have shown that there is a UQTM that can simulate every other QTM for an arbitrary, but preassigned number of time steps. As a corollary to this result, we give a rigorous proof that quantum Kolmogorov complexity as defined by Berthiaume et al. is invariant, i.e. depends on the choice of the UQTM only up to an additive constant. Our proof is based on a new mathematical framework for QTMs, including a thorough analysis of their halting behaviour. We introduce the notion of mutually orthogonal halting spaces and show that the information encoded in an input qubit string can always be effectively decomposed into a classical and a quantum part.

Abstract: Ping-pong protocols with recursive definitions of agents, but without any active intruder, are a Turing powerful model. We show that under the environment sensitive semantics (i.e. by adding an active intruder capable of storing all exchanged messages including full analysis and synthesis of messages) some verification problems become decidable. In particular we give an algorithm to decide control state reachability, a problem related to security properties like secrecy and authenticity. The proof is via a reduction to a new prefix rewriting model called Monotonic Set-extended Prefix rewriting (MSP). We demonstrate further applicability of the introduced model by encoding a fragment of the ccp (concurrent constraint programming) language into MSP.

Abstract: In this chapter we present a general framework to provide efficient solutions to decision problems through families of cell-like membrane systems constructed in a semi-uniform way (associating with each instance of the problem one P system solving it) or a uniform way (all instances of a decision problem having the same size are processed by the same system). We also show a brief compendium of efficient semi-uniform and uniform solutions to hard problems in these systems, and we explicitly describe some of these solutions.

Abstract: Infinite Time Turing Machines (or Hamkins-Kidder machines) have been introduced in [HaLe00] and their computability theory has been investigated in comparison to the usual computability theory in a sequence of papers by Hamkins, Lewis, Welch and Seabold: [HaLe00], [We00a], [We00b], [HaSe01], [Hale02], [We04], [We05] (cf. also the survey papers [Ha02], [Ha04] and [Ha05]). Infinite Time Turing Machines have the same hardware as ordinary Turing Machines, and almost the same software. However, an Infinite Time Turing Machine can continue its computation if it still hasn't reached the Halt state after infinitely many steps (for details, see §, 1).

Abstract: A. M. Turing has bequeathed us a conceptulary including ‘Turing, or Turing-Church, thesis’, ‘Turing machine’, ‘universal Turing machine’, ‘Turing test’ and ‘Turing structures’, plus other unnamed achievements. These include a proof that any formal language adequate to express arithmetic contains undecidable formulas, as well as achievements in computer science, artificial intelligence, mathematics, biology, and cognitive science. Here it is argued that these achievements hang together and have prospered well in the 50 years since Turing's death.

Abstract: Recent experimental progress in DNA lattice construction, DNA robotics, and DNA computing provides the basis for designing DNA cellular computing devices, i.e. autonomous nano-mechanical DNA computing devices embedded in DNA lattices. Once assembled, DNA cellular computing devices can serve as reusable, compact computing devices that perform (universal) computation, and programmable robotics devices that demonstrate complex motion. As a prototype of such devices, we recently reported the design of an Autonomous DNA Turing Machine, which is capable of universal sequential computation, and universal translational motion, i.e. the motion of the head of a single tape universal mechanical Turing machine. In this paper, we describe the design of an Autonomous DNA Cellular Automaton (ADCA), which can perform parallel universal computation by mimicking a one-dimensional (ID) universal cellular automaton. In the computation process, this device, embedded in a ID DNA lattice, also demonstrates well coordinated parallel motion. The key technical innovation here is a molecular mechanism that synchronizes pipelined molecular reaction waves along a ID track, and in doing so, realizes parallel computation. We first describe the design of ADCA on an abstract level, and then present detailed DNA sequence level implementation using commercially available protein enzymes. We also discuss how to extend the ID design to 2D.

Abstract: Kucera and Gacs independently showed that every infinite sequence is Turing reducible to a Martin-Lof random sequence. We extend this result to show that every infinite sequence S is Turing reducible to a Martin-Lof random sequence R such that the asymptotic number of bits of R needed to compute n bits of S, divided by n, is precisely the constructive dimension of S. We show that this is the optimal ratio of query bits to computed bits achievable with Turing reductions. As an application of this result, we give a new characterization of constructive dimension in terms of Turing reduction compression ratios.