The random access hierarchy

The speed of copying on one-tape off-line Turing machines

We prove that functions over the reals computable in polynomial time can be characterised using discrete ordinary differential equations (ODE), also known as finite differences. We also provide a characterisation of functions computable in polynomial space over the reals.While existing characterisations could only cover time complexity or were restricted to functions over the integers, here we deal with real numbers and also cover space complexity. Furthermore, we prove that no artificial sign or test function is needed even for time complexity.At a technical level, this is obtained by proving that Turing machines can be simulated with analytic discrete ordinary differential equations. We believe this result opens the way to many applications, as it opens the possibility of programming with ODEs, with an underlying well-understood time and space complexity.

The probably almost exact model (PAExact) can be viewed as the exact model relaxed so that: 1. The counterexamples to equivalence queries are distributionally drawn rather than adversarially chosen. 2. The output hypothesis is equal to the target with negligible error (1//spl omega/(poly) for any poly). This model allows studying (almost) exact learnability of infinite classes and is in some sense analogous to the Exact-learning model for finite classes. It is known that PAExact-learnable/spl rArr/PAC-learnable [BJT02]. In this paper we show that if a class is PAC-learnable (in polynomial time) then it is PAExact-learnable (in polynomial time). Therefore, PAExact-learnable=PAC-learnable. It follows from this result that if a class is PAC-learnable then it is learnable in the probabilistic prediction model from examples with an algorithm that runs in polynomial time for each prediction (polynomial in log(the number of trials)) and that after polynomial number of mistakes achieves a hypothesis that predicts the target with probability 1-1/2/sup poly/. We also show that if a class is PAC-learnable in parallel then it is PAExact-learnable in parallel.

Currently, IEEE 802.16a wireless MAN supports the contention based OFDMA-CDMA ranging subsystem for ranging operation (Initial Ranging, Periodic Ranging, Bandwidth Request)[1]. This system uses essentially the slotted ALOHA protocol. However, the number of code-slots/frame for random access transmission tends to be much greater than one and a simple Markov chain analysis may not be numerically feasible. At the same time, the frame size in number of code-slots may be dynamically adjusted based on the traffic load. In order to evaluate delay-throughput performance and stability measure of the random access protocol, we first examine the possible traffic load to be carried throughput the ranging subchannel. We the present performance analysis and numerical examples.

Wang’s B machines are efficiently universal, as is Hasenjaeger’s small universal electromechanical toy

The Money Changing Problem (also known as Equality Constrained Integer Knapsack Problem) is as follows: Let a 1 < a 2 < ... < a k be fixed positive integers with \(\gcd(a_1, \dots, a_k) = 1\). Given some integer n, are there non-negative integers x 1, ..., x k such that ∑ i a i x i = n? The Frobenius numberg(a 1, ..., a k ) is the largest integer n that has no decomposition of the above form.There exist algorithms that, for fixed k, compute the Frobenius number in time polynomial in log a k . For variable k, one can compute a residue table of a 1 words which, in turn, allows to determine the Frobenius number. The best known algorithm for computing the residue table has runtime O(ka 1 log a 1) using binary heaps, and O(a 1 (k+log a 1)) using Fibonacci heaps. In both cases, O(a 1) extra memory in addition to the residue table is needed. Here, we present an intriguingly simple algorithm to compute the residue table in time O(ka 1) and extra memory O(1). In addition to computing the Frobenius number, we can use the residue table to solve the given instance of the Money Changing Problem in constant time, for any n.KeywordsPriority QueueGreat Common DivisorFrobenius NumberExtra MemoryLinear Diophantine EquationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Efficient parallel factorization and solution of structured and unstructured linear systems

In formal language theory, many families of languages are defined using either grammars or finite acceptors. For instance, context-sensitive languages are the languages generated by growing grammars, or equivalently those accepted by Turing machines whose work tape's size is proportional to that of their input. A few years ago, a new characterisation of context-sensitive languages as the sets of traces, or path labels, of rational graphs (infinite graphs defined by sets of finite-state transducers) was established. We investigate a similar characterisation in the more general framework of graphs defined by term transducers. In particular, we show that the languages of term-automatic graphs between regular sets of vertices coincide with the languages accepted by alternating linearly bounded Turing machines. As a technical tool, we also introduce an arborescent variant of tiling systems, which provides yet another characterisation of these languages.

We investigate the relative computing power of Turing machines with differences in the number of work tapes, heads pro work tape, instruction repertoire etc. We concentrate on the k-tape, k-head and k-head jump models as well as the 2-way multihead finite automata with and without jumps. Differences in computing power between machines of unlike specifications emerge under the real-time restriction. In particular it is shown that k+1 heads are more powerful than k heads for real-time Turing machines.

International organizations such as the 3rd Generation Partnership Project (3GPP) and the International Telecommunication Union (ITU) regard non-terrestrial networks (NTNs) as an essential component of the sixth-generation (6G) mobile communication technology and have advanced relevant standardization efforts. Low Earth orbit (LEO) satellite communication (SatCom) constitutes a key part of NTNs, and efficient uplink random access (RA) is crucial for establishing initial connections in LEO SatCom systems. However, the long propagation delay and wide coverage of LEO satellites substantially increase access latency and collision probability due to the limited number of beams and their constrained coverage areas. In addition, the highly non-uniform spatial distribution of user equipment (UE) further aggravates access inefficiency. To this end, this paper investigates joint beam position grouping and RA channel (RACH) occasions (ROs) allocation (JBPGRA) for LEO SatCom systems. Specifically, we develop a system model for RA under beam hopping and identify the key factors that influence RA performance. Furthermore, we derive expressions for both the instantaneous signal-to-interference-plus-noise ratio (SINR) and the average SINR under a given non-uniform UE spatial distribution. Building on this analysis, the JBPGRA problem is formulated as an integer linear programming problem that seeks to maximize RA success while conserving RO resources under non-uniform UE distribution. To achieve a practical solution, we propose an efficient JBPGRA algorithm composed of beam position classification, sparse beam position grouping, and RO allocation modules. Simulation results demonstrate that, under the same UE density, the proposed JBPGRA scheme achieves over 29% higher access success rate in dense beam positions compared with the uniform baseline adopted in existing SatCom systems, while reducing RO consumption by more than 49% and decreasing the number of beam position groups by over 57%.

A model of quantum computation based on unitary matrix operations was introduced by Feynman and Deutsch. It has been asked whether the power of this model exceeds that of classical Turing machines. We show here that a significant class of these quantum computations can be simulated classically in polynomial time. In particular we show that two-bit operations characterized by 4 \times 4 matrices in which the sixteen entries obey a set of five polynomial relations can be composed according to certain rules to yield a class of circuits that can be simulated classically in polynomial time. This contrasts with the known universality of two-bit operations, and demonstrates that efficient quantum computation of restricted classes is reconcilable with the Polynomial Time Turing Hypothesis. In other words it is possible that quantum phenomena can be used in a scalable fashion to make computers but that they do not have superpolynomial speedups compared to Turing machines for any problem. The techniques introduced bring the quantum computational model within the realm of algebraic complexity theory. In a manner consistent will one view of quantum physics, the wave function is simulated deterministically, and randomization arises only in the course of making measurements. The results generalize the quantum model in that they do not require the matrices to be unitary. In a different direction these techniques also yield deterministic polynomial time algorithms for the decision and parity problems for certain classes of read-twice Boolean formulae. All our results are based on the use of gates that are defined in terms of their graph matching properties.

We consider a classical multiple access system with a single transmission channel, finite number of users (users), and randomized transmission protocol (ALOHA). We assume that every user sends messages to the base station with various intensities. Due to the overlapping of messages during their sending there are restrictions on the time between the messages and a mathematical model adequate to the physical one becomes quite complicated. In this work, we propose a simplified mathematical model that is easy to analyze and takes into account the properties of real systems.

With the rapid growth in the size and complexity of digital circuits, the possibility of bug occurrence has significantly increased. In order to avoid the enormous financial loss due to the production of buggy circuits, using scalable formal verification methods is essential. The scalability of a verification method for a specific design is proven by showing that the method has polynomial space and time complexities. Unfortunately, not all verification methods have a polynomial complexity, particularly when it comes to the verification of large and complex designs. In this paper, we propose a divide-and-conquer strategy for Polynomial Formal Verification (PFV) of complex circuits. Instead of using a monolithic proof engine to verify the entire design, we break the verification task down into several problems, which can be solved in polynomial space and time using a hybrid proof engine. As a case study, we investigate the PFV of the ALU in a RISC-V processor using our divide-and-conquer strategy.

Rational verification involves checking which temporal logic properties hold of a concurrent and multiagent system, under the assumption that agents in the system choose strategies in game theoretic equilibrium. Rational verification can be understood as a counterpart of model checking for multiagent systems, but while model checking can be done in polynomial time for some temporal logic specification languages such as CTL, and polynomial space with LTL specifications, rational verification is much more intractable: it is 2EXPTIME-complete with LTL specifications, even when using explicit-state system representations. In this paper we show that the complexity of rational verification can be greatly reduced by restricting specifications to GR(1), a fragment of LTL that can represent most response properties of reactive systems. We also provide improved complexity results for rational verification when considering players' goals given by mean-payoff utility functions -- arguably the most widely used quantitative objective for agents in concurrent and multiagent systems. In particular, we show that for a number of relevant settings, rational verification can be done in polynomial space or even in polynomial time.

On uniform circuit complexity