Abstract. We give an overview of a quantum adiabatic algorithm for Hilbert’s tenth problem, including some discussions on its fundamental aspects and the emphasis on the probabilistic correctness of its findings. For the purpose of illustration, the numerical simulation results of some simple Diophantine equations are presented. We also discuss some prejudicial misunderstandings as well as some plausible difficulties faced by the algorithm in its physical implementations. “To believe otherwise is merely to cling to a prejudice which only gives rise to further prejudices... ” 1

In 1950, Turing suggested that intelligent behavior might require “a departure from the completely disciplined behaviour involved in computation”, but nothing that a digital computer could not do. In this paper, I want to explore Turing’s suggestion by asking what it is, beyond computation, that intelligence might require, why it might require it and what knowing the answers to the first two questions might do to help us understand artificial and natural intelligence.

Uncertainty is an inherent property of all living systems. Curiously enough, computational models inspired by biological systems do not take, in general, under consideration this essential aspect of living systems. In this paper, after introducing the notion of a multi-fuzzy set (i.e., multisets where objects are repeated to some degree), we introduce two variants of P systems with fuzzy components: P systems with fuzzy data and P systems with fuzzy multiset rewriting rules. By silently assuming that fuzzy data are not the result of some fuzzification process, P systems with fuzzy data are shown to be a step towards real hypercomputation.

This paper offers an account of what it is for a physical system to be a computing mechanism—a system that performs computations. A computing mechanism is a mechanism whose function is to generate output strings from input strings and (possibly) internal states, in accordance with a general rule that applies to all relevant strings and depends on the input strings and (possibly) internal states for its application. This account is motivated by reasons endogenous to the philosophy of computing, namely, doing justice to the practices of computer scientists and computability theorists. It is also an application of recent literature on mechanisms, because it assimilates computational explanation to mechanistic explanation. The account can be used to individuate computing mechanisms and the functions they compute and to taxonomize computing mechanisms based on their computing power. 1. Introduction. This

Digital circuits with feedback loops can solve some instances of NP-hard problems by relaxation: the circuit will either oscillate or settle down to a stable state that represents a solution to the problem instance. This approach differs from using hardware accelerators to speed up the execution of deterministic algorithms, as it exploits stabilisation properties of circuits with feedback, and it allows a variety of hardware techniques that do not have counterparts in software. A feedback circuit that solves many instances of Boolean satisfiability problems is described, with experimental results from a preliminary simulation using a hardware accelerator. Keywords: NP-hard problem, Boolean satisfiability, digital circuit with feedback, relaxation, simulated annealing

The accelerated Turing machine (ATM) is the work-horse of hypercomputation. In certain cases, a machine having run through a countably infinite number of steps is supposed to have decided some interesting question such as the Twin Prime conjecture. One is, however, careful to avoid unnecessary discussion of either the possible actual use by such a machine of an infinite amount of space, or the difficulty (even if only a finite amount of space is used) of defining an outcome for machines acting like Thomson’s lamp. It is the authors ’ impression that insufficient attention has been paid to introducing a clearly defined counterpart for ATMs of the halting/non-halting dichotomy for classical Turing computation. This paper tackles the problem of defining the output, or final message, of a machine which has run for a countably infinite number of steps. Non-standard integers appear quite useful in this regard and we describe several models of computation using filters.

The accelerated Turing machine (ATM) is the work-horse of hypercomputation. In certain cases, a machine having run through a countably infinite number of steps is supposed to have decided some interesting question such as the Twin Prime conjecture. One is, however, careful to avoid unnecessary discussion of either the possible actual use by such a machine of an infinite amount of space, or the difficulty (even if only a finite amount of space is used) of defining an outcome for machines acting like Thomson’s lamp. It is the authors’ impression that insufficient attention has been paid to introducing a clearly defined counterpart for ATMs of the halting/non-halting dichotomy for classical Turing computation. This paper tackles the problem of defining the output, or final message, of a machine which has run for a countably infinite number of steps. Non-standard integers appear quite useful in this regard and we describe several models of computation using filters.

This paper investigates the view that digital hypercomputing is a good reason for rejection or re-interpretation of the Church-Turing thesis. After suggestion that such re-interpretation is historically problematic and often involves attack on a straw man (the ‘maximality thesis’), it discusses proposals for digital hypercomputing with “Zeno-machines”, i.e. computing machines that compute an infinite number of computing steps in finite time, thus performing supertasks. It argues that effective computing with Zeno-machines falls into a dilemma: either they are specified such that they do not have output states, or they are specified such that they do have output states, but involve contradiction. Repairs though noneffective methods or special rules for semi-decidable problems are sought, but not found. The paper concludes that hypercomputing supertasks are impossible in the actual world and thus no reason for rejection of the Church-Turing thesis in its traditional interpretation. 1