AND and/or OR: Uniform Polynomial-Size Circuits

We investigate the complexity of uniform OR circuits and AND circuits of polynomial-size and depth. As their name suggests, OR circuits have OR gates as their computation gates, as well as the usual input, output and constant (0/1) gates. As is the norm for Boolean circuits, our circuits have multiple sink gates, which implies that an OR circuit computes an OR function on some subset of its input variables. Determining that subset amounts to solving a number of reachability questions on a polynomial-size directed graph (which input gates are connected to the output gate?), taken from a very sparse set of graphs. However, it is not obvious whether or not this (restricted) reachability problem can be solved, by say, uniform AC^0 circuits (constant depth, polynomial-size, AND, OR, NOT gates). This is one reason why characterizing the power of these simple-looking circuits in terms of uniform classes turns out to be intriguing. Another is that the model itself seems particularly natural and worthy of study. Our goal is the systematic characterization of uniform polynomial-size OR circuits, and AND circuits, in terms of known uniform machine-based complexity classes. In particular, we consider the languages reducible to such uniform families of OR circuits, and AND circuits, under a variety of reduction types. We give upper and lower bounds on the computational power of these language classes. We find that these complexity classes are closely related to tallyNL, the set of unary languages within NL, and to sets reducible to tallyNL. Specifically, for a variety of types of reductions (many-one, conjunctive truth table, disjunctive truth table, truth table, Turing) we give characterizations of languages reducible to OR circuit classes in terms of languages reducible to tallyNL classes. Then, some of these OR classes are shown to coincide, and some are proven to be distinct. We give analogous results for AND circuits. Finally, for many of our OR circuit classes, and analogous AND circuit classes, we prove whether or not the two classes coincide, although we leave one such inclusion open.Comment: In Proceedings MCU 2013, arXiv:1309.104

After The Crash-The Ground and the Sublime in the Works of Hito Steyerl and Trevor Paglen

An analysis into how fact, fiction, and perspective are affected in contemporary digital economies, through the work of artists Hito Steyerl and Trevor Paglen

Detecting large-scale structure in the era of petabyte/gigaparsec Astronomy

In this thesis, we present a study of the identification of large-scale structure in optical astronomical surveys. This encompasses the detection of large connected structures of alaxies in spectroscopic datasets and of galaxy clusters in deep photometric surveys. Beginning with a survey featuring full 3D galaxy data, in chapter 2 we present a method to identify filamentary structure after accounting for the line-of-sight velocity distortions characteristic of the virialised systems we search for. We compare data from a real galaxy survey to a series of realistic mocks. Despite broad similarities between the two, we find models do not reproduce the argest observed structures. To evaluate the exploration of a multi-band survey lacking spectroscopy, we simulate the effects of photometric redshift uncertainties on galaxy redshifts. Our findings provide limits on the accuracy of photometric redshift estimators required to recover the same diverse range of structures detected in the original spectroscopic survey. As an alternative means of exploiting the deep multi-band photometric data common to wide-area observational campaigns, in chapter 3 we present a red sequence-based algorithm to detect galaxy clusters with Voronoi diagrams. This algorithm makes no prior assumptions about cluster properties other than the similarity in colour of their members, and an enhanced projected surface density. Testing the algorithm with mock galaxy survey data reveals a detection performance equalling or exceeding that of alternative detection algorithms. Chapter 4 describes the application of this algorithm to a $270{\rm deg^2}$ survey with deep SDSS photometry. The scientific exploitation of $4,000\ {\rm z}\ \leq\ 0.6$ cluster detections from this survey is ongoing, but work presented here shows: agreement with the red sequence slope evolution derived from semi-analytic galaxy models, evidence stellar age is not responsible for responsible for the sequence slope, and a well-defined colour-colour track of potential use in photometric cluster redshift stimation. We detail improvements made to the cluster algorithm in chapter 5. Through a series of case studies we verify our approach successfully identifies galaxy clusters in a diverse range of surveys, from volumes spanning $2h^{-3}{\rm Gpc}^{3}$ to deep near-IR detections at ${\rm z}\sim 1$. based on our findings, we expect the Pan-STARRS $3\pi$ large-area survey to identify over $10^{5}$ clusters and groups. In chapter 6, we explore the characteristics of randomly-distributed noise in Voronoi diagrams. We verify the model traditionally used to describe the distribution of Voronoi cell areas in Poisson data fails to describe the frequency of high-density random cells. Because high-density cells resemble those expected from a population of galaxy cluster members, using a large dataset generated in this study we propose an alternative model that better estimates the frequency of their areas. This new model may in the future be used to improve Voronoi-based recovery of clustered data in a diverse range of applications, both astronomical and otherwise

Uniformity is weaker than semi-uniformity for some membrane systems

We investigate computing models that are presented as families of finite computing devices with a uniformity condition on the entire family. Examples of such models include Boolean circuits, membrane systems, DNA computers, chemical reaction networks and tile assembly systems, and there are many others. However, in such models there are actually two distinct kinds of uniformity condition. The first is the most common and well-understood, where each input length is mapped to a single computing device (e.g. a Boolean circuit) that computes on the finite set of inputs of that length. The second, called semi-uniformity, is where each input is mapped to a computing device for that input (e.g. a circuit with the input encoded as constants). The former notion is well-known and used in Boolean circuit complexity, while the latter notion is frequently found in literature on nature-inspired computation from the past 20 years or so. Are these two notions distinct? For many models it has been found that these notions are in fact the same, in the sense that the choice of uniformity or semi-uniformity leads to characterisations of the same complexity classes. In other related work, we showed that these notions are actually distinct for certain classes of Boolean circuits. Here, we give analogous results for membrane systems by showing that certain classes of uniform membrane systems are strictly weaker than the analogous semi-uniform classes. This solves a known open problem in the theory of membrane systems. We then go on to present results towards characterising the power of these semi-uniform and uniform membrane models in terms of NL and languages reducible to the unary languages in NL, respectively.Comment: 28 pages, 1 figur

On acceptance conditions for membrane systems: characterisations of L and NL

In this paper we investigate the affect of various acceptance conditions on recogniser membrane systems without dissolution. We demonstrate that two particular acceptance conditions (one easier to program, the other easier to prove correctness) both characterise the same complexity class, NL. We also find that by restricting the acceptance conditions we obtain a characterisation of L. We obtain these results by investigating the connectivity properties of dependency graphs that model membrane system computations

First Steps Towards Linking Membrane Depth and the Polynomial Hierarchy

In this paper we take the first steps in studying possible connections between non-elementary division with limited membrane depth and the levels of the Polynomial Hierarchy. We present a uniform family with a membrane structure of depth d + 1 that solves a problem complete for level d of the Polynomial Hierarchy

Computational Processes and Incompleteness

We introduce a formal definition of Wolfram's notion of computational process based on cellular automata, a physics-like model of computation. There is a natural classification of these processes into decidable, intermediate and complete. It is shown that in the context of standard finite injury priority arguments one cannot establish the existence of an intermediate computational process

The Pagoda Sequence: a Ramble through Linear Complexity, Number Walls, D0L Sequences, Finite State Automata, and Aperiodic Tilings

We review the concept of the number wall as an alternative to the traditional linear complexity profile (LCP), and sketch the relationship to other topics such as linear feedback shift-register (LFSR) and context-free Lindenmayer (D0L) sequences. A remarkable ternary analogue of the Thue-Morse sequence is introduced having deficiency 2 modulo 3, and this property verified via the re-interpretation of the number wall as an aperiodic plane tiling

A Concrete View of Rule 110 Computation

Rule 110 is a cellular automaton that performs repeated simultaneous updates of an infinite row of binary values. The values are updated in the following way: 0s are changed to 1s at all positions where the value to the right is a 1, while 1s are changed to 0s at all positions where the values to the left and right are both 1. Though trivial to define, the behavior exhibited by Rule 110 is surprisingly intricate, and in (Cook, 2004) we showed that it is capable of emulating the activity of a Turing machine by encoding the Turing machine and its tape into a repeating left pattern, a central pattern, and a repeating right pattern, which Rule 110 then acts on. In this paper we provide an explicit compiler for converting a Turing machine into a Rule 110 initial state, and we present a general approach for proving that such constructions will work as intended. The simulation was originally assumed to require exponential time, but surprising results of Neary and Woods (2006) have shown that in fact, only polynomial time is required. We use the methods of Neary and Woods to exhibit a direct simulation of a Turing machine by a tag system in polynomial time

On the boundaries of solvability and unsolvability in tag systems. Theoretical and Experimental Results

Several older and more recent results on the boundaries of solvability and unsolvability in tag systems are surveyed. Emphasis will be put on the significance of computer experiments in research on very small tag systems