Asymptotic Proportion of Hard Instances of the Halting Problem

Although the halting problem is undecidable, imperfect testers that fail on some instances are possible. Such instances are called hard for the tester. One variant of imperfect testers replies "I don't know" on hard instances, another variant fails to halt, and yet another replies incorrectly "yes" or "no". Also the halting problem has three variants: does a given program halt on the empty input, does a given program halt when given itself as its input, or does a given program halt on a given input. The failure rate of a tester for some size is the proportion of hard instances among all instances of that size. This publication investigates the behaviour of the failure rate as the size grows without limit. Earlier results are surveyed and new results are proven. Some of them use C++ on Linux as the computational model. It turns out that the behaviour is sensitive to the details of the programming language or computational model, but in many cases it is possible to prove that the proportion of hard instances does not vanish.Comment: 18 pages. The differences between this version and arXiv:1307.7066v1 are significant. They have been listed in the last paragraph of Section 1. Excluding layout, this arXiv version is essentially identical to the Acta Cybernetica versio

Randomised algorithms for counting and generating combinatorial structures

SIGLEAvailable from British Library Document Supply Centre- DSC:D85048 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Simplicity, scientific inference and econometric modelling

Economic Schools;Econometric Models;Economic Methodology

On the performance and programming of reversible molecular computers

If the 20th century was known for the computational revolution, what will the 21st be known for? Perhaps the recent strides in the nascent fields of molecular programming and biological computation will help bring about the ‘Coming Era of Nanotechnology’ promised in Drexler’s ‘Engines of Creation’. Though there is still far to go, there is much reason for optimism. This thesis examines the underlying principles needed to realise the computational aspects of such ‘engines’ in a performant way. Its main body focusses on the ways in which thermodynamics constrains the operation and design of such systems, and it ends with the proposal of a model of computation appropriate for exploiting these constraints. These thermodynamic constraints are approached from three different directions. The first considers the maximum possible aggregate performance of a system of computers of given volume, V, with a given supply of free energy. From this perspective, reversible computing is imperative in order to circumvent the Landauer limit. A result of Frank is refined and strengthened, showing that the adiabatic regime reversible computer performance is the best possible for any computer—quantum or classical. This therefore shows a universal scaling law governing the performance of compact computers of ~V^(5/6), compared to ~V^(2/3) for conventional computers. For the case of molecular computers, it is shown how to attain this bound. The second direction extends this performance analysis to the case where individual computational particles or sub-units can interact with one another. The third extends it to interactions with shared, non-computational parts of the system. It is found that accommodating these interactions in molecular computers imposes a performance penalty that undermines the earlier scaling result. Nonetheless, scaling superior to that of irreversible computers can be preserved, and appropriate mitigations and considerations are discussed. These analyses are framed in a context of molecular computation, but where possible more general computational systems are considered. The proposed model, the א-calculus, is appropriate for programming reversible molecular computers taking into account these constraints. A variety of examples and mathematical analyses accompany it. Moreover, abstract sketches of potential molecular implementations are provided. Developing these into viable schemes suitable for experimental validation will be a focus of future work

Why solutions can be hard to find: a featural theory of cost for a local search algorithm on random satisfiability instances

The local search algorithm WSat is one of the most successful algorithms for solving the archetypal NP-complete problem of satisfiability (SAT). It is notably effective at solving Random-3-SAT instances near the so-called 'satisfiability threshold', which are thought to be universally hard. However, WSat still shows a peak in search cost near the threshold and large variations in cost over different instances. Why are solutions to the threshold instances so hard to find using WSat? What features characterise threshold instances which make them difficult for WSat to solve? We make a number of significant contributions to the analysis of WSat on these high-cost random instances, using the recently-introduced concept of the backbone of a SAT instance. The backbone is the set of literals which are implicates of an instance. We find that the number of solutions predicts the cost well for small-backbone instances but is much less relevant for the large-backbone instances which appear near the threshold and dominate in the overconstrained region. We undertake a detailed study of the behaviour of the algorithm during search and uncover some interesting patterns. These patterns lead us to introduce a measure of the backbone fragility of an instance, which indicates how persistent the backbone is as clauses are removed. We propose that high-cost random instances for WSat are those with large backbones which are also backbone-fragile. We suggest that the decay in cost for WSat beyond the satisfiability threshold, which has perplexed a number of researchers, is due to the decreasing backbone fragility. Our hypothesis makes three correct predictions. First, that a measure of the backbone robustness of an instance (the opposite to backbone fragility) is negatively correlated with the WSat cost when other factors are controlled for. Second, that backbone-minimal instances (which are 3-SAT instances altered so as to be more backbone-fragile) are unusually hard for WSat. Third, that the clauses most often unsatisfied during search are those whose deletion has the most effect on the backbone. Our analysis of WSat on random-3-SAT threshold instances can be seen as a featural theory of WSat cost, predicting features of cost behaviour from structural features of SAT instances. In this thesis, we also present some initial studies which investigate whether the scope of this featural theory can be broadened to other kinds of random SAT instance. random-2+p-SAT interpolates between the polynomial-time problem Random-2-SAT when p = 0 and Random-3-SAT when p = 1. At some value p ~ pq ~ 0.41, a dramatic change in the structural nature of instances is predicted by statistical mechanics methods, which may imply the appearance of backbone fragile instances. We tested NovELTY+, a recent variant of WSat, on rand o m- 2 +p-SAT and find some evidence that growth of its median cost changes from polynomial to superpolynomial between p = 0.3 and p = 0.5. We also find evidence that it is the onset of backbone fragility which is the cause of this change in cost scaling: typical instances at p — 0.5 are more backbone-fragile than their counterparts at p — 0.3. Not-All-Equal (NAE) 3-SAT is a variant of the SAT problem which is similar to it in most respects. However, for NAE 3-SAT instances no implicate literals are possible. Hence the backbone for NAE 3-SAT must be redefined. We show that under a redefinition of the backbone, the pattern of factors influencing WSat cost at the NAE Random-3-SAT threshold is much the same as in Random-3-SAT, including the role of backbone fragility

The biology and behaviour of a free-living population of black rats (Rattus rattus)

A population of wild Rattus rattus living in the roofs of the laboratory buildings was studied by supplying food every evening and watching the behaviour of the animals at the feeding place. Some observations were also made on caged animals. The rats were predominantly of the black rattus variety but white-bellied greys appeared now and then. In breeding tests the grey colour behaved as though determined by a single recessive gene. The study covered two periods of approximately 9 months each, separated by an interval of 3 months during which a reduced quantity of food was provided and the rat population underwent a major decline. During the two periods of richer feeding the population first increased and then stabilized at a level where the animals remained in good condition and there was no starvation. In the first 9-month period, stabilization was achieved by emigration of young adults who colonized neighbouring buildings. Towards the end of the second period, stabilization was achieved by limitation of breeding. The rats accepted a wide variety of foods, including meat, and a number of instances of predation were seen. Small vertebrates as well as insects were killed and eaten. Small pieces of food were usually eaten in situ but large bits were taken up to the nests in the roof. Such differential treatment in relation to size may be a factor of some importance in the evolution of hoarding. The rats visiting the feeding place formed a unit with a definite social structure. A single dominant male and never more than one, was always present and in certain circumstances a linear male hierarchy was formed. There were usually two or three mutually tolerant top ranking females who were subordinate to the top male but dominant to all other members of the group. Within the group attacks were directed downwards in the social scale. An attacked subordinate either fled or appeased and serious fights therefore did not develop. The most essential component of the appease. ment appeared to be a mouth to mouth contact which may be derived from the infantile pattern of 'mouth suckling'. Appeasement permitted superior rats to maintain their status without the necessity of carrying attacks on subordinates to the point where actual hurt was inflicted. A group territory round the feeding place was defended against interlopers. Both sexes took part in chasing out intruders but since males showed inhibition in attacking females, the exclusion of strange females was due principally to the activities of the home females. The point at which pursuit of an intruder stopped was regarded as the territorial boundary. This was also the limit beyond which a group member would not allow himself to be chased but it was not a prison wall. When agonistic tendencies were not aroused the animals no longer always I turned back at the boundary and foraging beyond its limits allowed them to become familiar with an area larger than the territory. Although intruders were normally driven out, it was occasionally possible for a particularly determined animal of either sex to force its way in and ultimately become a member of the group. The patterns of behaviour seen are described, particularly those concerned with hostile encounters and with mating. Scent marking with urine drip trails was not seen but adults of both sexes marked by rubbing the cheeks and ventral surface on branches. The circumstances in which tooth gnashing was heard suggest that this behaviour is not a form of threat but a response to unfamiliar auditory or visual stimuli. There was some evidence that it functioned as an alarm signal within the group. Pilo-erection and a gait or posture with the hind legs much extended ('stegosauring') are considered to function as threats. Pilo-erection occurred in situations where there was little to suggest conflict and is considered to represent a form of threat which has undergone emancipation. Various forms of displacement and ambivalent behaviour were seen. Rapid vibration of the tail occurred in thwarting situations, either during mating or when a defeated opponent suddenly vanished. There was no evidence that it acted as a signal. The common form of amicable behaviour was social grooming. Another amicable action was sitting together with the bodies in contact. Animals reared in cages remained shy and wary and even hand reared young developed the usual alarm responses to movement and noises. Females had their first litters at ages of 3 to 5 months. For first litters gestation periods were 21 to 22 days but in females that were simultaneously lactating they ranged from 23 to 29 days. Eight was the commonest litter number and ten the highest recorded. At birth the tail is very much shorter than the body but has outstripped it by the time the youngster emerges from the nest. This was found to be the result of a period of extremely rapid tail growth immediately preceding emergence. In Rattus norvegicus the peak in tail growth rate was found to be later and less striking. The difference is interpreted as related to the importance of the tail in climbing in the more arboreal R. rattus. During the second week of life an edge response (retreat from a declivity) and a clinging response made their appearance: these have the function of preventing accidental falls from a nest situated above ground level. Mouth suckling was seen only during a period of a few days towards the end of lactation. Play developed within a few days of emergence from the nest: locomotor and fighting play were the common types. Older animals occasionally joined in play with the young. In problem solving tests, first solutions were not insightful but once a solution had been found, the successful technique was at once adopted and subsequently perfected. There was no evidence of learning by imitation but the rats did learn from each other's behaviour that food could be obtained at a certain location and thus the solution of a problem by one rat accelerated its independent solution by others. The reasons for the differences between the behaviour of the free living population and the caged animals studied by other authors are discussed