Trading inference effort versus size in CNF Knowledge Compilation

Knowledge Compilation (KC) studies compilation of boolean functions f into some formalism F, which allows to answer all queries of a certain kind in polynomial time. Due to its relevance for SAT solving, we concentrate on the query type "clausal entailment" (CE), i.e., whether a clause C follows from f or not, and we consider subclasses of CNF, i.e., clause-sets F with special properties. In this report we do not allow auxiliary variables (except of the Outlook), and thus F needs to be equivalent to f. We consider the hierarchies UC_k <= WC_k, which were introduced by the authors in 2012. Each level allows CE queries. The first two levels are well-known classes for KC. Namely UC_0 = WC_0 is the same as PI as studied in KC, that is, f is represented by the set of all prime implicates, while UC_1 = WC_1 is the same as UC, the class of unit-refutation complete clause-sets introduced by del Val 1994. We show that for each k there are (sequences of) boolean functions with polysize representations in UC_{k+1}, but with an exponential lower bound on representations in WC_k. Such a separation was previously only know for k=0. We also consider PC < UC, the class of propagation-complete clause-sets. We show that there are (sequences of) boolean functions with polysize representations in UC, while there is an exponential lower bound for representations in PC. These separations are steps towards a general conjecture determining the representation power of the hierarchies PC_k < UC_k <= WC_k. The strong form of this conjecture also allows auxiliary variables, as discussed in depth in the Outlook.Comment: 43 pages, second version with literature updates. Proceeds with the separation results from the discontinued arXiv:1302.442

Genetic diversity and phylogenetic analysis of the native mountain ponies of Britain and Ireland reveal a novel rare population

The conservation of unique populations of animals is critical in order to preserve valuable genetic diversity and, where populations are free-living, maintain their irreplaceable influence upon habitat ecology. An accurate assessment of genetic diversity and structure within and between populations is crucial in order to design and implement conservation strategies in natural and domesticated species. Moreover, where it is possible to identify relic populations that are related to a structured breed an ideal opportunity presents itself to model processes that reveal historical factors that have shaped genetic diversity. The origins of native UK mountain and moorland ponies are uncertain, but they may have directly descended from prehistoric populations and potentially harbour specific adaptations to the uplands of Britain and Ireland. To date, there have been no studies of population structure and genetic diversity present within a free-living group of ponies in the Carneddau mountain range of North Wales. Herein, we describe the use of microsatellites and SNPs together with analysis of the mitochondrial control region to quantify the extent and magnitude of genetic diversity present in the feral Carneddau pony and relate this to several recognised British and Irish pony breeds. Our results establish that the feral Carneddau ponies represent a unique and distinctive population that merits recognition as a defined population and conservation priority. We discuss the implications for conservation of this population as a unique pool of genetic diversity adapted to the British uplands and potentially of particular value in maintaining the biodiversity of these habitats

Generalising unit-refutation completeness and SLUR via nested input resolution

We introduce two hierarchies of clause-sets, SLUR_k and UC_k, based on the classes SLUR (Single Lookahead Unit Refutation), introduced in 1995, and UC (Unit refutation Complete), introduced in 1994. The class SLUR, introduced in [Annexstein et al, 1995], is the class of clause-sets for which unit-clause-propagation (denoted by r_1) detects unsatisfiability, or where otherwise iterative assignment, avoiding obviously false assignments by look-ahead, always yields a satisfying assignment. It is natural to consider how to form a hierarchy based on SLUR. Such investigations were started in [Cepek et al, 2012] and [Balyo et al, 2012]. We present what we consider the "limit hierarchy" SLUR_k, based on generalising r_1 by r_k, that is, using generalised unit-clause-propagation introduced in [Kullmann, 1999, 2004]. The class UC, studied in [Del Val, 1994], is the class of Unit refutation Complete clause-sets, that is, those clause-sets for which unsatisfiability is decidable by r_1 under any falsifying assignment. For unsatisfiable clause-sets F, the minimum k such that r_k determines unsatisfiability of F is exactly the "hardness" of F, as introduced in [Ku 99, 04]. For satisfiable F we use now an extension mentioned in [Ansotegui et al, 2008]: The hardness is the minimum k such that after application of any falsifying partial assignments, r_k determines unsatisfiability. The class UC_k is given by the clause-sets which have hardness <= k. We observe that UC_1 is exactly UC. UC_k has a proof-theoretic character, due to the relations between hardness and tree-resolution, while SLUR_k has an algorithmic character. The correspondence between r_k and k-times nested input resolution (or tree resolution using clause-space k+1) means that r_k has a dual nature: both algorithmic and proof theoretic. This corresponds to a basic result of this paper, namely SLUR_k = UC_k.Comment: 41 pages; second version improved formulations and added examples, and more details regarding future directions, third version further examples, improved and extended explanations, and more on SLUR, fourth version various additional remarks and editorial improvements, fifth version more explanations and references, typos corrected, improved wordin

The Effect of Diet Quality and Wing Morph on Male and Female Reproductive Investment in a Nuptial Feeding Ground Cricket

A common approach in the study of life-history trade-off evolution is to manipulate the nutrient content of diets during the life of an individual in order observe how the acquisition of resources influences the relationship between reproduction, lifespan and other life-history parameters such as dispersal. Here, we manipulate the quality of diet that replicate laboratory populations received as a thorough test of how diet quality influences the life-history trade-offs associated with reproductive investment in a nuptial feeding Australian ground cricket (Pteronemobius sp.). In this species, both males and females make significant contributions to the production of offspring, as males provide a nuptial gift by allowing females to chew on a modified tibial spur during copulation and feed directing on their haemolymph. Individuals also have two distinct wing morphs, a short-winged flightless morph and a long-winged morph that has the ability to disperse. By manipulating the quality of diet over seven generations, we found that the reproductive investment of males and females were affected differently by the diet quality treatment and wing morph of the individual. We discuss the broader implications of these findings including the differences in how males and females balance current and future reproductive effort in nuptial feeding insects, the changing nature of sexual selection when diets vary, and how the life-history trade-offs associated with the ability to disperse are expected to differ among populations