Uniform Diagonalization Theorem for Complexity Classes of Promise Problems including Randomized and Quantum Classes

Diagonalization in the spirit of Cantor's diagonal arguments is a widely used tool in theoretical computer sciences to obtain structural results about computational problems and complexity classes by indirect proofs. The Uniform Diagonalization Theorem allows the construction of problems outside complexity classes while still being reducible to a specific decision problem. This paper provides a generalization of the Uniform Diagonalization Theorem by extending it to promise problems and the complexity classes they form, e.g. randomized and quantum complexity classes. The theorem requires from the underlying computing model not only the decidability of its acceptance and rejection behaviour but also of its promise-contradicting indifferent behaviour - a property that we will introduce as "total decidability" of promise problems. Implications of the Uniform Diagonalization Theorem are mainly of two kinds: 1. Existence of intermediate problems (e.g. between BQP and QMA) - also known as Ladner's Theorem - and 2. Undecidability if a problem of a complexity class is contained in a subclass (e.g. membership of a QMA-problem in BQP). Like the original Uniform Diagonalization Theorem the extension applies besides BQP and QMA to a large variety of complexity class pairs, including combinations from deterministic, randomized and quantum classes.Comment: 15 page

A three-dimensional approach to visualize pairwise morphological variation and its application to fragmentary palaeontological specimens

Classifying isolated vertebrate bones to a high level of taxonomic precision can be difficult. Many of Australia's Cretaceous terrestrial vertebrate fossil-bearing deposits, for example, produce large numbers of isolated bones and very few associated or articulated skeletons. Identifying these often fragmentary remains beyond high-level taxonomic ranks, such as Ornithopoda or Theropoda, is difficult and those classified to lower taxonomic levels are often debated. The ever-increasing accessibility to 3D-based comparative techniques has allowed palaeontologists to undertake a variety of shape analyses, such as geometric morphometrics, that although powerful and often ideal, require the recognition of diagnostic landmarks and the generation of sufficiently large data sets to detect clusters and accurately describe major components of morphological variation. As a result, such approaches are often outside the scope of basic palaeontological research that aims to simply identify fragmentary specimens. Herein we present a workflow in which pairwise comparisons between fragmentary fossils and better known exemplars are digitally achieved through three-dimensional mapping of their surface profiles and the iterative closest point (ICP) algorithm. To showcase this methodology, we compared a fragmentary theropod ungual (NMV P186153) from Victoria, Australia, identified as a neovenatorid, with the manual unguals of the megaraptoran Australovenator wintonensis (AODF604). We discovered that NMV P186153 was a near identical match to AODF604 manual ungual II-3, differing only in size, which, given their 10–15Ma age difference, suggests stasis in megaraptoran ungual morphology throughout this interval. Although useful, our approach is not free of subjectivity; care must be taken to eliminate the effects of broken and incomplete surfaces and identify the human errors incurred during scaling, such as through replication. Nevertheless, this approach will help to evaluate and identify fragmentary remains, adding a quantitative perspective to an otherwise qualitative endeavour

The dinosaur tracks of Tyrants Aisle: An Upper Cretaceous ichnofauna from Unit 4 of the Wapiti Formation (upper Campanian), Alberta, Canada

The Wapiti Formation of northwest Alberta and northeast British Columbia, Canada, preserves an Upper Cretaceous terrestrial vertebrate fauna that is latitudinally situated between those documented further north in Alaska and those from southern Alberta and the contiguous U.S.A. Therefore, the Wapiti Formation is important for identifying broad patterns in vertebrate ecology, diversity, and distribution across Laramidia during the latest Cretaceous. Tracksites are especially useful as they provide a range of palaeoecological, palaeoenvironmental, and behavioural data that are complementary to the skeletal record. Here, we describe the Tyrants Aisle locality, the largest in-situ tracksite known from the Wapiti Formation. The site occurs in the lower part of Unit 4 of the formation (~72.5 Ma, upper Campanian), exposed along the southern bank of the Redwillow River. More than 100 tracks are documented across at least three distinct track-bearing layers, which were deposited on an alluvial floodplain. Hadrosaurid tracks are most abundant, and are referable to Hadrosauropodus based on track width exceeding track length, broad digits, and rounded or bilobed heel margins. We suggest the hadrosaurid trackmaker was Edmontosaurus regalis based on stratigraphic context. Tyrannosaurids, probable troodontids, possible ornithomimids, and possible azhdarchid pterosaurs represent minor but notable elements of the ichnofauna, as the latter is unknown from skeletal remains within the Wapiti Formation, and all others are poorly represented. Possible social behaviour is inferred for some of the hadrosaurid and small theropod-like trackmakers based on trackway alignment, suitable spacing and consistent preservation. On a broad taxonomic level (i.e., family or above), ichnofaunal compositions indicate that hadrosaurids were palaeoecologically dominant across Laramidia during the late Campanian within both high-and low-latitude deposits, although the role of depositional environment requires further testing

Cranial Growth and Variation in Edmontosaurs (Dinosauria: Hadrosauridae): Implications for Latest Cretaceous Megaherbivore Diversity in North America

The well-sampled Late Cretaceous fossil record of North America remains the only high-resolution dataset for evaluating patterns of dinosaur diversity leading up to the terminal Cretaceous extinction event. Hadrosaurine hadrosaurids (Dinosauria: Ornithopoda) closely related to Edmontosaurus are among the most common megaherbivores in latest Campanian and Maastrichtian deposits of western North America. However, interpretations of edmontosaur species richness and biostratigraphy have been in constant flux for almost three decades, although the clade is generally thought to have undergone a radiation in the late Maastrichtian. We address the issue of edmontosaur diversity for the first time using rigorous morphometric analyses of virtually all known complete edmontosaur skulls. Results suggest only two valid species, Edmontosaurus regalis from the late Campanian, and E. annectens from the late Maastrichtian, with previously named taxa, including the controversial Anatotitan copei, erected on hypothesized transitional morphologies associated with ontogenetic size increase and allometric growth. A revision of North American hadrosaurid taxa suggests a decrease in both hadrosaurid diversity and disparity from the early to late Maastrichtian, a pattern likely also present in ceratopsid dinosaurs. A decline in the disparity of dominant megaherbivores in the latest Maastrichtian interval supports the hypothesis that dinosaur diversity decreased immediately preceding the end Cretaceous extinction event

The accuracy and precision of body mass estimation in non‐avian dinosaurs

Inferring the body mass of fossil taxa, such as non‐avian dinosaurs, provides a powerful tool for interpreting physiological and ecological properties, as well as the ability to study these traits through deep time and within a macroevolutionary context. As a result, over the past 100 years a number of studies advanced methods for estimating mass in dinosaurs and other extinct taxa. These methods can be categorized into two major approaches: volumetric‐density (VD) and extant‐scaling (ES). The former receives the most attention in non‐avian dinosaurs and advanced appreciably over the last century: from initial physical scale models to three‐dimensional (3D) virtual techniques that utilize scanned data obtained from entire skeletons. The ES approach is most commonly applied to extinct members of crown clades but some equations are proposed and utilized in non‐avian dinosaurs. Because both approaches share a common goal, they are often viewed in opposition to one another. However, current palaeobiological research problems are often approach specific and, therefore, the decision to utilize a VD or ES approach is largely question dependent. In general, biomechanical and physiological studies benefit from the full‐body reconstruction provided through a VD approach, whereas large‐scale evolutionary and ecological studies require the extensive data sets afforded by an ES approach. This study summarizes both approaches to body mass estimation in stem‐group taxa, specifically non‐avian dinosaurs, and provides a comparative quantitative framework to reciprocally illuminate and corroborate VD and ES approaches. The results indicate that mass estimates are largely consistent between approaches: 73% of VD reconstructions occur within the expected 95% prediction intervals of the ES relationship. However, almost three quarters of outliers occur below the lower 95% prediction interval, indicating that VD mass estimates are, on average, lower than would be expected given their stylopodial circumferences. Inconsistencies (high residual and per cent prediction deviation values) are recovered to a varying degree among all major dinosaurian clades along with an overall tendency for larger deviations between approaches among small‐bodied taxa. Nonetheless, our results indicate a strong corroboration between recent iterations of the VD approach based on 3D specimen scans suggesting that our current understanding of size in dinosaurs, and hence its biological correlates, has improved over time. We advance that VD and ES approaches have fundamentally (metrically) different advantages and, hence, the comparative framework used and advocated here combines the accuracy afforded by ES with the precision provided by VD and permits the rapid identification of discrepancies with the potential to open new areas of discussion

A universal scaling relationship between body mass and proximal limb bone dimensions in quadrupedal terrestrial tetrapods

Abstract Background Body size is intimately related to the physiology and ecology of an organism. Therefore, accurate and consistent body mass estimates are essential for inferring numerous aspects of paleobiology in extinct taxa, and investigating large-scale evolutionary and ecological patterns in the history of life. Scaling relationships between skeletal measurements and body mass in birds and mammals are commonly used to predict body mass in extinct members of these crown clades, but the applicability of these models for predicting mass in more distantly related stem taxa, such as non-avian dinosaurs and non-mammalian synapsids, has been criticized on biomechanical grounds. Here we test the major criticisms of scaling methods for estimating body mass using an extensive dataset of mammalian and non-avian reptilian species derived from individual skeletons with live weights. Results Significant differences in the limb scaling of mammals and reptiles are noted in comparisons of limb proportions and limb length to body mass. Remarkably, however, the relationship between proximal (stylopodial) limb bone circumference and body mass is highly conserved in extant terrestrial mammals and reptiles, in spite of their disparate limb postures, gaits, and phylogenetic histories. As a result, we are able to conclusively reject the main criticisms of scaling methods that question the applicability of a universal scaling equation for estimating body mass in distantly related taxa. Conclusions The conserved nature of the relationship between stylopodial circumference and body mass suggests that the minimum diaphyseal circumference of the major weight-bearing bones is only weakly influenced by the varied forces exerted on the limbs (that is, compression or torsion) and most strongly related to the mass of the animal. Our results, therefore, provide a much-needed, robust, phylogenetically corrected framework for accurate and consistent estimation of body mass in extinct terrestrial quadrupeds, which is important for a wide range of paleobiological studies (including growth rates, metabolism, and energetics) and meta-analyses of body size evolution

A universal scaling relationship between body mass and proximal limb bone dimensions in quadrupedal terrestrial tetrapods

Abstract Background Body size is intimately related to the physiology and ecology of an organism. Therefore, accurate and consistent body mass estimates are essential for inferring numerous aspects of paleobiology in extinct taxa, and investigating large-scale evolutionary and ecological patterns in the history of life. Scaling relationships between skeletal measurements and body mass in birds and mammals are commonly used to predict body mass in extinct members of these crown clades, but the applicability of these models for predicting mass in more distantly related stem taxa, such as non-avian dinosaurs and non-mammalian synapsids, has been criticized on biomechanical grounds. Here we test the major criticisms of scaling methods for estimating body mass using an extensive dataset of mammalian and non-avian reptilian species derived from individual skeletons with live weights. Results Significant differences in the limb scaling of mammals and reptiles are noted in comparisons of limb proportions and limb length to body mass. Remarkably, however, the relationship between proximal (stylopodial) limb bone circumference and body mass is highly conserved in extant terrestrial mammals and reptiles, in spite of their disparate limb postures, gaits, and phylogenetic histories. As a result, we are able to conclusively reject the main criticisms of scaling methods that question the applicability of a universal scaling equation for estimating body mass in distantly related taxa. Conclusions The conserved nature of the relationship between stylopodial circumference and body mass suggests that the minimum diaphyseal circumference of the major weight-bearing bones is only weakly influenced by the varied forces exerted on the limbs (that is, compression or torsion) and most strongly related to the mass of the animal. Our results, therefore, provide a much-needed, robust, phylogenetically corrected framework for accurate and consistent estimation of body mass in extinct terrestrial quadrupeds, which is important for a wide range of paleobiological studies (including growth rates, metabolism, and energetics) and meta-analyses of body size evolution.</p

Head size, weaponry, and cervical adaptation: Testing craniocervical evolutionary hypotheses in Ceratopsia

The anterior cervical vertebrae form the skeletal connection between the cranial and postcranial skeletons in higher tetrapods. As a result, the morphology of the atlas-axis complex is likely to be shaped by selection pressures acting on either the head or neck. The neoceratopsian (Reptilia:Dinosauria) syncervical represents one of the most highly modified atlas-axis regions in vertebrates, being formed by the complete coalescence of the three most anterior cervical vertebrae. In ceratopsids, the syncervical has been hypothesized to be an adaptation to support a massive skull, or to act as a buttress during intraspecific head-to-head combat. Here, we test these functional/adaptive hypotheses within a phylogenetic framework and critically examine the previously proposed methods for quantifying relative head size in the fossil record for the first time. Results indicate that neither the evolution of cranial weaponry nor large head size correlates with the origin of cervical fusion in ceratopsians, and we, therefore, reject both adaptive hypotheses for the origin of the syncervical. Anterior cervical fusion has evolved independently in a number of amniote clades, and further research on extant groups with this peculiar anatomy is needed to understand the evolutionary basis for cervical fusion in Neoceratopsia