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

Early Frasnian thelodont scales from central Iran and their implications for turiniid taxonomy, systematics and distribution

<p>The early Frasnian turiniid thelodont <i>Neoturinia hutkensis</i> gen. nov. is re-described on the basis of a new large scale set from the Chahriseh section in central Iran. Detailed morphological and histological information supports a new generic affinity for this species, which was previously assigned to <i>Turinia</i>. The generic affinity of other turiniid thelodonts from Gondwana is tested by a phylogenetic analysis based on scales, which proposes that most taxa from Gondwana form a clade separate from that comprising the first-studied Laurussian forms, including the type species <i>Turinia pagei</i> Powrie, <a href="#cit0064" target="_blank">1870</a>. This suggests that many of the <i>Turinia</i> species from Gondwana should be separated from this genus and need to be revisited. On the strength of the analysis, younger Gondwanan taxa in the mid–early Late Devonian should probably be referred to the new genus; one Late Silurian taxon from eastern Gondwana is removed. A biogeographic analysis, using a parsimony ancestral state method, is also conducted in order to discuss dispersal patterns in relation to the achieved model of interrelationships.</p> <p>http://zoobank.org/urn:lisd:zoobank.org:pub:03499C12-344D-4296-A633-201CDB982445</p> <p>SUPPLEMENTAL DATA—Supplemental materials are available for this article for free at <a href="http://www.tandfonline.com/UJVP" target="_blank">www.tandfonline.com/UJVP</a></p> <p>Citation for this article: Hairapetian, V., H. Blom, and S. Turner. 2016. Early Frasnian thelodont scales from central Iran and their implications for turiniid taxonomy, systematics and distribution. Journal of Vertebrate Paleontology. DOI: 10.1080/02724634.2016.1100632.</p

Resolution of the Early Jurassic actinopterygian fish <i>Pachycormus</i> and a dispersal hypothesis for Pachycormiformes

<p>The Early Jurassic (Toarcian) actinopterygian <i>Pachycormus</i> is a basal taxon within Pachycormiformes, a Mesozoic marine neopterygian radiation that evolved an extreme ecomorph divergence between hyperspecialized ‘billfish-like’ macrocarnivores and gigantic suspension feeders, including the largest fish of all time. Current phylogenies place <i>Pachycormus</i> as an early member of the suspension-feeding lineage; however, species disparity renders character states uncertain and potential exists for considerable intraspecific variability. Given its importance for resolution of pachycormiform phylogenetic topology, we comprehensively reassessed <i>Pachycormus</i> fossils housed in collections across Europe and found that the proportional traits traditionally used to discriminate between species are actually consistent with an ontogenetic size morphocline. Our cladistic analyses further show that the monotypic senior synonym, <i>Pachycormus macropterus</i>, is a wildcard that manifests a mosaic of transitional states. This has significant implications for hypothesized Toarcian marine vertebrate provincialism because <i>P. macropterus</i> had a ubiquitous Boreal Tethyan distribution. Moreover, our tree-based palaeobiogeographical optimizations infer that the western Tethyan region was a pachycormiform dissemination center, with global dispersals occurring through transoceanic migration and invasion of epeiric basins.</p> <p>Citation for this article: Wretman, L., H. Blom, and B. P. Kear. 2016. Resolution of the Early Jurassic actinopterygian fish <i>Pachycormus</i> and a dispersal hypothesis for Pachycormiformes. Journal of Vertebrate Paleontology. DOI: 10.1080/02724634.2016.1206022.</p

Two acanthodian tooth whorls (SEM photos with line drawings).

<p><b>A–C,</b> GIT 658-1; <b>D–F,</b> GIT 658-2. <b>A,D.</b> Oral views, labial to the right. <b>B,E.</b> Sketch interpretations of A,D. <b>C,F.</b> Lateral views, labial to the right. <b>Abbreviations: t1-t3,</b> tooth element 1–3 belonging to the linear growth serial; <b>t0?,</b> possible origination element of the linear growth serial; <b>o,</b> disorganized odontodes on the labial base.</p

The scanned scale of <i>Andreolepis hedei</i> Gross, 1968.

<p><b>A.</b> Scanned part of the scale (PMU 24786) rendered in VG Studio 2.1, crown view, red line marking the position of the slice in <b>B</b>. <b>B.</b> Longitudinal virtual thin section from the synchrotron scanning data, the red rectangle marking the region in <b>C</b>. <b>C.</b> Close-up of <b>B</b>, showing the segmentation of odontodes and the tissue composition of the crown (picture exported from VG Studio 2.1). The numbers in circles correspond to the odontodes in following <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0071890#pone-0071890-g003" target="_blank">Figures 3</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0071890#pone-0071890-g004" target="_blank">4</a>. <b>D</b>. A real thin section (PMU 24784) from an <i>Andreolepis</i> body scale made in the anteroposterior plane for comparison (DIC, optical microscopy); the scale is slender and comes from the posterior region of the body <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0071890#pone.0071890-Chen1" target="_blank">[28]</a>. <b>Abbreviations: e</b>, enamel; <b>d</b>, dentine; <b>hc</b>, horizontal vascular canal, <b>O1-O5</b>, odontodes 1–5, <b>Shb,</b> Sharpey’s fibers.</p

A simplified phylogeny of jawed vertebrates modified from Brazeau [14].

<p>‘Placoderms’ and ‘Acanthodians’ are extinct, probably paraphyletic, groups. Representative scales of osteichthyans (<i>Cheirolepis</i>), acanthodians (<i>Nostolepis</i>), stem chondrichthyans (<i>Cladoselache</i>) and ‘placoderms’ (<i>Ohiolepis</i>) are modified from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0071890#pone.0071890-Gross5" target="_blank">[52]</a>, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0071890#pone.0071890-rvig6" target="_blank">[53]</a>, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0071890#pone.0071890-rvig5" target="_blank">[44]</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0071890#pone.0071890-Gross6" target="_blank">[54]</a> respectively.</p

The reconstructed growth pattern of odontodes in the scanned scale of <i>Andreolepis</i>, anterolateral view.

<p>For explanations see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0071890#pone-0071890-g003" target="_blank">Figure 3</a>.</p

Bazzi et al_Data_Dryad_Supplementary_File_3

Tps file containing two-dimensional landmark data

The reconstructed growth pattern of odontodes in the scanned scale of <i>Andreolepis</i>, crown view.

<p><b>A–J.</b> The referred sequential addition of odontodes – in the crown of the scale. The first generation odontodes (odontodes – , see text) form a growth series, but the other younger odontodes ( – ) do not necessarily fall neatly into the same sequence even though they generally continue to get larger; the yellow horizontal vascular canal system is used as landmark to show the positions of the odontodes <b>K</b>. Crown view of the scale with buried odontodes, showing the actual surface composition of the scale. Note that the most dorsal denticles compose the enamel layers from both odontode and , odontode is partially overlapped by ; odontode is only overlapped by and posteriorly and exposed to the surface otherwise.</p

R source code

Programming commands essential for replicating our geometric morphometric and statistical analyses are here given. The main source code (Bazzi et al_Dryad_MasterRScript) is accompanied by separate R scripts (functions and analytical loops) which can be sourced directly provided the working directory has been set