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

<i>Tlamaspis inopinatus</i> gen. et sp. nov. Associated skull roofs, cheek and thoracic plates.

<p>(A), NM Lc 11; (B), NM Lc 166; (C), posterior part of endocranial base associated with both shoulder girdles, NM Lc 148; (D), posterior part of endocranial base (counterpart of NM Lc 148), NM Lc 467. Scale bars = 10 mm. Abbreviations: AL anterior lateral plate; C central plate; end endocranium; MD median dorsal plate; N nuchal plate; p.pop posterior postorbital process; scp scapulocoracoid; SM submarginal plate.</p

<i>Sudaspis chlupaci</i> gen. et sp. nov.

<p>(A), holotype, NM Lc 27, posterior part of skull roof attached to endocranium; (B), endocranial base, NM Lc 4; (C), endocranial base, NM Lc 496. Scale bars = 10 mm. Abbreviations: ao.l lateral aorta; ch.c chordal canal; d.end.e external foramen for the endolymphatic duct; hyp.f hypophyseal fenestra; j.v.s secondary jugular vein; M marginal plate, N nuchal plate; occ.cd occipital condyle; Pn.m medial paranuchal plate; Pn.p posterior paranuchal plate; pse.a efferent pseudobranchial artery; soc supraorbital sensory line groove.</p

<i>Tlamaspis inopinatus</i> gen. et sp. nov. Endocrania.

<p>(A), holotype, NM Lc 35, endocranial base with rostronasal region; (B), endocranial base (counterpart of NM Lc 7), NM Lc 6; (C), posterior part of endocranial base, NM Lc 9; (D), endocranial base with rostronasal region and inner cast of submarginal plate, NM Lc 7. Scale bars = 10 mm. Abbreviations: ao.l lateral aorta; a.pop anterior postorbital process; ch.c chordal canal; gr.eff.a groove for the common efferent artery; hyp.f hypophyseal fenestra; j.v.s secondary jugular vein; occ.cd occipital condyle; p.pop posterior postorbital process; pse.a efferent pseudobranchial artery; SM submarginal plate.</p

<i>Radotina tesselata</i> Gross, 1958; holotype, NM Lc 95.

<p>(A), dorsal view; (B), left lateral view; Scale bars = 10 mm. Abbreviations: C central plate; cc central sensory line groove; e.s eye stalk; ioc infraorbital sensory line groove; na nasal openings; orb orbit; P pineal plate; PrO preorbital plate; PtN postnasal plate; PtO postorbital plate; R rostral plate; sac sacculus of the inner ear; Sbo suborbital plate; soc supraorbital sensory line groove.</p

<i>Tlamaspis inopinatus</i> gen. et sp. nov. Skull roofs.

<p>(A), holotype, NM Lc 34, skull roof attached to endocranium; (B), central part of skull roof, NM Lc 29; (C), nearly complete skull roof, NM Lc 22; (D), skull roof, NM Lc 491. Scale bars = 10 mm. Abbreviations: att.RoNa attachment for rostronasal capsule; C central plate; N nuchal plate; PP postpineal plate; ppl posterior pitline; soc supraorbital sensory line groove.</p

Devonian of the Prague Basin.

<p>(A) Upper left: Location of Devonian sediments (brown) in the central part of the Bohemian Massif (yellow). Right: Simplified map of the Devonian distribution. Tectonics omitted, the marine Cretaceous cover is displayed with respect to the known extent of the Devonian, younger continental units omitted; (B) Stratigraphic chart of the Lower Devonian in the Prague Basin. Numbers represent the fossil sites: 1 Černá rokle near Kosoř; 2 Koněprusy. Modified from [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0174794#pone.0174794.ref031" target="_blank">31</a>] after [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0174794#pone.0174794.ref041" target="_blank">41</a>].</p

<i>Radotina kosorensis</i> Gross, 1950.

<p>(A), holotype, NM Lc 25, skull roof; (B), natural mould of external surface of skull roof, NHM P.12829; (C), anterior part of skull roof with endocranial base, NM Lc 461. Scale bars = 10 mm. Abbreviations: C central plate; cc central sensory line groove; ch.c chordal canal; end endocranium; j.v.s secondary jugular vein; lc lateral sensory line groove; na nasal openings; P pineal plate; pi pineal organ; Pn paranuchal plate; ppl posterior pitline; PrM premedian plate; PrO preorbital plate; pse.a efferent pseudobranchial artery; PtN postnasal plate; PtO postorbital plate; R rostral plate; sepi internasal septum; soc supraorbital sensory line groove.</p

<i>Holopetalichthys primus</i> (Barrande, 1872).

<p>(A), holotype, NM Lc 40, posterior part of the skull roof, inner cast of submarginal plate; (B), skull roof, NM Lc 136; (C), median dorsal plate complex, NM Lc 139 (D), head and thoracic shield, NM Lc 33 and 138. Scale bars = 10 mm. Abbreviations: ADL anterior dorsolateral plate; AL anterior lateral plate; AV anterior ventral plate; AVL anterior ventrolateral plate; C central plate; cc central sensory line groove; d.end.e external foramen for the endolymphatic duct; IL interolateral plate; lc lateral sensory line groove; M marginal plate; MDa anterior median dorsal plate; MDp posterior median dorsal plate; N nuchal plate; P pineal plate; PDL posterior dorsolateral plate; pi pineal organ; Pn.a anterior paranuchal plate; PP postpineal plate; ppl posterior pitline; R rostral plate; Sbo suborbital plate; SM submarginal plate; soc supraorbital sensory line groove; Sp spinal plate; syn synarcuale.</p

Labyrinth region morphology of eight dipnomorphs.

<p>A, <i>Youngolepis praecursor</i> (from Chang <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0113898#pone.0113898-Chang2" target="_blank">[25]</a>, fig. 19); B, <i>Dipnorhynchus sussmilchi</i> (from Campbell and Barwick <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0113898#pone.0113898-Campbell1" target="_blank">[22]</a>, fig. 25); C, ‘<i>Chirodipterus’ australis</i> (from Miles <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0113898#pone.0113898-Miles1" target="_blank">[52]</a>, fig. 48); D, <i>Chirodipterus wildungensis</i> (from Säve-Söderbergh <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0113898#pone.0113898-SveSderbergh1" target="_blank">[21]</a>, fig. 9); E, <i>Griphognathus whitei</i> (from Miles <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0113898#pone.0113898-Miles1" target="_blank">[52]</a>, fig. 46); F, <i>Rhinodipterus kimberleyensis</i> (WAM 09.6.149); G, <i>Neoceratodus forsteri</i>; and H, <i>Protopterus annectens</i> (both from Retzius <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0113898#pone.0113898-Retzius1" target="_blank">[51]</a>, plate XXIV). Not to scale, anterior is to the right.</p