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

Right premaxilla and maxilla (PMU 24705/1b [formerly PMU R 233 a]) of <i>Euhelopus zdanskyi</i> in rostral (A), dorsal (B), right lateral (C), ventral (D), caudal (E) and medial (F) views.

<p>Right premaxilla and maxilla (PMU 24705/1b [formerly PMU R 233 a]) of <i>Euhelopus zdanskyi</i> in rostral (A), dorsal (B), right lateral (C), ventral (D), caudal (E) and medial (F) views.</p

Left squamosal (PMU 24705/1 h [formerly PMU R 233 a]) of <i>Euhelopus zdanskyi</i> in rostral (A), dorsal (B), left lateral (C), ventral (D), caudal (E), and medial (F) views.

<p>Left squamosal (PMU 24705/1 h [formerly PMU R 233 a]) of <i>Euhelopus zdanskyi</i> in rostral (A), dorsal (B), left lateral (C), ventral (D), caudal (E), and medial (F) views.</p

Right angular (PMU 24705/1q [formerly PMU R 233 s]) of <i>Euhelopus zdanskyi</i> in rostral (A), dorsal (B), right lateral (C), ventral (D), caudal (E), and medial (F) views.

<p>Right angular (PMU 24705/1q [formerly PMU R 233 s]) of <i>Euhelopus zdanskyi</i> in rostral (A), dorsal (B), right lateral (C), ventral (D), caudal (E), and medial (F) views.</p

Right palatine (PMU 24705/1n [formerly PMU R 233 γ]) of <i>Euhelopus zdanskyi</i> in rostral (A), dorsal (B), right lateral (C), ventral (D), caudal (E), and medial (F) views.

<p>Right palatine (PMU 24705/1n [formerly PMU R 233 γ]) of <i>Euhelopus zdanskyi</i> in rostral (A), dorsal (B), right lateral (C), ventral (D), caudal (E), and medial (F) views.</p

Left premaxilla and maxilla (PMU 24705/1a [formerly PMU R 233 a]) of <i>Euhelopus zdanskyi</i> in rostral (A), dorsal (B), left lateral (C), ventral (D), caudal (E), and medial (F) views.

<p>Note the distinctive buttresses on the lingual side of the teeth, located near the base of the crown.</p

Right quadrate, pterygoid, and vomer (PMU 24705/1k [formerly PMU R 233 a]) of <i>Euhelopus zdanskyi</i> in rostral (A), dorsal (B), right lateral (C), ventral (D), caudal (E), and medial (F) views.

<p>Right quadrate, pterygoid, and vomer (PMU 24705/1k [formerly PMU R 233 a]) of <i>Euhelopus zdanskyi</i> in rostral (A), dorsal (B), right lateral (C), ventral (D), caudal (E), and medial (F) views.</p

Right jugal (PMU 24705/1u [formerly PMU R 233 u]) of <i>Euhelopus zdanskyi</i> in lateral (A) and medial (B) views prior to additional preparation, and in lateral (C), anterior (D) and medial (E) views after additional preparation.

<p>As is clearly shown by comparing the two sets of photographs, the most distinctive feature of this element, the groove for articulation with the lacrimal, was only revealed after additional preparation.</p

Left lacrimal (PMU 24705/1e [formerly PMU R 233 α]) of <i>Euhelopus zdanskyi</i> in rostral (A), dorsal (B), left lateral (C), ventral (D), caudal (E), and medial (F) views.

<p>Left lacrimal (PMU 24705/1e [formerly PMU R 233 α]) of <i>Euhelopus zdanskyi</i> in rostral (A), dorsal (B), left lateral (C), ventral (D), caudal (E), and medial (F) views.</p

Left quadratojugal (PMU 24705/1i [formerly PMU R 233 z]) of <i>Euhelopus zdanskyi</i> in rostral (A), dorsal (B), left lateral (C), ventral (D), caudal (E), and medial (F) views.

<p>Left quadratojugal (PMU 24705/1i [formerly PMU R 233 z]) of <i>Euhelopus zdanskyi</i> in rostral (A), dorsal (B), left lateral (C), ventral (D), caudal (E), and medial (F) views.</p