The Subpower Membership Problem for Finite Algebras with Cube Terms

The subalgebra membership problem is the problem of deciding if a given element belongs to an algebra given by a set of generators. This is one of the best established computational problems in algebra. We consider a variant of this problem, which is motivated by recent progress in the Constraint Satisfaction Problem, and is often referred to as the Subpower Membership Problem (SMP). In the SMP we are given a set of tuples in a direct product of algebras from a fixed finite set $\mathcal{K}$ of finite algebras, and are asked whether or not a given tuple belongs to the subalgebra of the direct product generated by a given set. Our main result is that the subpower membership problem SMP($\mathcal{K}$) is in P if $\mathcal{K}$ is a finite set of finite algebras with a cube term, provided $\mathcal{K}$ is contained in a residually small variety. We also prove that for any finite set of finite algebras $\mathcal{K}$ in a variety with a cube term, each one of the problems SMP($\mathcal{K}$), SMP($\mathbb{HS} \mathcal{K}$), and finding compact representations for subpowers in $\mathcal{K}$, is polynomial time reducible to any of the others, and the first two lie in NP

Linear Datalog and Bounded Path Duality of Relational Structures

In this paper we systematically investigate the connections between logics with a finite number of variables, structures of bounded pathwidth, and linear Datalog Programs. We prove that, in the context of Constraint Satisfaction Problems, all these concepts correspond to different mathematical embodiments of a unique robust notion that we call bounded path duality. We also study the computational complexity implications of the notion of bounded path duality. We show that every constraint satisfaction problem \csp(\best) with bounded path duality is solvable in NL and that this notion explains in a uniform way all families of CSPs known to be in NL. Finally, we use the results developed in the paper to identify new problems in NL

Monumenta Monodica Medii Aevi - Responsorien = Monumenta Monodica Medii Aevi - The Responsory

A munkafolyamat tárgya: a responzórium-műfaj összkiadásának elkészítése. A responzórium a gregorián repertoár egyik legartisztikusabb műfaja, antik gyökerekre megy vissza, a középkorban helyi változatokban élt és 1500-ig gyarapodott, egyebek mellett helyi (ez esetben: magyar) kompozíciókkal Anyaga: két reprezentatív magyar forráscsoport: 1) a fővonalat képviselő esztergomi hagyomány, és 2) az itáliai stílust képviselő magyar ferences hagyomány responzórium-anyaga Koncepció: 1) egy nemzetközi viszonylatban is kiadatlan, ezer éven át fejlődött magas értékű műzenei repertoár közzététele; 2) tipológiai és stiláris osztályozás révén zenei sajátságuk feltárása; 3) a középkori magyar kultúrának és megoszlásának bemutatása egy műfajon keresztül Elvégzett munka: Több, mint 1000 tétel átírása egyenként 4-6 kódexből; ezek elemzése és osztályozása; a kéziratok szövegi és dallami változatainak feldolgozása kritikai jegyzetekben; terjedelmes bevezető, kommentár és indexek elkészítése; az egész anyag grafikázása és korrektúrája, a szövegek angolra fordítása A munka terjedelme és jövője: A teljes kézirat mintegy 1500 oldal terjedelmű, 3 kötetre oszlik, és a Bärenreiter kiadó (Basel, Kassel, etc.) tekintélyes Monumenta Monodica Medii Aevi c. sorozatában jelenik meg, előre láthatóan 2009-ben | The object of the working procedure: a complete edition of the Responsories. The responsory is one of the most elaborated genres of the chant repertory; its origins go back to the Christian antiquity, lived in local variants during the Middle Ages, and many local (in this case: Hungarian) compositions were added to it. The material: two representative group of Hungarian music manuscripts 1) the mainstream Esztergom tradition 2) the Italy-based tradition of the Hungarian Franciscans. The concept: 1) To make accessible a high-quality art music repertory, developed over thousand years, that has been published neither on international environment; 2) to demonstrate through typological-stylistic classification its music qualities; 3) to present a remarkable product of the medieval culture of Hungary in its differentiation in time and space. The work done: Transcription of more than thousand samples, each from 4-6 manuscripts; analysis and classification; registration of all textual and melodic variants summed in the critical notes; introduction and commentary to melodies and sources; indexing all pieces; graphical realization, its correction; English translation of the full text. Size and destination: The full manuscripts is 1500 pages in 3 volumes ; it is to be published by the Bärenreiter editor in series Monumenta Monodica Medii Aevi, presumably in 2009