9 research outputs found

    On the complexity of solving linear congruences and computing nullspaces modulo a constant

    Full text link
    We consider the problems of determining the feasibility of a linear congruence, producing a solution to a linear congruence, and finding a spanning set for the nullspace of an integer matrix, where each problem is considered modulo an arbitrary constant k>1. These problems are known to be complete for the logspace modular counting classes {Mod_k L} = {coMod_k L} in special case that k is prime (Buntrock et al, 1992). By considering variants of standard logspace function classes --- related to #L and functions computable by UL machines, but which only characterize the number of accepting paths modulo k --- we show that these problems of linear algebra are also complete for {coMod_k L} for any constant k>1. Our results are obtained by defining a class of functions FUL_k which are low for {Mod_k L} and {coMod_k L} for k>1, using ideas similar to those used in the case of k prime in (Buntrock et al, 1992) to show closure of Mod_k L under NC^1 reductions (including {Mod_k L} oracle reductions). In addition to the results above, we briefly consider the relationship of the class FUL_k for arbitrary moduli k to the class {F.coMod_k L} of functions whose output symbols are verifiable by {coMod_k L} algorithms; and consider what consequences such a comparison may have for oracle closure results of the form {Mod_k L}^{Mod_k L} = {Mod_k L} for composite k.Comment: 17 pages, one Appendix; minor corrections and revisions to presentation, new observations regarding the prospect of oracle closures. Comments welcom

    Logspace self-reducibility

    Get PDF
    A definition of self-reducibility is proposed to deal with logarithmic space complexity classes. A general property derived from the definition is used to prove known results comparing uniform and nonuniform complexity classes below polynomial time, and to obtain novel ones regarding nondeterministic nonuniform classes and reducibility to context-free languages.Peer ReviewedPostprint (published version

    Alternating and empty alternating auxiliary stack automata

    Get PDF
    AbstractWe consider variants of alternating auxiliary stack automata and characterize their computational power when the number of alternations is bounded by a constant or unlimited. In this way we get new characterizations of NP, the polynomial hierarchy, PSpace, and bounded query classes like co-DP=NL〈NP[1]〉 and Θ2P=PNP[O(logn)], in a uniform framework

    Upper Bounds on Recognition of a Hierarchy of Non-Context-Free Languages

    Get PDF
    Control grammars, a generalization of context-free grammars recently introduced for use in natural language recognition, are investigated. In particular, it is shown that a hierarchy of non-context-free languages, called the Control Language Hierarchy (CLH), generated by control grammars can be recognized in polynomial time. Previously, the best known upper bound was exponential time. It is also shown that CLH is in NC(2) the class of languages recognizable by uniform boolean circuits of polynomial size and O(log2 n) depth

    Space-efficient informational redundancy

    Get PDF
    AbstractWe study the relation of autoreducibility and mitoticity for polylog-space many-one reductions and log-space many-one reductions. For polylog-space these notions coincide, while proving the same for log-space is out of reach. More precisely, we show the following results with respect to nontrivial sets and many-one reductions:1.polylog-space autoreducible ⇔ polylog-space mitotic,2.log-space mitotic ⇒ log-space autoreducible ⇒ (logn⋅loglogn)-space mitotic,3.relative to an oracle, log-space autoreducible ⇏ log-space mitotic. The oracle is an infinite family of graphs whose construction combines arguments from Ramsey theory and Kolmogorov complexity

    Two Applications of Inductive Counting for Complementation Problems

    Full text link

    The parallel complexity of certain algorithmic problems in group theory

    Get PDF
    In this thesis, we study the parallel complexity of certain problems in algorithmic group theory. These problems are the word problem, the geodesic and normal form problem and the conjugacy problem. We study these problems for some products of groups, namely direct products, free products and graph products. For all those we consider the problems for a fixed group as well as the uniform versions. Uniform means that the group is part of the input. Among the studied problems, the word problem is the most important one and necessary to solve any of the other problems. For direct products, solving any of the mentioned problems reduces directly to the problems in the base groups of the product. Some of the solutions for the direct product are required for solving the problems of the more complicated products. For free products, we show that the word problem reduces in AC^0 to the word problem of the base groups and the word problem of the free group of rank two. This does hold for the word problem of a fixed free product as well as for the uniform version. For the geodesic and normal form problem of free products, we introduce an equivalence relation. This relation can be decided in AC^0 by using oracle calls to the word problems of the base groups. The solution of the word and geodesic problem can then be used to solve the conjugacy problem. In free products, two cyclically reduced words are conjugate if and only if they are transposed. Direct products and free products are special cases of graph products. A graph product can be written as an amalgamated product of smaller graph products. We first solve the word problem of some restricted amalgamated product. This solution can then be used to solve the word problem of a fixed graph product inductively. We obtain that the word problem of a fixed graph product is AC^0-reducible to the word problem of its base groups and the word problem of the free group of rank two. Unfortunately, this method cannot be used to solve the uniform word problem. We show that the uniform word problem of graph products is NL-hard. For solving it, we introduce an embedding of the graph product into the automorphism group of some (possibly infinite dimensional) vector space. We show that the evaluation of these automorphisms can be realized in GapL and that verifying its result is in CL by using oracle calls to the word problem in the base groups. The uniform word problem of graph products can be reduced to the evaluation of these automorphisms. For the geodesic problem, we introduce another equivalence relation. As for free products, this relation can be decided in AC^0 by using oracle calls to the (uniform) word problem. In graph products the normal form of some word is the length-lexicographic first equivalent word. For solving the normal form problem, first a geodesic and then the lexicographic normal form of this geodesic is computed. We show that for a fixed graph product the computation of the lexicographic normal form is in TC^0 and TC^0-complete for most graph products. We further show that the uniform version is FNL-complete. The solution of the word and geodesic problem can then be used to solve the conjugacy problem. First, we show how to compute cyclically reduced words in AC^0 by using oracle calls to the word problem. Then we show that in graph products two cyclically reduced words are conjugate if and only if they are obtained by a sequence of transpositions. This problem can then be solved by verifying whether the first word is a factor of some power of the second word. For a fixed graph product the factor problem can be decided in AC^0 by using oracle calls to the word problem. For the uniform factor problem we show that it can be decided in NL by using oracle calls to the uniform word problem. Combining all this gives a solution to the (uniform) conjugacy problem of graph products.In dieser Arbeit untersuchen wir die parallele KomplexitĂ€t gewisser Probleme in der algorithmischen Gruppentheorie. Diese Probleme sind das Wortproblem (engl. word problem), das GeodĂ€ten- und Normalformenproblem (engl. geodesic and normal form problem) und das Konjugationsproblem (engl. conjugacy problem). Wir untersuchen diese Probleme fĂŒr Produkte von Gruppen, genauer fĂŒr direkte Produkte, freie Produkte und Graphprodukte. FĂŒr all jene betrachten wir die Probleme sowohl fĂŒr eine feste Gruppe als auch ihre uniforme Variante. Uniform bedeutet, dass die Gruppe Teil der Eingabe ist. Unter den untersuchten Problemen ist das Wortproblem das wichtigste und notwendig fĂŒr die Lösung der anderen Probleme. Die erwĂ€hnten Probleme lassen sich fĂŒr direkte Produkte unmittelbar durch reduzieren auf die Probleme in den Basisgruppen lösen. Manche der Lösungen fĂŒr direkte Produkte werden benötigt, um die Probleme in den komplizierteren Produkten zu lösen. FĂŒr freie Produkte können wir zeigen, dass sich das Wortproblem in AC^0 auf das Wortproblem der Basisgruppen und das Wortproblem der freien Gruppe vom Rang zwei reduzieren lĂ€sst. Dies gilt sowohl fĂŒr das Wortproblem eines festen freien Produkts als auch fĂŒr die uniforme Variante. FĂŒr das GeodĂ€ten- und Normalformenproblem freier Produkte fĂŒhren wir eine Äquivalenzrelation ein. Diese Relation kann in AC^0 durch Orakelanfragen an das Wortproblem des freien Produkts entschieden werden. Die Lösung des Wort- und GeodĂ€tenproblems kann schließlich genutzt werden, um das Konjugationsproblem zu lösen. In freien Produkten sind zwei zyklisch reduzierte (engl. cyclically reduced) Wörter genau dann konjugiert, wenn sie transponiert zueinander sind. Direkte Produkte und freie Produkte sind SpezialfĂ€lle des Graphprodukts. Ein Graphprodukt kann als amalgamiertes Produkt kleinerer Graphprodukte aufgefasst werden. Wir lösen zuerst das Wortproblem dieser eingeschrĂ€nkten amalgamierten Produkte. Diese Lösung kann schließlich genutzt werden, um das Wortproblem eines festen Graphprodukts induktiv zu lösen. Wir erhalten, dass das Wortproblem eines festen Graphprodukts AC^0-reduzierbar auf das Wortproblem in den Basisgruppen und das Wortproblem der freien Gruppe vom Rang zwei ist. Diese Methode lĂ€sst sich nicht fĂŒr das uniforme Wortproblem in Graphprodukten nutzen. Wir zeigen, dass das uniforme Wortproblem von Graphprodukten NL-schwer ist. Um dieses zu lösen fĂŒhren wir eine Einbettung des Graphprodukts in die Automorphismengruppe eines (möglicherweise unendlich dimensionalen) Vektorraums ein. Wir zeigen, dass durch Orakelanfragen an das Wortproblem in den Basisgruppen, die Auswertung dieser Automorphismen in GapL realisiert werden kann und, dass die Verifikation des Ergebnisses in CL ist. Das uniforme Wortproblem kann auf die Auswertung dieser Automorphismen reduziert werden. FĂŒr das GeodĂ€tenproblem fĂŒhren wir eine weitere Äquivalenzrelation ein. Wie schon fĂŒr freie Produkte kann diese Relation in AC^0 durch Orakelanfragen an das (uniforme) Wortproblem entschieden werden. Die Normalform eines Wortes ist das lĂ€ngenlexikographisch erste Ă€quivalente Wort. Um das Normalformenproblem zu lösen wird zuerst eine GeodĂ€tische und anschließend die lexikographische Normalform dieser GeodĂ€tischen berechnet. Wir zeigen, dass die Berechnung der lexiographischen Normalform in TC^0 möglich ist und TC^0-vollstĂ€ndig fĂŒr die meisten Graphprodukte ist. Wir zeigen außerdem, dass die uniforme Variante FNL-vollstĂ€ndig ist. Die Lösung des Wort- und GeodĂ€tenproblems kann schließlich genutzt werden, um das Konjugationsproblem zu lösen. Wir zeigen zuerst, wie sich zyklisch reduzierte Wörter in AC^0 durch Orakelanfragen an das Wortproblem berechnen lassen. Anschließend zeigen wir, dass in Graphprodukten zwei zyklisch reduzierte Wörter genau dann konjugiert sind, wenn sie durch eine Folge von Transpositionen auseinander hervorgehen. Dieses Problem kann schließlich gelöst werden, indem geprĂŒft wird, ob das erste Wort ein Faktor einer Potenz des zweiten Worts ist. FĂŒr ein festes Graphprodukt kann das Faktorproblem in AC^0 durch Orakelanfragen an das Wortproblem entschieden werden. FĂŒr das uniforme Faktorproblem zeigen wir, dass es in NL durch Orakelanfragen an das uniforme Wortproblem entschieden werden kann. Zusammengesetzt ergibt sich eine Lösung des (uniformen) Konjugationsproblems
    corecore