A note on closure properties of logspace MOD classes

Hertrampf, Ulrich; Reith, Steffen; Vollmer, Heribert

doi:10.1016/s0020-0190(00)00091-0

Cited by 18 publications

(22 citation statements)

References 8 publications

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“…We show below that if G is planar and bipartite, then the second stage can be performed in L GGR . Since GGR is known to be in UL∩co-UL [11] which is contained in ⊕L, and since ⊕L ⊕L = L ⊕L = ⊕L ( [21]), it then follows that Planar-B-UPM is in ⊕L.…”

Section: Planar-b-upm Is In ⊕Lmentioning

confidence: 99%

Planarity, Determinants, Permanents, and (Unique) Matchings

Datta

Kulkarni

Limaye³

et al. 2010

ACM Trans. Comput. Theory

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Abstract. Viewing the computation of the determinant and the permanent of integer matrices as combinatorial problems on associated graphs, we explore the restrictiveness of planarity on their complexities and show that both problems remain as hard as in the general case, i.e. GapL and #P complete. On the other hand, both bipartite planarity and bimodal planarity bring the complexity of permanents down (but no further) to that of determinants. The permanent or the determinant modulo 2 is complete for ⊕L, and we show that parity of paths in a layered grid graph (which is bimodal planar) is also complete for this class. We also relate the complexity of grid graph reachability to that of testing existence/uniqueness of a perfect matching in a planar bipartite graph.

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Section: Planar-b-upm Is In ⊕Lmentioning

confidence: 99%

Planarity, Determinants, Permanents, and (Unique) Matchings

Datta

Kulkarni

Limaye³

et al. 2010

ACM Trans. Comput. Theory

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“…For a function f (x) : Σ * → Σ * and x ∈ Σ * , let | f (x)| denote the length of the representation of f (x); and let f (x) j denote the j th symbol in that representation. Following Hertrampf, Reith and Vollmer [8], for a function f : Σ * → Σ * on some alphabet Σ, and for some symbol • / ∈ Σ, we may define the decision problem…”

Section: Preliminariesmentioning

confidence: 99%

“…[7] and explicitly shown in Ref. [8] may be interpreted as saying that the characteristic function of any L ∈ Mod p L may be computed by a FUL p machine. That is, a Mod p L oracle can be directly simulated in a Mod p L algorithm, by simulating the corresponding FUL p machine as a subroutine.…”

Section: Natural Function Classes For Modular Logspacementioning

confidence: 99%

“…Without loss of generality, as in Ref. [8,Corollary 3.2] each machine T j,b accepts on a number of branches…”

Section: Natural Function Classes For Modular Logspacementioning

confidence: 99%

“…[8], albeit extended beyond the characteristic functions of L ∈ Mod p L: when k is a prime power, any function whose bits are verifiable by coMod k L algorithms, can also be evaluated naturally as a subroutine of a coMod k L algorithm. The importance of this result to us lies in the consequence for # k L, as the prototypical class of functions verifiable in coMod k L:…”

Section: Natural Function Classes For Modular Logspacementioning

confidence: 99%

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Untitled

Beaudrap¹

2013

Chicago J. of Theoretical Comp. Sci.

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Abstract: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 2. 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 [7]. 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 2.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 2, using ideas similar to those used in the case of k prime in Ref. [7] 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. † This work was supported by the EC project QCS.

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On the Complexity of Some Equivalence Problems for Propositional Calculi

Reith

2003

Mathematical Foundations of Computer Science 2003

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A note on closure properties of logspace MOD classes

Cited by 18 publications

References 8 publications

Planarity, Determinants, Permanents, and (Unique) Matchings

Planarity, Determinants, Permanents, and (Unique) Matchings

Untitled

On the Complexity of Some Equivalence Problems for Propositional Calculi

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