Counting Classes and the Fine Structure between NC 1 and L

Datta, Samir; Mahajan, Meena; Rao, B. V. Raghavendra; Thomas, Michael E.; Vollmer, Heribert

doi:10.1007/978-3-642-15155-2_28

Cited by 5 publications

(2 citation statements)

References 19 publications

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“…The GapNC 1 hierarchy comprises of languages accepted by polynomial-size constant depth unbounded fanin circuits (AC 0 ) with oracle access to bits of GapNC 1 functions, and is known to be contained in DLOG. (See [9,10] for more details.) Miscellaneous Notation.…”

Section: Preliminariesmentioning

confidence: 99%

Identity Testing, Multilinearity Testing, and Monomials in Read-Once/Twice Formulas and Branching Programs

Mahajan

Rao

Sreenivasaiah

2012

Mathematical Foundations of Computer Science 2012

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Abstract. We study the problem of testing if the polynomial computed by an arithmetic circuit is identically zero (ACIT). We give a deterministic polynomial time algorithm for this problem when the inputs are read-twice formulas. This algorithm also computes the MLIN predicate, testing if the input circuit computes a multilinear polynomial. We further study two related computational problems on arithmetic circuits. Given an arithmetic circuit C, 1) ZMC: test if a given monomial in C has zero coefficient or not, and 2) MonCount: compute the number of monomials in C. These problems were introduced by Fournier, Malod and Mengel [STACS 2012], and shown to characterize various levels of the counting hierarchy (CH). We address the above problems on read-restricted arithmetic circuits and branching programs. We prove several complexity characterizations for the above problems on these restricted classes of arithmetic circuits.

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Section: Preliminariesmentioning

confidence: 99%

Identity Testing, Multilinearity Testing, and Monomials in Read-Once/Twice Formulas and Branching Programs

Mahajan

Rao

Sreenivasaiah

2012

Mathematical Foundations of Computer Science 2012

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“…Further down the complexity hierarchy, Caussinus et al [17] introduced counting versions of NC 1 based on various characterisations of NC 1 . The counting and probabilistic analogues of NC 1 exhibit properties similar to their logspace counterparts [18]. Moreover, counting and gap variants of the class AC 0 were defined by Agrawal et al [19].…”

Section: Introductionmentioning

confidence: 96%

Parameterised Counting in Logspace

et al. 2023

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Logarithmic space-bounded complexity classes such as $$\textbf{L} $$ L and $$\textbf{NL} $$ NL play a central role in space-bounded computation. The study of counting versions of these complexity classes have lead to several interesting insights into the structure of computational problems such as computing the determinant and counting paths in directed acyclic graphs. Though parameterised complexity theory was initiated roughly three decades ago by Downey and Fellows, a satisfactory study of parameterised logarithmic space-bounded computation was developed only in the last decade by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). In this paper, we introduce a new framework for parameterised counting in logspace, inspired by the parameterised space-bounded models developed by Elberfeld, Stockhusen and Tantau. They defined the operators $$\textbf{para}_{\textbf{W}}$$ para W and $$\textbf{para}_\beta $$ para β for parameterised space complexity classes by allowing bounded nondeterminism with multiple-read and read-once access, respectively. Using these operators, they characterised the parameterised complexity of natural problems on graphs. In the spirit of the operators $$\textbf{para}_{\textbf{W}}$$ para W and $$\textbf{para}_\beta $$ para β by Stockhusen and Tantau, we introduce variants based on tail-nondeterminism, $$\textbf{para}_{{\textbf{W}}[1]}$$ para W [ 1 ] and $$\textbf{para}_{\beta {\textbf{tail}}}$$ para β tail . Then, we consider counting versions of all four operators and apply them to the class $$\textbf{L} $$ L . We obtain several natural complete problems for the resulting classes: counting of paths in digraphs, counting first-order models for formulas, and counting graph homomorphisms. Furthermore, we show that the complexity of a parameterised variant of the determinant function for (0, 1)-matrices is $$\#\textbf{para}_{\beta {\textbf{tail}}}\textbf{L} $$ # para β tail L -hard and can be written as the difference of two functions in $$\#\textbf{para}_{\beta {\textbf{tail}}}\textbf{L} $$ # para β tail L . These problems exhibit the richness of the introduced counting classes. Our results further indicate interesting structural characteristics of these classes. For example, we show that the closure of $$\#\textbf{para}_{\beta {\textbf{tail}}}\textbf{L} $$ # para β tail L under parameterised logspace parsimonious reductions coincides with $$\#\textbf{para}_\beta \textbf{L} $$ # para β L . In other words, in the setting of read-once access to nondeterministic bits, tail-nondeterminism coincides with unbounded nondeterminism modulo parameterised reductions. Initiating the study of closure properties of these parameterised logspace counting classes, we show that all introduced classes are closed under addition and multiplication, and those without tail-nondeterminism are closed under parameterised logspace parsimonious reductions. Finally, we want to emphasise the significance of this topic by providing a promising outlook highlighting several open problems and directions for further research.

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Collapsing Exact Arithmetic Hierarchies

Balaji

Datta

2014

Algorithms and Computation

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Counting Classes and the Fine Structure between NC 1 and L

Cited by 5 publications

References 19 publications

Identity Testing, Multilinearity Testing, and Monomials in Read-Once/Twice Formulas and Branching Programs

Identity Testing, Multilinearity Testing, and Monomials in Read-Once/Twice Formulas and Branching Programs

Parameterised Counting in Logspace

Collapsing Exact Arithmetic Hierarchies

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