Low-Depth Uniform Threshold Circuits and the Bit-Complexity of Straight Line Programs

Allender, Eric; Balaji, N.; Datta, Samir

doi:10.1007/978-3-662-44465-8_2

“…Hence, the algorithm cannot be directly implemented by a polynomial-space Turing machine (TM). One could try to adapt the method of Allender et al [20,2] based on an intricate use of the Chinese remainder representation (CRR) of integers. However, there is no known way of computing the max operation directly and efficiently on numbers in CRR.…”

Section: Challengesmentioning

confidence: 99%

On the Complexity of Value Iteration

Balaji,

Kiefer,

Novotný

et al. 2018

Preprint

Self Cite

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Value iteration is a fundamental algorithm for solving Markov Decision Processes (MDPs). It computes the maximal n-step payoff by iterating n times a recurrence equation which is naturally associated to the MDP. At the same time, value iteration provides a policy for the MDP that is optimal on a given finite horizon n. In this paper, we settle the computational complexity of value iteration. We show that, given a horizon n in binary and an MDP, computing an optimal policy is EXPTIME-complete, thus resolving an open problem that goes back to the seminal 1987 paper on the complexity of MDPs by Papadimitriou and Tsitsiklis. To obtain this main result, we develop several stepping stones that yield results of an independent interest. For instance, we show that it is EXPTIME-complete to compute the n-fold iteration (with n in binary) of a function given by a straight-line program over the integers with max and + as operators. We also provide new complexity results for the bounded halting problem in linear-update counter machines.

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“…4.1], one shows by induction on the depth of a gate that the problem whether a given gate of that circuit (the gate is specified by a bit string of length O(n)) evaluates to true is in the counting hierarchy, where the level in the counting hierarchy depends on the level of the gate in the circuit. 2 Hence we have to show that L belongs to the counting hierarchy. Let A be an SLP for a tree t, n = |A|, p ∈ P n , and 1 ≤ j ≤ 2n.…”

Section: Difficult Arithmetical Evaluation Problemsmentioning

confidence: 99%

Tree Compression Using String Grammars

GanardiMoses

¹

,

HuckeDanny

²

,

LohreyMarkus

³

et al. 2017

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We study the compressed representation of a ranked tree by a (string) straight-line program (SLP) for its preorder traversal, and compare it with the well-studied representation by straight-line context free tree grammars (which are also known as tree straight-line programs or TSLPs). Although SLPs turn out to be exponentially more succinct than TSLPs, we show that many simple tree queries can still be performed efficiently on SLPs, such as computing the height and Horton-Strahler number of a tree, tree navigation, or evaluation of Boolean expressions. Other problems on tree traversals turn out to be intractable, e.g. pattern matching and evaluation of tree automata.

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“…Computing a certain bit of the output number of an arithmetic circuit belongs to PH PP PP PP [2] (but no matching lower bound is known). In our situation, the level gets even higher, so we made no effort to compute it.…”

Section: Difficult Arithmetical Evaluation Problemsmentioning

confidence: 99%

Tree Compression Using String Grammars

Ganardi

¹

,

Hucke

²

,

Lohrey

³

et al. 2016

LATIN 2016: Theoretical Informatics

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We study the compressed representation of a ranked tree by a (string) straight-line program (SLP) for its preorder traversal, and compare it with the well-studied representation by straight-line context free tree grammars (which are also known as tree straight-line programs or TSLPs). Although SLPs turn out to be exponentially more succinct than TSLPs, we show that many simple tree queries can still be performed efficiently on SLPs, such as computing the height and Horton-Strahler number of a tree, tree navigation, or evaluation of Boolean expressions. Other problems on tree traversals turn out to be intractable, e.g. pattern matching and evaluation of tree automata. * The third and fourth author are supported by the DFG-project LO 748/10-1 (QUANT-KOMP).

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Low-Depth Uniform Threshold Circuits and the Bit-Complexity of Straight Line Programs

Cited by 13 publications

References 22 publications

On the Complexity of Value Iteration

On the Complexity of Value Iteration

Tree Compression Using String Grammars

Tree Compression Using String Grammars

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