All reports by Author James Cook:

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TR23-174
| 15th November 2023
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James Cook, Ian Mertz#### Tree Evaluation is in Space O(log n ยท log log n)

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TR22-026
| 17th February 2022
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James Cook, Ian Mertz#### Trading Time and Space in Catalytic Branching Programs

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TR21-054
| 14th April 2021
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James Cook, Ian Mertz#### Encodings and the Tree Evaluation Problem

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TR20-056
| 17th April 2020
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James Cook, Ian Mertz#### Catalytic Approaches to the Tree Evaluation Problem

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TR12-175
| 13th December 2012
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James Cook, Omid Etesami, Rachel Miller, Luca Trevisan#### On the One-Way Function Candidate Proposed by Goldreich

Revisions: 1

James Cook, Ian Mertz

The Tree Evaluation Problem ($TreeEval$) (Cook et al. 2009) is a central candidate for separating polynomial time ($P$) from logarithmic space ($L$) via composition. While space lower bounds of $\Omega(\log^2 n)$ are known for multiple restricted models, it was recently shown by Cook and Mertz (2020) that TreeEval can be ... more >>>

James Cook, Ian Mertz

An $m$-catalytic branching program (Girard, Koucky, McKenzie 2015) is a set of $m$ distinct branching programs for $f$ which are permitted to share internal (i.e. non-source non-sink) nodes. While originally introduced as a non-uniform analogue to catalytic space, this also gives a natural notion of amortized non-uniform space complexity for ... more >>>

James Cook, Ian Mertz

We show that the Tree Evaluation Problem with alphabet size $k$ and height $h$ can be solved by branching programs of size $k^{O(h/\log h)} + 2^{O(h)}$. This answers a longstanding challenge of Cook et al. (2009) and gives the first general upper bound since the problem's inception.

more >>>James Cook, Ian Mertz

The study of branching programs for the Tree Evaluation Problem, introduced by S. Cook et al. (TOCT 2012), remains one of the most promising approaches to separating L from P. Given a label in $[k]$ at each leaf of a complete binary tree and an explicit function in $[k]^2 \to ... more >>>

James Cook, Omid Etesami, Rachel Miller, Luca Trevisan

A function $f$ mapping $n$-bit strings to $m$-bit strings can be constructed from a bipartite graph with $n$ vertices on the left and $m$ vertices on the right having right-degree $d$ together with a predicate $P:\{0,1\}^d\rightarrow\{0,1\}$. The vertices on the left correspond to the bits of the input to the ... more >>>