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TR21-054 | 14th April 2021
James Cook, Ian Mertz

#### Encodings and the Tree Evaluation Problem

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.

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TR21-053 | 13th April 2021
Jan Krajicek

#### Information in propositional proofs and algorithmic proof search

We study from the proof complexity perspective the (informal) proof search problem:
Is there an optimal way to search for propositional proofs?
We note that for any fixed proof system there exists a time-optimal proof search algorithm. Using classical proof complexity results about reflection principles we prove that a time-optimal ... more >>>

TR21-052 | 12th April 2021
Benny Applebaum, Oded Nir

#### Upslices, Downslices, and Secret-Sharing with Complexity of $1.5^n$

A secret-sharing scheme allows to distribute a secret $s$ among $n$ parties such that only some predefined authorized'' sets of parties can reconstruct the secret, and all other unauthorized'' sets learn nothing about $s$.
The collection of authorized/unauthorized sets can be captured by a monotone function $f:\{0,1\}^n\rightarrow \{0,1\}$.
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