5 research outputs found
Busy beavers gone wild
We show some incompleteness results a la Chaitin using the busy beaver
functions. Then, with the help of ordinal logics, we show how to obtain a
theory in which the values of the busy beaver functions can be provably
established and use this to reveal a structure on the provability of the values
of these functions
Recovery from Non-Decomposable Distance Oracles
A line of work has looked at the problem of recovering an input from distance
queries. In this setting, there is an unknown sequence , and one chooses a set of queries and
receives for a distance function . The goal is to make as few
queries as possible to recover . Although this problem is well-studied for
decomposable distances, i.e., distances of the form for some function , which includes the important cases of
Hamming distance, -norms, and -estimators, to the best of our
knowledge this problem has not been studied for non-decomposable distances, for
which there are important special cases such as edit distance, dynamic time
warping (DTW), Frechet distance, earth mover's distance, and so on. We initiate
the study and develop a general framework for such distances. Interestingly,
for some distances such as DTW or Frechet, exact recovery of the sequence
is provably impossible, and so we show by allowing the characters in to be
drawn from a slightly larger alphabet this then becomes possible. In a number
of cases we obtain optimal or near-optimal query complexity. We also study the
role of adaptivity for a number of different distance functions. One motivation
for understanding non-adaptivity is that the query sequence can be fixed and
the distances of the input to the queries provide a non-linear embedding of the
input, which can be used in downstream applications involving, e.g., neural
networks for natural language processing.Comment: This work has been presented at conference The 14th Innovations in
Theoretical Computer Science (ITCS 2023) and accepted for publishing in the
journal IEEE Transactions on Information Theor
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum