364 research outputs found
Automated theory formation in pure mathematics
The automation of specific mathematical tasks such as theorem proving and algebraic
manipulation have been much researched. However, there have only been a few isolated
attempts to automate the whole theory formation process. Such a process involves
forming new concepts, performing calculations, making conjectures, proving theorems
and finding counterexamples. Previous programs which perform theory formation are
limited in their functionality and their generality. We introduce the HR program
which implements a new model for theory formation. This model involves a cycle of
mathematical activity, whereby concepts are formed, conjectures about the concepts
are made and attempts to settle the conjectures are undertaken.HR has seven general production rules for producing a new concept from old ones and
employs a best first search by building new concepts from the most interesting old
ones. To enable this, HR has various measures which estimate the interestingness of a
concept. During concept formation, HR uses empirical evidence to suggest conjectures
and employs the Otter theorem prover to attempt to prove a given conjecture. If this
fails, HR will invoke the MACE model generator to attempt to disprove the conjecture
by finding a counterexample. Information and new knowledge arising from the attempt
to settle a conjecture is used to assess the concepts involved in the conjecture, which
fuels the heuristic search and closes the cycle.The main aim of the project has been to develop our model of theory formation and
to implement this in HR. To describe the project in the thesis, we first motivate
the problem of automated theory formation and survey the literature in this area.
We then discuss how HR invents concepts, makes and settles conjectures and how
it assesses the concepts and conjectures to facilitate a heuristic search. We present
results to evaluate HR in terms of the quality of the theories it produces and the
effectiveness of its techniques. A secondary aim of the project has been to apply HR to
mathematical discovery and we discuss how HR has successfully invented new concepts
and conjectures in number theory
Discrete Bulk Reconstruction
According to the AdS/CFT correspondence, the geometries of certain spacetimes
are fully determined by quantum states that live on their boundaries -- indeed,
by the von Neumann entropies of portions of those boundary states. This work
investigates to what extent the geometries can be reconstructed from the
entropies in polynomial time. Bouland, Fefferman, and Vazirani (2019) argued
that the AdS/CFT map can be exponentially complex if one wants to reconstruct
regions such as the interiors of black holes. Our main result provides a sort
of converse: we show that, in the special case of a single 1D boundary, if the
input data consists of a list of entropies of contiguous boundary regions, and
if the entropies satisfy a single inequality called Strong Subadditivity, then
we can construct a graph model for the bulk in linear time. Moreover, the bulk
graph is planar, it has vertices (the information-theoretic minimum),
and it's ``universal,'' with only the edge weights depending on the specific
entropies in question. From a combinatorial perspective, our problem boils down
to an ``inverse'' of the famous min-cut problem: rather than being given a
graph and asked to find a min-cut, here we're given the values of min-cuts
separating various sets of vertices, and need to find a weighted undirected
graph consistent with those values. Our solution to this problem relies on the
notion of a ``bulkless'' graph, which might be of independent interest for
AdS/CFT. We also make initial progress on the case of multiple 1D boundaries --
where the boundaries could be connected via wormholes -- including an upper
bound of vertices whenever a planar bulk graph exists (thus putting
the problem into the complexity class ).Comment: 41 pages, 18 figures. Comments welcomed! v2: new corollaries 2.3 and
4.5 with more explicit discussions of computability, additional references
and discussio
Phase Transitions in Nonlinear Filtering
It has been established under very general conditions that the ergodic
properties of Markov processes are inherited by their conditional distributions
given partial information. While the existing theory provides a rather complete
picture of classical filtering models, many infinite-dimensional problems are
outside its scope. Far from being a technical issue, the infinite-dimensional
setting gives rise to surprising phenomena and new questions in filtering
theory. The aim of this paper is to discuss some elementary examples,
conjectures, and general theory that arise in this setting, and to highlight
connections with problems in statistical mechanics and ergodic theory. In
particular, we exhibit a simple example of a uniformly ergodic model in which
ergodicity of the filter undergoes a phase transition, and we develop some
qualitative understanding as to when such phenomena can and cannot occur. We
also discuss closely related problems in the setting of conditional Markov
random fields.Comment: 51 page
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