7 research outputs found
Synthesising Graphical Theories
In recent years, diagrammatic languages have been shown to be a powerful and
expressive tool for reasoning about physical, logical, and semantic processes
represented as morphisms in a monoidal category. In particular, categorical
quantum mechanics, or "Quantum Picturalism", aims to turn concrete features of
quantum theory into abstract structural properties, expressed in the form of
diagrammatic identities. One way we search for these properties is to start
with a concrete model (e.g. a set of linear maps or finite relations) and start
composing generators into diagrams and looking for graphical identities.
Naively, we could automate this procedure by enumerating all diagrams up to a
given size and check for equalities, but this is intractable in practice
because it produces far too many equations. Luckily, many of these identities
are not primitive, but rather derivable from simpler ones. In 2010, Johansson,
Dixon, and Bundy developed a technique called conjecture synthesis for
automatically generating conjectured term equations to feed into an inductive
theorem prover. In this extended abstract, we adapt this technique to
diagrammatic theories, expressed as graph rewrite systems, and demonstrate its
application by synthesising a graphical theory for studying entangled quantum
states.Comment: 10 pages, 22 figures. Shortened and one theorem adde
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Enumerating molecules.
This report is a comprehensive review of the field of molecular enumeration from early isomer counting theories to evolutionary algorithms that design molecules in silico. The core of the review is a detail account on how molecules are counted, enumerated, and sampled. The practical applications of molecular enumeration are also reviewed for chemical information, structure elucidation, molecular design, and combinatorial library design purposes. This review is to appear as a chapter in Reviews in Computational Chemistry volume 21 edited by Kenny B. Lipkowitz
Generation of Graph Classes with Efficient Isomorph Rejection
In this thesis, efficient isomorph-free generation of graph classes with the method of
generation by canonical construction path(GCCP) is discussed. The method GCCP
has been invented by McKay in the 1980s. It is a general method to recursively generate
combinatorial objects avoiding isomorphic copies. In the introduction chapter, the
method of GCCP is discussed and is compared to other well-known methods of generation.
The generation of the class of quartic graphs is used as an example to explain
this method. Quartic graphs are simple regular graphs of degree four. The programs,
we developed based on GCCP, generate quartic graphs with 18 vertices more than two
times as efficiently as the well-known software GENREG does.
This thesis also demonstrates how the class of principal graph pairs can be generated
exhaustively in an efficient way using the method of GCCP. The definition and
importance of principal graph pairs come from the theory of subfactors where each
subfactor can be modelled as a principal graph pair. The theory of subfactors has
applications in the theory of von Neumann algebras, operator algebras, quantum algebras
and Knot theory as well as in design of quantum computers. While it was
initially expected that the classification at index 3 + √5 would be very complicated,
using GCCP to exhaustively generate principal graph pairs was critical in completing
the classification of small index subfactors to index 5¼.
The other set of classes of graphs considered in this thesis contains graphs without
a given set of cycles. For a given set of graphs, H, the Turán Number of H, ex(n,H),
is defined to be the maximum number of edges in a graph on n vertices without a
subgraph isomorphic to any graph in H. Denote by EX(n,H), the set of all extremal
graphs with respect to n and H, i.e., graphs with n vertices, ex(n,H) edges and no
subgraph isomorphic to any graph in H. We consider this problem when H is a set of
cycles. New results for ex(n, C) and EX(n, C) are introduced using a set of algorithms
based on the method of GCCP. Let K be an arbitrary subset of {C3, C4, C5, . . . , C32}.
For given n and a set of cycles, C, these algorithms can be used to calculate ex(n, C)
and extremal graphs in Ex(n, C) by recursively extending smaller graphs without any
cycle in C where C = K or C = {C3, C5, C7, . . .} ᴜ K and n≤64. These results are
considerably in excess of the previous results of the many researchers who worked on
similar problems. In the last chapter, a new class of canonical relabellings for graphs, hierarchical
canonical labelling, is introduced in which if the vertices of a graph, G, is canonically
labelled by {1, . . . , n}, then G\{n} is also canonically labelled. An efficient hierarchical
canonical labelling is presented and the application of this labelling in generation
of combinatorial objects is discussed
Efficient algorithms for listing unlabeled graphs
Available from British Library Document Supply Centre- DSC:4534.9203(EU-CSR--7-90) / BLDSC - British Library Document Supply CentreSIGLEGBUnited Kingdo
Efficient algorithms for listing unlabeled graphs
Available from British Library Document Supply Centre- DSC:4534.9203(EU-CSR--7-90) / BLDSC - British Library Document Supply CentreSIGLEGBUnited Kingdo