Atomtomate / EquivalenceClassesConstructor.jl

Construction of minimal set of representatives from a given set, under a number of equivalence relations. A mapping back to the original set is also computed.

Equivalence Classes Constructor

This Project aims to provide a tool package that can construct equivalence classes from symmetry predicates over a given set or mappings between elements. The reason being that subsequent calculations may be sped up by only considering one representative of each equivalence class.

This code contains collection of methods that are able to construct mappings between full and reduced sets. Specifically, for any indices x, y in some given set vl the known symmetries are given as input in the form of predicates, i.e. p(x,y) ∈ {true, false} or in form of mappings m(x) = y. The latter greatly increases performance but may lead to unexpected behavior with non monotonic mappings¹. As result, the original set vl is supplemented with a reduced set vl_red and a mapping expand_vl which can be used to expand subsequent calculations over the reduced set back to the full one.

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Setup

Basic Usage

By default, the constructor of the EquivalenceClasses will do the heavy lifting. You should therefore expect this line of the code to use up substential amount of compute time for large inputs. The first step is to define a seg S and a relationship between elements of S.

using EquivalenceClassesConstructor

S = -4:3
m1 = Mapping(x -> [-x])
p1 = Predicate((x,y) -> x == -y)

In cases where a mapping is known, it should be prefered over predicates since it requires substantially less evaluation time. Once the relationships and grid are defined we can compute a minimal set of representatives from all equivalence classes of S under m1 or p1.

eqc = EquivalenceClasses(m1, S)
eqc_2 = EquivalenceClasses(p1, S)

Both computations should yield a minimal set and a mapping back to S. Ordering is not guaranteed, but can be forced by adding the optional parameter sorted=true to the EquivalenceClasses constructor.

Since the output is most likely part of a larger computation, the package provides some IO options. For example as fixed width human readable files.

open("eqc_out.txt", "w") do io
    write_fixed_width(io, eqc)
end;

Examples

VertexReducedGrid.jl constructs a minimal set for the input vertex of 'LadderDGA.jl'.

1: The reason being, that m(m(x)) may lie in vl but m(x)) may not, forcing the algorithm to terminate.