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79 Commits

Author SHA1 Message Date
fa326a147f
add deprecation warning 2019-07-10 23:32:49 +02:00
f1d0ff9c14 \lambda = 50.0 got certified! 2018-11-24 13:59:32 +01:00
8666cc6b2c last minute functional changes 2018-11-24 13:58:49 +01:00
9fc23c43d7 ignore python files 2018-11-17 22:44:50 +01:00
c075b93313 split definitions of action to a separate file 2018-11-17 22:43:42 +01:00
bf8fc67bb4 update check_positivity 2018-11-17 22:42:42 +01:00
523d783614 add check_positivity.jl 2018-11-06 08:31:37 +01:00
7b06dc1eb2 fix upper-bound in Settings 2018-09-17 00:22:35 +02:00
7dd061cdb7 fix typos 2018-09-16 17:53:40 +02:00
b45adc2e00 add external constructors for SAut i SL 2018-09-16 17:53:10 +02:00
8ee8eebd9e GAPGroups are trivial structs 2018-09-09 13:15:13 +02:00
4cd2212554 setup paths and logger using fullpath(::Settings) 2018-09-09 13:14:07 +02:00
b84b23d486 make solver a runtime choice 2018-09-09 13:13:21 +02:00
c7b8776e2d dispatch main ::PropertyTGroup 2018-09-09 13:12:53 +02:00
1b45625a71 create Property.Settings from PropertyTGroup and args 2018-09-09 13:09:43 +02:00
1b6793f37c parametrize PropertyTGroups types by N 2018-09-09 13:05:49 +02:00
8a2aa6542e if no "nosymmetry" in args, assume symmetrization 2018-09-06 22:45:16 +02:00
aaa36b2d5c fix MCG constructor 2018-09-06 22:43:28 +02:00
88fa1ded31 remove files for running particular groups
everything is handled by run.jl
2018-09-05 18:30:41 +02:00
b41f2ccdc8 add default log name 2018-09-05 18:27:18 +02:00
1ea396330c fix typo in MCG constructor 2018-09-05 18:26:51 +02:00
a1e4a917a6 include all groups in runtests.jl 2018-09-05 17:57:25 +02:00
385d2d4f5e selection of PropertyTGroup happens inside PropertyTGroups 2018-09-05 17:53:26 +02:00
e02b410240 update main 2018-09-05 17:52:23 +02:00
c8f58e1c93 one global run file (for all groups) 2018-09-05 17:51:43 +02:00
0a0856bb51 update groups to the new input format 2018-09-05 17:48:59 +02:00
df4b63a6a6 export group names 2018-08-20 03:28:09 +02:00
01405cb3cc Group <-> Ring dispatch problem is solved when generating Laplacian 2018-08-20 03:27:25 +02:00
6abb2c4186 dispatch method of main is decided based on G 2018-08-20 03:24:49 +02:00
0cae4882f1 replace Standard -→ Naive (methods of build problem) 2018-08-20 03:22:16 +02:00
707efaca3a move the information on solvers 2018-08-20 02:55:19 +02:00
57e9e3c404 use external logging 2018-08-20 02:54:50 +02:00
000069961b fix SCSSolver params to the most stable ones 2018-08-18 23:25:47 +02:00
ab49755863 add tests! 2018-08-18 23:25:20 +02:00
b5b9917536 add generatingset(::MatSpace,...) 2018-08-18 23:24:57 +02:00
95066b9c56 fix the action of WreathProdElem on MatElem
mul! is not correct for NmodRing??
2018-08-18 23:23:52 +02:00
42f915b91d fix use of Nemo's FiniteField 2018-08-15 17:20:25 +02:00
1e54e25672 pass the right Id element for rings 2018-08-15 17:20:02 +02:00
45d8d85189 add perm action from the right (by inverse) on MatElems 2018-08-15 17:19:37 +02:00
7257c02820 rename SymmetricGroup → SymmetrizedGroup 2018-08-15 17:19:28 +02:00
2c7bd86418 Merge branch 'fix/new_type_system' of git.wmi.amu.edu.pl:kalmar/GroupsWithPropertyT into fix/new_type_system 2018-08-08 19:43:12 +02:00
97f64fdee3 small fixes to GAPGroups 2018-08-08 19:38:22 +02:00
87658ea46e separate solver selection 2018-08-08 19:37:57 +02:00
3d59ecc874 update FPgroup.jl to PropertyTGroup interface 2018-08-08 19:37:28 +02:00
16bc76e950 update FPgroup.jl to PropertyTGroup interface 2018-08-08 00:37:08 +02:00
5ea5093f42 remove main_gapgroup.jl 2018-08-08 00:30:06 +02:00
7f4824492c initial adjust of GAPGroups files 2018-08-08 00:29:22 +02:00
02825a8ec0 dispatch main on SymmetricGroup, GAPGroup 2018-08-08 00:28:15 +02:00
34066a7128 global usings 2018-08-08 00:26:39 +02:00
52b0e598c6 use Allgroups.jl, separate params and summarize functions 2018-08-08 00:26:15 +02:00
fd29784a00 more efficient actions of autS 2018-08-08 00:24:17 +02:00
1d82aeed8d implement the SymmetricGroup interface for SLn and AutFn 2018-08-08 00:21:13 +02:00
7dc95cf93a add PropertyTGroup, SymmetricGroup, GAPGroup hierarchy 2018-08-08 00:17:44 +02:00
788da99d82 separate set_parallel_mthread(N::Int, workers::Bool) 2018-08-08 00:16:05 +02:00
6f0f1d5cc6 mv MCGn.jl to FPgroup.jl 2018-07-31 16:55:36 +02:00
3859a26c22 generalize MCG to all FP GAP groups 2018-07-31 16:54:35 +02:00
b5cf3946a2 move MCG.jl to MCGn.jl 2018-07-31 16:44:41 +02:00
8a2d6086cb use AbstractAlgebra in mcg 2018-07-31 15:52:16 +02:00
0d2f15c887 remove the unused Orbit.jl 2018-07-31 15:51:20 +02:00
80f181da11 make ignores more specific 2018-07-31 12:43:45 +02:00
b23a98d23e update to AbstractAlgebra 2018-07-31 12:42:58 +02:00
df940c8c07 rework starting scripts 2018-07-31 12:41:48 +02:00
6acdff2b70 increase O.maxstates to 1000*$maxeqns 2018-03-22 11:41:35 +01:00
ed12d87b7a Add tools to compute in mapping class groups 2018-03-22 11:09:03 +01:00
e0d3cb607b standardise usage of GROUP modules 2018-03-22 11:07:52 +01:00
10cbd4c6f3 use CPUselect.jl 2018-03-22 11:05:36 +01:00
4c26db6ee8 standardise interface and fix comments 2018-03-22 11:03:11 +01:00
98b2417ae1 move specific group-related code to groups/... 2018-03-22 10:59:27 +01:00
a75ee1f645 use --warmstart option 2018-01-07 16:12:51 +01:00
2ae11d9f33 indentation and cosmetics 2018-01-04 22:09:53 +01:00
76ff889366 update for enh/rework-logging of PropertyT 2018-01-04 22:09:13 +01:00
cb16c78690 rewrite 2018-01-04 22:08:32 +01:00
9cc5bcfc00 rename parsed_args 2018-01-04 10:50:26 +01:00
961f04b964 integrate with SLNs.jl 2018-01-04 10:49:49 +01:00
98df554624 add leading "o" to groupname in Orbit.jl 2018-01-04 10:48:54 +01:00
476d532d93 use CPUSelect 2018-01-04 10:47:00 +01:00
122ebe92bf save GAP output to GAP.log 2018-01-04 10:46:40 +01:00
86cadd0440 set alpha=1.95, acceleration_lookback=1 for numerical stability 2017-12-30 00:26:36 +01:00
403d7f7921 add MCG code 2017-12-29 17:45:09 +01:00
23 changed files with 1206 additions and 657 deletions

17
.gitignore vendored
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@ -1,8 +1,15 @@
Articles
Higman
MCG*
notebooks
Oldies
oSAutF*
oSL*
SAutF*
SL*_*
*ipynb* *ipynb*
*.gws *.gws
*/*.jld
*/*.log
Old*
.* .*
o* tests*
*test* *.py
*.pyc

118
AutFN.jl
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@ -1,118 +0,0 @@
using ArgParse
using Nemo
import SCS.SCSSolver
using PropertyT
using Groups
Nemo.setpermstyle(:cycles)
#=
Note that the element
α(i,j,k) = ϱ(i,j)*ϱ(i,k)*inv(ϱ(i,j))*inv(ϱ(i,k)),
which surely belongs to ball of radius 4 in Aut(F₄) becomes trivial under the representation
Aut(F₄) GL₄()ℤ⁴ GL₅().
Moreover, due to work of Potapchik and Rapinchuk [1] every real representation of Aut(Fₙ) into GLₘ() (for m 2n-2) factors through GLₙ()ℤⁿ, so will have the same problem.
We need a different approach: Here we actually compute in Aut(𝔽₄)
=#
###############################################################################
#
# Generating set
#
###############################################################################
function SOutFN_generating_set(N::Int)
SOutFN = AutGroup(FreeGroup(N), special=true, outer=true)
S = gens(SOutFN);
S = [S; [inv(s) for s in S]]
return SOutFN, unique(S)
end
###############################################################################
#
# Parsing command line
#
###############################################################################
function cpuinfo_physicalcores()
maxcore = -1
for line in eachline("/proc/cpuinfo")
if startswith(line, "core id")
maxcore = max(maxcore, parse(Int, split(line, ':')[2]))
end
end
maxcore < 0 && error("failure to read core ids from /proc/cpuinfo")
return maxcore + 1
end
function parse_commandline()
s = ArgParseSettings()
@add_arg_table s begin
"--tol"
help = "set numerical tolerance for the SDP solver"
arg_type = Float64
default = 1e-6
"--iterations"
help = "set maximal number of iterations for the SDP solver"
arg_type = Int
default = 20000
"--upper-bound"
help = "Set an upper bound for the spectral gap"
arg_type = Float64
default = Inf
"--cpus"
help = "Set number of cpus used by solver (default: auto)"
arg_type = Int
required = false
"-N"
help = "Consider automorphisms of free group on N generators"
arg_type = Int
default = 2
end
return parse_args(s)
end
function main()
parsed_args = parse_commandline()
if parsed_args["cpus"] nothing
if parsed_args["cpus"] > cpuinfo_physicalcores()
warn("Number of specified cores exceeds the physical core cound. Performance will suffer.")
end
Blas.set_num_threads(parsed_args["cpus"])
end
N = parsed_args["N"]
radius = 2
tol = parsed_args["tol"]
iterations = parsed_args["iterations"]
upper_bound = parsed_args["upper-bound"]
dirname = "SOutF$(N)_$(upper_bound)_r=$radius"
logger = PropertyT.setup_logging(dirname)
info(logger, "Group: $dirname")
info(logger, "Iterations: $iterations")
info(logger, "Precision: $tol")
info(logger, "Upper bound: $upper_bound")
G, S = SOutFN_generating_set(N)
info(logger, G)
info(logger, "Symmetric generating set of size $(length(S))")
info(logger, S)
Id = G()
solver = SCSSolver(eps=tol, max_iters=iterations, linearsolver=SCS.Direct)
@time PropertyT.check_property_T(dirname, S, Id, solver, upper_bound, tol, 2)
return 0
end
main()

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@ -1,54 +0,0 @@
using ArgParse
###############################################################################
#
# Parsing command line
#
###############################################################################
function parse_commandline()
settings = ArgParseSettings()
@add_arg_table settings begin
"--tol"
help = "set numerical tolerance for the SDP solver"
arg_type = Float64
default = 1e-14
"--iterations"
help = "set maximal number of iterations for the SDP solver (default: 20000)"
arg_type = Int
default = 60000
"--upper-bound"
help = "Set an upper bound for the spectral gap"
arg_type = Float64
default = Inf
"--cpus"
help = "Set number of cpus used by solver (default: auto)"
arg_type = Int
required = false
"-N"
help = "Consider automorphisms of free group on N generators"
arg_type = Int
default = 2
"--radius"
help = "Radius of ball B_r(e,S) to find solution over"
arg_type = Int
default = 2
"--warmstart"
help = "Use warmstart.jl as the initial guess for SCS"
action = :store_true
end
return parse_args(settings)
end
parsed_args = parse_commandline()
include("CPUselect.jl")
set_parallel_mthread(parsed_args, workers=true)
include("SAutFNs.jl")
include("Orbit.jl")
main(SAutFNs, parsed_args)

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@ -9,16 +9,10 @@ function cpuinfo_physicalcores()
return maxcore + 1 return maxcore + 1
end end
function set_parallel_mthread(parsed_args; workers=false) function set_parallel_mthread(N::Int, workers::Bool)
if parsed_args["cpus"] == nothing
N = cpuinfo_physicalcores()
else
N = parsed_args["cpus"]
if N > cpuinfo_physicalcores() if N > cpuinfo_physicalcores()
warn("Number of specified cores exceeds the physical core count. Performance may suffer.") warn("Number of specified cores exceeds the physical core count. Performance may suffer.")
end end
end
if workers if workers
addprocs(N) addprocs(N)
@ -28,5 +22,14 @@ function set_parallel_mthread(parsed_args; workers=false)
BLAS.set_num_threads(N) BLAS.set_num_threads(N)
info("Using $N threads in BLAS.") info("Using $N threads in BLAS.")
return N end
function set_parallel_mthread(parsed_args::Dict; workers=false)
if parsed_args["cpus"] == nothing
N = cpuinfo_physicalcores()
else
N = parsed_args["cpus"]
end
set_parallel_mthread(N, workers)
end end

157
FPGroups_GAP.jl Normal file
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@ -0,0 +1,157 @@
using JLD
function GAP_code(group_code, dir, R; maxeqns=10_000, infolevel=2)
code = """
LogTo("$(dir)/GAP.log");
RequirePackage("kbmag");
SetInfoLevel(InfoRWS, $infolevel);
MetricBalls := function(rws, R)
local l, basis, sizes, i;
l := EnumerateReducedWords(rws, 0, R);;
SortBy(l, Length);
sizes := [1..R];
Apply(sizes, i -> Number(l, w -> Length(w) <= i));
return [l, sizes];
end;;
ProductMatrix := function(rws, basis, len)
local result, dict, g, tmpList, t;
result := [];
dict := NewDictionary(basis[1], true);
t := Runtime();
for g in [1..Length(basis)] do;
AddDictionary(dict, basis[g], g);
od;
Print("Creating dictionary: \t\t", StringTime(Runtime()-t), "\\n");
for g in basis{[1..len]} do;
tmpList := List(Inverse(g)*basis{[1..len]}, w->ReducedForm(rws, w));
#t := Runtime();
tmpList := List(tmpList, x -> LookupDictionary(dict, x));
#Print(Runtime()-t, "\\n");
Assert(1, ForAll(tmpList, x -> x <> fail));
Add(result, tmpList);
od;
return result;
end;;
SaveCSV := function(fname, pm)
local file, i, j, k;
file := OutputTextFile(fname, false);;
for i in pm do;
k := 1;
for j in i do;
if k < Length(i) then
AppendTo(file, j, ", ");
else
AppendTo(file, j, "\\n");
fi;
k := k+1;
od;
od;
CloseStream(file);
end;;
$group_code
# G:= SimplifiedFpGroup(G);
RWS := KBMAGRewritingSystem(G);
# ResetRewritingSystem(RWS);
O:=OptionsRecordOfKBMAGRewritingSystem(RWS);;
O.maxeqns := $maxeqns;
O.maxstates := 1000*$maxeqns;
#O.maxstoredlen := [100,100];
before := Runtimes();;
KnuthBendix(RWS);
after := Runtimes();;
delta := after.user_time_children - before.user_time_children;;
Print("Knuth-Bendix completion: \t", StringTime(delta), "\\n");
t := Runtime();
res := MetricBalls(RWS,$(2*R));;
Print("Metric-Balls generation: \t", StringTime(Runtime()-t), "\\n");
B := res[1];; sizes := res[2];;
Print("Sizes of generated Balls: \t", sizes, "\\n");
t := Runtime();
pm := ProductMatrix(RWS, B, sizes[$R]);;
Print("Computing ProductMatrix: \t", StringTime(Runtime()-t), "\\n");
S := EnumerateReducedWords(RWS, 1, 1);
S := List(S, s -> Position(B,s));
SaveCSV("$(dir)/pm.csv", pm);
SaveCSV("$(dir)/S.csv", [S]);
SaveCSV("$(dir)/sizes.csv", [sizes]);
Print("DONE!\\n");
quit;""";
return code
end
function GAP_groupcode(S, rels)
F = "FreeGroup("*join(["\"$v\""for v in S], ", ") *");"
m = match(r".*(\[.*\])$", string(rels))
rels = replace(m.captures[1], " ", "\n")
code = """
F := $F;
AssignGeneratorVariables(F);;
relations := $rels;;
G := F/relations;
"""
return code
end
function GAP_execute(gap_code, dir)
isdir(dir) || mkdir(dir)
GAP_file = joinpath(dir, "GAP_code.g")
@show dir
@show GAP_file;
open(GAP_file, "w") do io
write(io, gap_code)
end
run(pipeline(`cat $(GAP_file)`, `gap -q`))
end
function prepare_pm_delta_csv(name, group_code, R; maxeqns=10_000, infolevel=2)
info("Preparing multiplication table using GAP (via kbmag)")
gap_code = GAP_code(group_code, name, R, maxeqns=maxeqns, infolevel=infolevel)
GAP_execute(gap_code, name)
end
function prepare_pm_delta(name, group_code, R; maxeqns=100_000, infolevel=2)
pm_fname = joinpath(name, "pm.csv")
S_fname = joinpath(name, "S.csv")
sizes_fname = joinpath(name, "sizes.csv")
delta_fname = joinpath(name, "delta.jld")
if !isfile(pm_fname) || !isfile(S_fname) || !isfile(sizes_fname)
prepare_pm_delta_csv(name, group_code, R, maxeqns=maxeqns, infolevel=infolevel)
end
if isfile(sizes_fname)
sizes = readcsv(sizes_fname, Int)[1,:]
if 2R > length(sizes)
prepare_pm_delta_csv(name, group_code, R, maxeqns=maxeqns, infolevel=infolevel)
end
end
pm = readcsv(pm_fname, Int)
S = readcsv(S_fname, Int)[1,:]
sizes = readcsv(sizes_fname, Int)[1,:]
Δ = spzeros(sizes[2R])
Δ[S] .= -1
Δ[1] = length(S)
pm = pm[1:sizes[R], 1:sizes[R]]
save(joinpath(name, "pm.jld"), "pm", pm)
save(joinpath(name, "delta.jld"), "Δ", Δ)
end

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@ -1,53 +0,0 @@
using SCS.SCSSolver
# using Mosek
# using CSDP
# using SDPA
using Nemo
using PropertyT
using Groups
function main(GROUP, parsed_args)
radius = parsed_args["radius"]
tol = parsed_args["tol"]
iterations = parsed_args["iterations"]
upper_bound = parsed_args["upper-bound"]
warm = parsed_args["warmstart"]
name, N = GROUP.groupname(parsed_args)
G, S = GROUP.generatingset(parsed_args)
autS = GROUP.autS(parsed_args)
name = "$(name)_r$radius"
isdir(name) || mkdir(name)
logger = PropertyT.setup_logging(joinpath(name, "$(upper_bound)"))
info(logger, "Group: $name")
info(logger, "Iterations: $iterations")
info(logger, "Precision: $tol")
info(logger, "Upper bound: $upper_bound")
info(logger, "Threads: $(Threads.nthreads())")
info(logger, "Workers: $(workers())")
info(logger, G)
info(logger, "Symmetric generating set of size $(length(S))")
# info(logger, S)
solver = SCSSolver(eps=tol, max_iters=iterations, linearsolver=SCS.Direct, alpha=1.9, acceleration_lookback=1)
# solver = Mosek.MosekSolver(
# MSK_DPAR_INTPNT_CO_TOL_REL_GAP=tol,
# MSK_IPAR_INTPNT_MAX_ITERATIONS=iterations,
# QUIET=false)
# solver = CSDP.CSDPSolver(axtol=tol, atytol=tol, objtol=tol, minstepp=tol*10.0^-1, minstepd=tol*10.0^-1)
# solver = SDPA.SDPASolver(epsilonStar=tol, epsilonDash=tol)
sett = Settings(name, N, G, S, autS, radius, solver, upper_bound, tol, warm)
PropertyT.check_property_T(sett)
end

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@ -1,4 +1,10 @@
This repository contains code for computations in [Certifying Numerical Estimates of Spectral Gaps](https://arxiv.org/abs/1703.09680). # DEPRECATED!
This repository has not been updated for a while!
If You are interested in replicating results for [1712.07167](https://arxiv.org/abs/1712.07167) please check [these instruction](https://kalmar.faculty.wmi.amu.edu.pl/post/1712.07176/)
Also [this notebook](https://nbviewer.jupyter.org/gist/kalmarek/03510181bc1e7c98615e86e1ec580b2a) could be of some help. If everything else fails the [zenodo dataset](https://zenodo.org/record/1133440) should contain the last-resort instructions.
This repository contains some legacy code for computations in [Certifying Numerical Estimates of Spectral Gaps](https://arxiv.org/abs/1703.09680).
# Installing # Installing
To run the code You need `julia-v0.5` (should work on `v0.6`, but with warnings). To run the code You need `julia-v0.5` (should work on `v0.6`, but with warnings).

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@ -1,80 +0,0 @@
module SAutFNs
using Nemo
using Groups
if VERSION >= v"0.6.0"
import Nemo.Generic.perm
end
###############################################################################
#
# Generating set
#
###############################################################################
function generatingset(N::Int)
SAutFN = AutGroup(FreeGroup(N), special=true)
S = gens(SAutFN);
S = [S; [inv(s) for s in S]]
return SAutFN, unique(S)
end
function generatingset(parsed_args)
N = parsed_args["N"]
return generatingset(N)
end
###############################################################################
#
# Action of WreathProductElems on AutGroupElem
#
###############################################################################
function AutFG_emb(A::AutGroup, g::WreathProductElem)
isa(A.objectGroup, FreeGroup) || throw("Not an Aut(Fₙ)")
parent(g).P.n == length(A.objectGroup.gens) || throw("No natural embedding of $(parent(g)) into $A")
powers = [(elt == parent(elt)() ? 0: 1) for elt in g.n.elts]
elt = reduce(*, [A(Groups.flip_autsymbol(i))^pow for (i,pow) in enumerate(powers)])
Groups.r_multiply!(elt, [Groups.perm_autsymbol(g.p)])
return elt
end
function AutFG_emb(A::AutGroup, p::perm)
isa(A.objectGroup, FreeGroup) || throw("Not an Aut(Fₙ)")
parent(p).n == length(A.objectGroup.gens) || throw("No natural embedding of $(parent(g)) into $A")
return A(Groups.perm_autsymbol(p))
end
function (g::WreathProductElem)(a::AutGroupElem)
g = AutFG_emb(parent(a),g)
return g*a*g^-1
end
function (p::perm)(a::AutGroupElem)
g = AutFG_emb(parent(a),p)
return g*a*g^-1
end
autS(N::Int) = WreathProduct(PermutationGroup(2), PermutationGroup(N))
function autS(parsed_args)
return autS(parsed_args["N"])
end
###############################################################################
#
# Misc
#
###############################################################################
function groupname(parsed_args)
N = parsed_args["N"]
return groupname(N), N
end
groupname(N::Int) = "oSAutF$(N)"
end # of module SAutFNs

151
SL.jl
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@ -1,151 +0,0 @@
using ArgParse
using Nemo
import SCS.SCSSolver
using PropertyT
###############################################################################
#
# Generating set
#
###############################################################################
function E(i::Int, j::Int, M::MatSpace, val=one(M.base_ring))
@assert i≠j
m = one(M)
m[i,j] = val
return m
end
function SLsize(n,p)
result = BigInt(1)
for k in 0:n-1
result *= p^n - p^k
end
return div(result, p-1)
end
function SL_generatingset(n::Int, X::Bool=false)
indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
G = MatrixSpace(ZZ, n, n)
if X
S = [E(i,j,G,v) for (i,j) in indexing for v in [1, 100]]
else
S = [E(i,j,G,v) for (i,j) in indexing for v in [1]]
end
S = vcat(S, [inv(x) for x in S])
return G, unique(S)
end
function SL_generatingset(n::Int, p::Int, X::Bool=false)
p == 0 && return SL_generatingset(n, X)
(p > 1 && n > 1) || throw("Both n and p should be positive integers!")
info("Size(SL($n,$p)) = $(SLsize(n,p))")
F = ResidueRing(ZZ, p)
G = MatrixSpace(F, n, n)
indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
S = [E(i, j, G) for (i,j) in indexing]
S = vcat(S, [inv(x) for x in S])
return G, unique(S)
end
###############################################################################
#
# Parsing command line
#
###############################################################################
function cpuinfo_physicalcores()
maxcore = -1
for line in eachline("/proc/cpuinfo")
if startswith(line, "core id")
maxcore = max(maxcore, parse(Int, split(line, ':')[2]))
end
end
maxcore < 0 && error("failure to read core ids from /proc/cpuinfo")
return maxcore + 1
end
function parse_commandline()
settings = ArgParseSettings()
@add_arg_table settings begin
"--tol"
help = "set numerical tolerance for the SDP solver"
arg_type = Float64
default = 1e-6
"--iterations"
help = "set maximal number of iterations for the SDP solver (default: 20000)"
arg_type = Int
default = 50000
"--upper-bound"
help = "Set an upper bound for the spectral gap"
arg_type = Float64
default = Inf
"--cpus"
help = "Set number of cpus used by solver"
arg_type = Int
required = false
"-N"
help = "Consider elementary matrices EL(N)"
arg_type = Int
default = 2
"-p"
help = "Matrices over field of p-elements (p=0 => over ZZ)"
arg_type = Int
default = 0
"--radius"
help = "Radius of ball B_r(e,S) to find solution over"
arg_type = Int
default = 2
end
return parse_args(settings)
end
function main()
parsed_args = parse_commandline()
if parsed_args["cpus"] nothing
if parsed_args["cpus"] > cpuinfo_physicalcores()
warn("Number of specified cores exceeds the physical core cound. Performance will suffer.")
end
BLAS.set_num_threads(parsed_args["cpus"])
end
N = parsed_args["N"]
p = parsed_args["p"]
if p == 0
dirname = "SL$(N)Z"
else
dirname = "SL$(N)_$p"
end
  radius = parsed_args["radius"]
tol = parsed_args["tol"]
iterations = parsed_args["iterations"]
upper_bound = parsed_args["upper-bound"]
dirname = "$(dirname)_$(upper_bound)_r=$radius"
logger = PropertyT.setup_logging(dirname)
info(logger, "Group: $dirname")
info(logger, "Iterations: $iterations")
info(logger, "Precision: $tol")
info(logger, "Upper bound: $upper_bound")
G, S = SL_generatingset(N, p)
info(logger, G)
info(logger, "Symmetric generating set of size $(length(S))")
Id = one(G)
solver = SCSSolver(eps=tol, max_iters=iterations, linearsolver=SCS.Direct)
@time PropertyT.check_property_T(dirname, S, Id, solver, upper_bound, tol, radius)
return 0
end
main()

124
SLNs.jl
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module SLNs
using Nemo
using Groups
if VERSION >= v"0.6.0"
import Nemo.Generic.perm
end
###############################################################################
#
# Generating set
#
###############################################################################
function E(i::Int, j::Int, M::MatSpace, val=one(M.base_ring))
@assert i≠j
m = one(M)
m[i,j] = val
return m
end
function SLsize(n,p)
result = BigInt(1)
for k in 0:n-1
result *= p^n - p^k
end
return div(result, p-1)
end
function generatingset(n::Int, X::Bool=false)
indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
G = MatrixSpace(ZZ, n, n)
if X
S = [E(i,j,G,v) for (i,j) in indexing for v in [1, 100]]
else
S = [E(i,j,G,v) for (i,j) in indexing for v in [1]]
end
S = vcat(S, [inv(x) for x in S])
return G, unique(S)
end
function generatingset(n::Int, p::Int, X::Bool=false)
(p > 1 && n > 1) || throw("Both n and p should be positive integers!")
info("Size(SL($n,$p)) = $(SLsize(n,p))")
F = ResidueRing(ZZ, p)
G = MatrixSpace(F, n, n)
indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
S = [E(i, j, G) for (i,j) in indexing]
S = vcat(S, [inv(x) for x in S])
return G, unique(S)
end
function generatingset(parsed_args)
N = parsed_args["N"]
p = parsed_args["p"]
X = parsed_args["X"]
if p == 0
return generatingset(N, X)
else
return generatingset(N, p, X)
end
end
###############################################################################
#
# Action of WreathProductElems on Nemo.MatElem
#
###############################################################################
function matrix_emb(MM::MatSpace, g::WreathProductElem)
parent(g).P.n == MM.cols == MM.rows || throw("No natural embedding of $(parent(g)) in ")
powers = [(elt == parent(elt)() ? 0: 1) for elt in g.n.elts]
elt = diagm([(-1)^(elt == parent(elt)() ? 0: 1) for elt in g.n.elts])
return MM(elt)*MM(Nemo.matrix_repr(g.p)')
end
function (g::WreathProductElem)(A::MatElem)
G = matrix_emb(parent(A), g)
inv_G = matrix_emb(parent(A), inv(g))
return G*A*inv_G
end
function (p::perm)(A::MatElem)
length(p.d) == A.r == A.c || throw("Can't act via $p on matrix of size ($(A.r), $(A.c))")
R = parent(A)
return p*A*inv(p)
end
autS(N::Int) = WreathProduct(PermutationGroup(2), PermutationGroup(N))
function autS(parsed_args)
return autS(parsed_args["N"])
end
###############################################################################
#
# Misc
#
###############################################################################
function groupname(parsed_args)
N = parsed_args["N"]
p = parsed_args["p"]
X = parsed_args["X"]
return groupname(N, p, X), N
end
function groupname(N, p=0, X=false)
if p == 0
if X
name = "oSL$(N)Z⟨X⟩"
else
name = "oSL$(N)Z"
end
else
name = "oSL$(N)_$p"
end
return name
end
end # of module SLNs

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using ArgParse
###############################################################################
#
# Parsing command line
#
###############################################################################
function parse_commandline()
settings = ArgParseSettings()
@add_arg_table settings begin
"--tol"
help = "set numerical tolerance for the SDP solver"
arg_type = Float64
default = 1e-14
"--iterations"
help = "set maximal number of iterations for the SDP solver (default: 20000)"
arg_type = Int
default = 60000
"--upper-bound"
help = "Set an upper bound for the spectral gap"
arg_type = Float64
default = Inf
"--cpus"
help = "Set number of cpus used by solver"
arg_type = Int
required = false
"-N"
help = "Consider elementary matrices EL(N)"
arg_type = Int
default = 2
"-p"
help = "Matrices over field of p-elements (p=0 => over ZZ)"
arg_type = Int
default = 0
"--radius"
help = "Radius of ball B_r(e,S) to find solution over"
arg_type = Int
default = 2
"-X"
help = "Consider EL(N, ZZ⟨X⟩)"
action = :store_true
"--warmstart"
help = "Use warmstart.jl as the initial guess for SCS"
action = :store_true
end
return parse_args(settings)
end
parsed_args = parse_commandline()
include("CPUselect.jl")
set_parallel_mthread(parsed_args, workers=true)
include("SLNs.jl")
include("Orbit.jl")
main(SLNs, parsed_args)

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module PropertyTGroups
using PropertyT
using AbstractAlgebra
using Nemo
using Groups
using GroupRings
export PropertyTGroup, SymmetrizedGroup, GAPGroup,
SpecialLinearGroup,
SpecialAutomorphismGroup,
HigmanGroup,
CapraceGroup,
MappingClassGroup
export PropertyTGroup
abstract type PropertyTGroup end
abstract type SymmetrizedGroup <: PropertyTGroup end
abstract type GAPGroup <: PropertyTGroup end
function PropertyTGroup(args)
if haskey(args, "SL")
G = PropertyTGroups.SpecialLinearGroup(args)
elseif haskey(args, "SAut")
G = PropertyTGroups.SpecialAutomorphismGroup(args)
elseif haskey(args, "MCG")
G = PropertyTGroups.MappingClassGroup(args)
elseif haskey(args, "Higman")
G = PropertyTGroups.HigmanGroup(args)
elseif haskey(args, "Caprace")
G = PropertyTGroups.CapraceGroup(args)
else
throw("You must provide one of --SL, --SAut, --MCG, --Higman, --Caprace")
end
return G
end
include("autfreegroup.jl")
include("speciallinear.jl")
Comm(x,y) = x*y*x^-1*y^-1
function generatingset(G::GAPGroup)
S = gens(group(G))
return unique([S; inv.(S)])
end
include("mappingclassgroup.jl")
include("higman.jl")
include("caprace.jl")
include("actions.jl")
end # of module PropertyTGroups

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function (p::perm)(A::GroupRingElem)
RG = parent(A)
result = zero(RG, eltype(A.coeffs))
for (idx, c) in enumerate(A.coeffs)
if c!= zero(eltype(A.coeffs))
result[p(RG.basis[idx])] = c
end
end
return result
end
###############################################################################
#
# Action of WreathProductElems on Nemo.MatElem
#
###############################################################################
function matrix_emb(n::DirectProductGroupElem, p::perm)
Id = parent(n.elts[1])()
elt = diagm([(-1)^(el == Id ? 0 : 1) for el in n.elts])
return elt[:, p.d]
end
function (g::WreathProductElem)(A::MatElem)
g_inv = inv(g)
G = matrix_emb(g.n, g_inv.p)
G_inv = matrix_emb(g_inv.n, g.p)
M = parent(A)
return M(G)*A*M(G_inv)
end
import Base.*
doc"""
*(x::AbstractAlgebra.MatElem, P::Generic.perm)
> Apply the pemutation $P$ to the rows of the matrix $x$ and return the result.
"""
function *(x::AbstractAlgebra.MatElem, P::Generic.perm)
z = similar(x)
m = rows(x)
n = cols(x)
for i = 1:m
for j = 1:n
z[i, j] = x[i,P[j]]
end
end
return z
end
function (p::perm)(A::MatElem)
length(p.d) == A.r == A.c || throw("Can't act via $p on matrix of size ($(A.r), $(A.c))")
return p*A*inv(p)
end
###############################################################################
#
# Action of WreathProductElems on AutGroupElem
#
###############################################################################
function AutFG_emb(A::AutGroup, g::WreathProductElem)
isa(A.objectGroup, FreeGroup) || throw("Not an Aut(Fₙ)")
parent(g).P.n == length(A.objectGroup.gens) || throw("No natural embedding of $(parent(g)) into $A")
elt = A()
Id = parent(g.n.elts[1])()
flips = Groups.AutSymbol[Groups.flip_autsymbol(i) for i in 1:length(g.p.d) if g.n.elts[i] != Id]
Groups.r_multiply!(elt, flips, reduced=false)
Groups.r_multiply!(elt, [Groups.perm_autsymbol(g.p)])
return elt
end
function AutFG_emb(A::AutGroup, p::perm)
isa(A.objectGroup, FreeGroup) || throw("Not an Aut(Fₙ)")
parent(p).n == length(A.objectGroup.gens) || throw("No natural embedding of $(parent(p)) into $A")
return A(Groups.perm_autsymbol(p))
end
function (g::WreathProductElem)(a::Groups.Automorphism)
A = parent(a)
g = AutFG_emb(A,g)
res = A()
Groups.r_multiply!(res, g.symbols, reduced=false)
Groups.r_multiply!(res, a.symbols, reduced=false)
Groups.r_multiply!(res, [inv(s) for s in reverse!(g.symbols)])
return res
end
function (p::perm)(a::Groups.Automorphism)
g = AutFG_emb(parent(a),p)
return g*a*inv(g)
end

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struct SpecialAutomorphismGroup{N} <: SymmetrizedGroup
group::AutGroup
end
function SpecialAutomorphismGroup(args::Dict)
N = args["SAut"]
return SpecialAutomorphismGroup{N}(AutGroup(FreeGroup(N), special=true))
end
name(G::SpecialAutomorphismGroup{N}) where N = "SAutF$(N)"
group(G::SpecialAutomorphismGroup) = G.group
function generatingset(G::SpecialAutomorphismGroup)
S = gens(group(G));
return unique([S; inv.(S)])
end
function autS(G::SpecialAutomorphismGroup{N}) where N
return WreathProduct(PermutationGroup(2), PermutationGroup(N))
end

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struct CapraceGroup <: GAPGroup end
name(G::CapraceGroup) = "CapraceGroup"
function group(G::CapraceGroup)
caprace_group = Groups.FPGroup(["x","y","z","t","r"])
x,y,z,t,r = gens(caprace_group)
relations = [
x^7,
y^7,
t^2,
r^73,
t*r*t*r,
Comm(x,y)*z^-1,
Comm(x,z),
Comm(y,z),
Comm(x^2*y*z^-1, t),
Comm(x*y*z^3, t*r),
Comm(x^3*y*z^2, t*r^17),
Comm(x, t*r^-34),
Comm(y, t*r^-32),
Comm(z, t*r^-29),
Comm(x^-2*y*z, t*r^-25),
Comm(x^-1*y*z^-3, t*r^-19),
Comm(x^-3*y*z^-2, t*r^-11)
];
relations = [relations; [inv(rel) for rel in relations]]
Groups.add_rels!(caprace_group, Dict(rel => caprace_group() for rel in relations))
return caprace_group
end

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groups/higman.jl Normal file
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struct HigmanGroup <: GAPGroup end
name(G::HigmanGroup) = "HigmanGroup"
function group(G::HigmanGroup)
higman_group = Groups.FPGroup(["a","b","c","d"]);
a,b,c,d = gens(higman_group)
relations = [
b*Comm(b,a),
c*Comm(c,b),
d*Comm(d,c),
a*Comm(a,d)
];
relations = [relations; [inv(rel) for rel in relations]]
Groups.add_rels!(higman_group, Dict(rel => higman_group() for rel in relations))
return higman_group
end

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struct MappingClassGroup{N} <: GAPGroup end
MappingClassGroup(args::Dict) = MappingClassGroup{args["MCG"]}()
name(G::MappingClassGroup{N}) where N = "MCG(N)"
function group(G::MappingClassGroup{N}) where N
if N < 2
throw("Genus must be at least 2!")
elseif N == 2
MCGroup = Groups.FPGroup(["a1","a2","a3","a4","a5"]);
S = gens(MCGroup)
n = length(S)
A = prod(reverse(S))*prod(S)
relations = [
[Comm(S[i], S[j]) for i in 1:n for j in 1:n if abs(i-j) > 1]...,
[S[i]*S[i+1]*S[i]*inv(S[i+1]*S[i]*S[i+1]) for i in 1:G.n-1]...,
(S[1]*S[2]*S[3])^4*inv(S[5])^2,
Comm(A, S[1]),
A^2
]
relations = [relations; [inv(rel) for rel in relations]]
Groups.add_rels!(MCGroup, Dict(rel => MCGroup() for rel in relations))
return MCGroup
else
MCGroup = Groups.FPGroup(["a$i" for i in 0:2N])
S = gens(MCGroup)
a0 = S[1]
A = S[2:end]
k = length(A)
relations = [
[Comm(A[i], A[j]) for i in 1:k for j in 1:k if abs(i-j) > 1]...,
[Comm(a0, A[i]) for i in 1:k if i != 4]...,
[A[i]*A[(i+1)]*A[i]*inv(A[i+1]*A[i]*A[i+1]) for i in 1:k-1]...,
A[4]*a0*A[4]*inv(a0*A[4]*a0)
]
# 3-chain relation
c = prod(reverse(A[1:4]))*prod(A[1:4])
b0 = c*a0*inv(c)
push!(relations, (A[1]*A[2]*A[3])^4*inv(a0*b0))
# Lantern relation
b1 = inv(A[4]*A[5]*A[3]*A[4])*a0*(A[4]*A[5]*A[3]*A[4])
b2 = inv(A[2]*A[3]*A[1]*A[2])*b1*(A[2]*A[3]*A[1]*A[2])
u = inv(A[6]*A[5])*b1*(A[6]*A[5])
x = prod(reverse(A[2:6]))*u*prod(inv.(A[1:4]))
b3 = x*a0*inv(x)
push!(relations, a0*b2*b1*inv(A[1]*A[3]*A[5]*b3))
# Hyperelliptic relation
X = prod(reverse(A))*prod(A)
function n(i::Int, b=b0)
if i == 1
return A[1]
elseif i == 2
return b
else
return w(i-2)*n(i-2)*w(i-2)
end
end
function w(i::Int)
(A[2i+4]*A[2i+3]*A[2i+2]* n(i+1))*(A[2i+1]*A[2i] *A[2i+2]*A[2i+1])*
(A[2i+3]*A[2i+2]*A[2i+4]*A[2i+3])*( n(i+1)*A[2i+2]*A[2i+1]*A[2i] )
end
# push!(relations, X*n(N)*inv(n(N)*X))
relations = [relations; [inv(rel) for rel in relations]]
Groups.add_rels!(MCGroup, Dict(rel => MCGroup() for rel in relations))
return MCGroup
end
end

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struct SpecialLinearGroup{N} <: SymmetrizedGroup
group::AbstractAlgebra.Group
p::Int
X::Bool
end
function SpecialLinearGroup(args::Dict)
N = args["SL"]
p = args["p"]
X = args["X"]
if p == 0
G = MatrixSpace(Nemo.ZZ, N, N)
else
R = Nemo.NmodRing(UInt(p))
G = MatrixSpace(R, N, N)
end
return SpecialLinearGroup{N}(G, p, X)
end
function name(G::SpecialLinearGroup{N}) where N
if G.p == 0
R = (G.X ? "Z[x]" : "Z")
else
R = "F$(G.p)"
end
return SL($(G.N),$R)
end
group(G::SpecialLinearGroup) = G.group
function generatingset(G::SpecialLinearGroup{N}) where N
G.p > 0 && G.X && throw("SL(n, F_p[x]) not implemented")
SL = group(G)
return generatingset(SL, G.X)
end
# r is the injectivity radius of
# SL(n, Z[X]) -> SL(n, Z) induced by X -> 100
function generatingset(SL::MatSpace, X::Bool=false, r=5)
n = SL.cols
indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
if !X
S = [E(idx[1],idx[2],SL) for idx in indexing]
else
S = [E(i,j,SL,v) for (i,j) in indexing for v in [1, 100*r]]
end
return unique([S; inv.(S)])
end
function E(i::Int, j::Int, M::MatSpace, val=one(M.base_ring))
@assert i≠j
m = one(M)
m[i,j] = val
return m
end
function autS(G::SpecialLinearGroup{N}) where N
return WreathProduct(PermutationGroup(2), PermutationGroup(N))
end

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using Memento
function setup_logging(filename::String, handlername::Symbol=:log)
isdir(dirname(filename)) || mkdir(dirname(filename))
logger = Memento.config!("info", fmt="{date}| {msg}")
handler = DefaultHandler(filename, DefaultFormatter("{date}| {msg}"))
logger.handlers[String(handlername)] = handler
return logger
end
macro logtime(logger, ex)
quote
local stats = Base.gc_num()
local elapsedtime = Base.time_ns()
local val = $(esc(ex))
elapsedtime = Base.time_ns() - elapsedtime
local diff = Base.GC_Diff(Base.gc_num(), stats)
local ts = time_string(elapsedtime,
diff.allocd,
diff.total_time,
Base.gc_alloc_count(diff)
)
$(esc(info))($(esc(logger)), ts)
val
end
end
function time_string(elapsedtime, bytes, gctime, allocs)
str = @sprintf("%10.6f seconds", elapsedtime/1e9)
if bytes != 0 || allocs != 0
bytes, mb = Base.prettyprint_getunits(bytes, length(Base._mem_units), Int64(1024))
allocs, ma = Base.prettyprint_getunits(allocs, length(Base._cnt_units), Int64(1000))
if ma == 1
str*= @sprintf(" (%d%s allocation%s: ", allocs, Base._cnt_units[ma], allocs==1 ? "" : "s")
else
str*= @sprintf(" (%.2f%s allocations: ", allocs, Base._cnt_units[ma])
end
if mb == 1
str*= @sprintf("%d %s%s", bytes, Base._mem_units[mb], bytes==1 ? "" : "s")
else
str*= @sprintf("%.3f %s", bytes, Base._mem_units[mb])
end
if gctime > 0
str*= @sprintf(", %.2f%% gc time", 100*gctime/elapsedtime)
end
str*=")"
elseif gctime > 0
str*= @sprintf(", %.2f%% gc time", 100*gctime/elapsedtime)
end
return str
end
import Base: info, @time
Base.info(x) = info(getlogger(), x)
macro time(x)
return :(@logtime(getlogger(Main), $(esc(x))))
end

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using PropertyT
include("FPGroups_GAP.jl")
include("groups/Allgroups.jl")
using PropertyTGroups
import PropertyT.Settings
function summarize(sett::PropertyT.Settings)
info("Threads: $(Threads.nthreads())")
info("Workers: $(workers())")
info("GroupDir: $(PropertyT.prepath(sett))")
info(string(sett.G))
info("with generating set of size $(length(sett.S))")
info("Radius: $(sett.radius)")
info("Precision: $(sett.tol)")
info("Upper bound: $(sett.upper_bound)")
info("Solver: $(sett.solver)")
end
function Settings(Gr::PropertyTGroup, args, solver)
r = get(args, "radius", 2)
gr_name = PropertyTGroups.name(Gr)*"_r$r"
G = PropertyTGroups.group(Gr)
S = PropertyTGroups.generatingset(Gr)
sol = solver
ub = get(args,"upper-bound", Inf)
tol = get(args,"tol", 1e-10)
ws = get(args, "warmstart", false)
if get(args, "nosymmetry", false)
return PropertyT.Settings(gr_name, G, S, r, sol, ub, tol, ws)
else
autS = PropertyTGroups.autS(Gr)
return PropertyT.Settings(gr_name, G, S, r, sol, ub, tol, ws, autS)
end
end
function main(::PropertyTGroup, sett::PropertyT.Settings)
isdir(PropertyT.fullpath(sett)) || mkpath(PropertyT.fullpath(sett))
summarize(sett)
return PropertyT.check_property_T(sett)
end
function main(::GAPGroup, sett::PropertyT.Settings)
isdir(PropertyT.fullpath(sett)) || mkpath(PropertyT.fullpath(sett))
summarize(sett)
S = [s for s in sett.S if s.symbols[1].pow == 1]
relations = [k*inv(v) for (k,v) in sett.G.rels]
prepare_pm_delta(PropertyT.prepath(sett), GAP_groupcode(S, relations), sett.radius)
return PropertyT.check_property_T(sett)
end

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using AbstractAlgebra
using Groups
using GroupRings
using PropertyT
using SCS
solver(tol, iterations) =
SCSSolver(linearsolver=SCS.Direct,
eps=tol, max_iters=iterations,
alpha=1.95, acceleration_lookback=1)
include("../main.jl")
using PropertyTGroups
args = Dict("SAut" => 5, "upper-bound" => 50.0, "radius" => 2, "nosymmetry"=>false, "tol"=>1e-12, "iterations" =>200000, "warmstart" => true)
Gr = PropertyTGroups.PropertyTGroup(args)
sett = PropertyT.Settings(Gr, args,
solver(args["tol"], args["iterations"]))
@show sett
fullpath = PropertyT.fullpath(sett)
isdir(fullpath) || mkpath(fullpath)
# setup_logging(PropertyT.filename(fullpath, :fulllog), :fulllog)
function small_generating_set(RG::GroupRing{AutGroup{N}}, n) where N
indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
rmuls = [Groups.rmul_autsymbol(i,j) for (i,j) in indexing]
lmuls = [Groups.lmul_autsymbol(i,j) for (i,j) in indexing]
gen_set = RG.group.([rmuls; lmuls])
return [gen_set; inv.(gen_set)]
end
function computeX(RG::GroupRing{AutGroup{N}}) where N
Tn = small_generating_set(RG, N-1)
= Int64
Δn = length(Tn)*one(RG, ) - RG(Tn, );
Alt_N = [g for g in elements(PermutationGroup(N)) if parity(g) == 0]
@time X = sum(σ(Δn)*sum(τ(Δn) for τ Alt_N if τ σ) for σ in Alt_N);
return X
end
function Sq(RG::GroupRing{AutGroup{N}}) where N
T2 = small_generating_set(RG, 2)
= Int64
Δ₂ = length(T2)*one(RG, ) - RG(T2, );
Alt_N = [g for g in elements(PermutationGroup(N)) if parity(g) == 0]
elt = sum(σ(Δ₂)^2 for σ in Alt_N)
return elt
end
function Adj(RG::GroupRing{AutGroup{N}}) where N
T2 = small_generating_set(RG, 2)
= Int64
Δ₂ = length(T2)*one(RG, ) - RG(T2, );
Alt_N = [g for g in elements(PermutationGroup(N)) if parity(g) == 0]
adj(σ::perm, τ::perm, i=1, j=2) = Set([σ[i], σ[j]]) Set([τ[i], τ[j]])
adj(σ::perm) = [τ for τ in Alt_N if length(adj(σ, τ)) == 1]
@time elt = sum(σ(Δ₂)*sum(τ(Δ₂) for τ in adj(σ)) for σ in Alt_N);
# return RG(elt.coeffs÷factorial(N-2)^2)
return elt
end
function Op(RG::GroupRing{AutGroup{N}}) where N
T2 = small_generating_set(RG, 2)
= Int64
Δ₂ = length(T2)*one(RG, ) - RG(T2, );
Alt_N = [g for g in elements(PermutationGroup(N)) if parity(g) == 0]
adj(σ::perm, τ::perm, i=1, j=2) = Set([σ[i], σ[j]]) Set([τ[i], τ[j]])
adj(σ::perm) = [τ for τ in Alt_N if length(adj(σ, τ)) == 0]
@time elt = sum(σ(Δ₂)*sum(τ(Δ₂) for τ in adj(σ)) for σ in Alt_N);
# return RG(elt.coeffs÷factorial(N-2)^2)
return elt
end
const ELT_FILE = joinpath(dirname(PropertyT.filename(sett, )), "SqAdjOp_coeffs.jld")
const WARMSTART_FILE = PropertyT.filename(sett, :warmstart)
if isfile(PropertyT.filename(sett,)) && isfile(ELT_FILE) &&
isfile(PropertyT.filename(sett, :OrbitData))
# cached
Δ = PropertyT.loadGRElem(PropertyT.filename(sett,), sett.G)
RG = parent(Δ)
orbit_data = load(PropertyT.filename(sett, :OrbitData), "OrbitData")
sq_c, adj_c, op_c = load(ELT_FILE, "Sq", "Adj", "Op")
# elt = ELT_FILE, sett.G)
sq = GroupRingElem(sq_c, RG)
adj = GroupRingElem(adj_c, RG)
op = GroupRingElem(op_c, RG);
else
info("Compute Laplacian")
Δ = PropertyT.Laplacian(sett.S, sett.radius)
RG = parent(Δ)
info("Compute Sq, Adj, Op")
@time sq, adj, op = Sq(RG), Adj(RG), Op(RG)
PropertyT.saveGRElem(PropertyT.filename(sett, ), Δ)
save(ELT_FILE, "Sq", sq.coeffs, "Adj", adj.coeffs, "Op", op.coeffs)
info("Compute OrbitData")
if !isfile(PropertyT.filename(sett, :OrbitData))
orbit_data = PropertyT.OrbitData(parent(Y), sett.autS)
save(PropertyT.filename(sett, :OrbitData), "OrbitData", orbit_data)
else
orbit_data = load(PropertyT.filename(sett, :OrbitData), "OrbitData")
end
end;
orbit_data = PropertyT.decimate(orbit_data);
elt = adj+2op;
const SOLUTION_FILE = PropertyT.filename(sett, :solution)
if !isfile(SOLUTION_FILE)
SDP_problem, varλ, varP = PropertyT.SOS_problem(elt, Δ, orbit_data; upper_bound=sett.upper_bound)
begin
using SCS
scs_solver = SCS.SCSSolver(linearsolver=SCS.Direct,
eps=sett.tol,
max_iters=args["iterations"],
alpha=1.95,
acceleration_lookback=1)
JuMP.setsolver(SDP_problem, scs_solver)
end
λ = Ps = nothing
ws = PropertyT.warmstart(sett)
# using ProgressMeter
# @showprogress "Running SDP optimization step... " for i in 1:args["repetitions"]
while true
λ, Ps, ws = PropertyT.solve(PropertyT.filename(sett, :solverlog),
SDP_problem, varλ, varP, ws);
if all((!isnan).(ws[1]))
save(WARMSTART_FILE, "warmstart", ws, "λ", λ, "Ps", Ps)
save(WARMSTART_FILE[1:end-4]*"_$(now()).jld", "warmstart", ws, "λ", λ, "Ps", Ps)
else
warn("No valid solution was saved!")
end
end
info("Reconstructing P...")
@time P = PropertyT.reconstruct(Ps, orbit_data);
save(SOLUTION_FILE, "λ", λ, "P", P)
end
P, λ = load(SOLUTION_FILE, "P", "λ")
@show λ;
@time const Q = real(sqrtm(P));
function SOS_residual(eoi::GroupRingElem, Q::Matrix)
RG = parent(eoi)
@time sos = PropertyT.compute_SOS(RG, Q);
return eoi - sos
end
info("Floating Point arithmetic:")
EOI = elt - λ*Δ
b = SOS_residual(EOI, Q)
@show norm(b, 1);
info("Interval arithmetic:")
using IntervalArithmetic
Qint = PropertyT.augIdproj(Q);
@assert all([zero(eltype(Q)) in sum(view(Qint, :, i)) for i in 1:size(Qint, 2)])
EOI_int = elt - @interval(λ)*Δ;
Q_int = PropertyT.augIdproj(Q);
@assert all([zero(eltype(Q)) in sum(view(Q_int, :, i)) for i in 1:size(Q_int, 2)])
b_int = SOS_residual(EOI_int, Q_int)
@show norm(b_int, 1);
info("λ is certified to be > ", (@interval(λ) - 2^2*norm(b_int,1)).lo)

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using ArgParse
###############################################################################
#
# Parsing command line
#
###############################################################################
function parse_commandline()
settings = ArgParseSettings()
@add_arg_table settings begin
"--tol"
help = "set numerical tolerance for the SDP solver"
arg_type = Float64
default = 1e-6
"--iterations"
help = "set maximal number of iterations for the SDP solver"
arg_type = Int
default = 50000
"--upper-bound"
help = "Set an upper bound for the spectral gap"
arg_type = Float64
default = Inf
"--cpus"
help = "Set number of cpus used by solver"
arg_type = Int
required = false
"--radius"
help = "Radius of ball B_r(e,S) to find solution over"
arg_type = Int
default = 2
"--warmstart"
help = "Use warmstart.jld as the initial guess for SCS"
action = :store_true
"--nosymmetry"
help = "Don't use symmetries of the Laplacian"
action = :store_true
"--SL "
help = "GROUP: the group generated by elementary matrices of size n by n"
arg_type = Int
required = false
"-p"
help = "Matrices over field of p-elements (p=0 => over ZZ) [only with --SL]"
arg_type = Int
default = 0
"-X"
help = "Consider EL(N, ZZ⟨X⟩) [only with --SL]"
action = :store_true
"--SAut"
help = "GROUP: the automorphisms group of the free group on N generators"
arg_type = Int
required = false
"--MCG"
help = "GROUP: mapping class group of surface of genus N"
arg_type = Int
required = false
"--Higman"
help = "GROUP: the Higman Group"
action = :store_true
"--Caprace"
help = "GROUP: for Caprace Group"
action = :store_true
end
return parse_args(settings)
end
const PARSEDARGS = parse_commandline()
set_parallel_mthread(PARSEDARGS, workers=false)
include("CPUselect.jl")
include("logging.jl")
include("main.jl")
using SCS.SCSSolver
# using Mosek
# using CSDP
# using SDPA
solver(tol, iterations) =
SCSSolver(linearsolver=SCS.Direct,
eps=tol, max_iters=iterations,
alpha=1.95, acceleration_lookback=1)
# Mosek.MosekSolver(
# MSK_DPAR_INTPNT_CO_TOL_REL_GAP=tol,
# MSK_IPAR_INTPNT_MAX_ITERATIONS=iterations,
# QUIET=false)
# CSDP.CSDPSolver(axtol=tol, atytol=tol, objtol=tol, minstepp=tol*10.0^-1, minstepd=tol*10.0^-1)
# SDPA.SDPASolver(epsilonStar=tol, epsilonDash=tol)
const Gr = PropertyTGroups.PropertyTGroup(PARSEDARGS)
const sett = PropertyT.Settings(Gr, PARSEDARGS,
solver(PARSEDARGS["tol"], PARSEDARGS["iterations"]))
fullpath = PropertyT.fullpath(sett)
isdir(fullpath) || mkpath(fullpath)
setup_logging(PropertyT.filename(fullpath, :fulllog), :fulllog)
main(Gr, sett)

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using Base.Test
include("main.jl")
testdir = "tests_"*string(now())
mkdir(testdir)
include("logging.jl")
logger=setup_logging(joinpath(testdir, "tests.log"))
info(testdir)
cd(testdir)
# groupname = name(G)
# ub = PARSEDARGS["upper-bound"]
#
# fullpath = joinpath(groupname, string(ub))
# isdir(fullpath) || mkpath(fullpath)
separator(n=60) = info("\n"*("\n"*"="^n*"\n"^3)*"\n")
function SL_tests(args)
args["SL"] = 2
args["p"] = 3
G = PropertyTGroup(args)
@test main(G) == true
separator()
let args = args
args["SL"] = 2
args["p"] = 5
G = PropertyTGroup(args)
@test main(G) == false
separator()
args["warmstart"] = true
G = PropertyTGroup(args)
@test main(G) == false
separator()
args["upper-bound"] = 0.1
G = PropertyTGroup(args)
@test main(G) == true
separator()
end
args["SL"] = 2
args["p"] = 7
G = PropertyTGroup(args)
@test main(G) == false
separator()
args["SL"] = 3
args["p"] = 7
G = PropertyTGroup(args)
@test main(G) == true
separator()
# begin
# args["iterations"] = 25000
# args["N"] = 3
# args["p"] = 0
# args["upper-bound"] = Inf
#
# G = PropertyTGroups.SpecialLinearGroup(args)
# @test main(G) == false
# separator()
#
# args["warmstart"] = false
# args["upper-bound"] = 0.27
# G = PropertyTGroups.SpecialLinearGroup(args)
# @test main(G) == false
# separator()
#
# args["warmstart"] = true
# G = PropertyTGroups.SpecialLinearGroup(args)
# @test main(G) == true
# separator()
# end
return 0
end
function SAut_tests(args)
G = PropertyTGroup(args)
@test main(G) == false
separator()
args["warmstart"] = true
G = PropertyTGroup(args)
@test main(G) == false
separator()
args["upper-bound"] = 0.1
G = PropertyTGroup(args)
@test main(G) == false
separator()
return 0
end
@testset "Groups with(out) (T)" begin
@testset "GAPGroups" begin
args = Dict(
"Higman" => true,
"iterations"=>5000,
"tol"=>1e-7,
"upper-bound"=>Inf,
"cpus"=>2,
"radius"=>2,
"warmstart"=>false,
"nosymmetry"=>true,
)
G = PropertyTGroup(args)
@test main(G) == false
args = Dict(
"Caprace" => true,
"iterations"=>5000,
"tol"=>1e-7,
"upper-bound"=>Inf,
"cpus"=>2,
"radius"=>2,
"warmstart"=>false,
"nosymmetry"=>true,
)
G = PropertyTGroup(args)
@test main(G) == false
args = Dict(
"MCG" => 3,
"iterations"=>5000,
"tol"=>1e-7,
"upper-bound"=>Inf,
"cpus"=>2,
"radius"=>2,
"warmstart"=>false,
"nosymmetry"=>true,
)
G = PropertyTGroup(args)
@test main(G) == false
end
@testset "SLn's" begin
@testset "Non-Symmetrized" begin
args = Dict(
"SL" => 2,
"p" => 3,
"X" => false,
"iterations"=>50000,
"tol"=>1e-7,
"upper-bound"=>Inf,
"cpus"=>2,
"radius"=>2,
"warmstart"=>false,
"nosymmetry"=>true,
)
SL_tests(args)
end
@testset "Symmetrized" begin
args = Dict(
"SL" => 2,
"p" => 3,
"X" => false,
"iterations"=>20000,
"tol"=>1e-7,
"upper-bound"=>Inf,
"cpus"=>2,
"radius"=>2,
"warmstart"=>false,
"nosymmetry"=>false,
)
SL_tests(args)
end
end
@testset "SAutF_n's" begin
@testset "Non-Symmetrized" begin
args = Dict(
"SAut" => 2,
"iterations"=>5000,
"tol"=>1e-7,
"upper-bound"=>Inf,
"cpus"=>2,
"radius"=>2,
"warmstart"=>false,
"nosymmetry"=>true,
)
SAut_tests(args)
end
@testset "Symmetrized" begin
args = Dict(
"SAut" => 3,
"iterations"=>500,
"tol"=>1e-7,
"upper-bound"=>Inf,
"cpus"=>2,
"radius"=>2,
"warmstart"=>false,
"nosymmetry"=>false,
)
SAut_tests(args)
end
end
end