GroupsWithPropertyT/SL.jl

using ArgParse
using GroupAlgebras
using PropertyT

using Mods
import Primes: isprime

import SCS.SCSSolver

function SL_generatingset(n::Int)

    indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]

    S = [E(i,j,N=n) for (i,j) in indexing];
    S = vcat(S, [convert(Array{Int,2},x') for x in S]);
    S = vcat(S, [convert(Array{Int,2},inv(x)) for x in S]);
    return unique(S)
end

function E(i::Int, j::Int; val=1, N::Int=3, mod=Inf)
   @assert i≠j
   m = eye(Int, N)
   m[i,j] = val
   if mod == Inf
       return m
   else
       return [Mod(x,mod) for x in m]
   end
end

function cofactor(i,j,M)
    z1 = ones(Bool,size(M,1))
    z1[i] = false

    z2 = ones(Bool,size(M,2))
    z2[j] = false

    return M[z1,z2]
end

import Base.LinAlg.det

function det(M::Array{Mod,2})
    if size(M,1) ≠ size(M,2)
        d = Mod(0,M[1,1].mod)
    elseif size(M,1) == 2
        d =  M[1,1]*M[2,2] - M[1,2]*M[2,1]
    else
        d = zero(eltype(M))
        for i in 1:size(M,1)
            d += (-1)^(i+1)*M[i,1]*det(cofactor(i,1,M))
        end
    end
#     @show (M, d)
    return d
end

function adjugate(M)
    K = similar(M)
    for i in 1:size(M,1), j in 1:size(M,2)
        K[j,i] = (-1)^(i+j)*det(cofactor(i,j,M))
    end
    return K
end

import Base: inv, one, zero, *

one(::Type{Mod}) = 1
zero(::Type{Mod}) = 0
zero(x::Mod) = Mod(x.mod)

function inv(M::Array{Mod,2})
    d = det(M)
    d ≠ 0*d || thow(ArgumentError("Matrix is not invertible!"))
    return inv(det(M))*adjugate(M)
    return adjugate(M)
end

function SL_generatingset(n::Int, p::Int)
    p == 0 && return SL_generatingset(n)
    (p > 1 && n > 0) || throw(ArgumentError("Both n and p should be positive integers!"))
    isprime(p) || throw(ArgumentError("p should be a prime number!"))

    indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
    S = [E(i,j, N=n, mod=p) for (i,j) in indexing]
    S = vcat(S, [inv(s) for s in S])
    S = vcat(S, [permutedims(x, [2,1]) for x in S]);
    return unique(S)
end

function products{T}(U::AbstractVector{T}, V::AbstractVector{T})
    result = Vector{T}()
    for u in U
        for v in V
            push!(result, u*v)
        end
    end
    return unique(result)
end

function ΔandSDPconstraints{T<:Number}(identity::Array{T,2}, S::Vector{Array{T,2}})
    B₁ = vcat([identity], S)
    B₂ = products(B₁, B₁);
    B₃ = products(B₁, B₂);
    B₄ = products(B₁, B₃);
    @assert B₄[1:length(B₂)] == B₂

    product_matrix = PropertyT.create_product_matrix(B₄,length(B₂));
    sdp_constraints = PropertyT.constraints_from_pm(product_matrix, length(B₄))
    L_coeff = PropertyT.splaplacian_coeff(S, B₂, length(B₄));
    Δ = GroupAlgebraElement(L_coeff, product_matrix)

    return Δ, sdp_constraints
end

ID(n::Int) = eye(Int, n)

function ID(n::Int, p::Int)
    p==0 && return ID(n)
    return [Mod(x,p) for x in eye(Int,N)]
end


#=
To use file property(T).jl (specifically: check_property_T function)
You need to define:

function ΔandSDPconstraints(identity, S):: (Δ, sdp_constraints)

=#

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-9
        "--iterations"
            help = "set maximal number of iterations for the SDP solver"
            arg_type = Int
            default = 100000
        "--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 matrices of size N"
            arg_type = Int
            default = 3
        "-p"
            help = "Matrices over filed of p-elements (0 = over ZZ)"
            arg_type = Int
            default = 0
    end

    return parse_args(s)
end

function main()
    parsed_args = parse_commandline()
    println("Parsed args:")

    # SL(3,Z)
    # upper_bound = 0.28-1e-5
    # tol = 1e-12
    # iterations = 500000

    # SL(4,Z)
    # upper_bound = 1.31
    # tol = 3e-11

    # upper_bound=0.738 # (N,p) = (3,7)

    tol = parsed_args["tol"]
    iterations = parsed_args["iterations"]

    # solver = MosekSolver(INTPNT_CO_TOL_REL_GAP=tol, QUIET=false)
    solver = SCSSolver(eps=tol, max_iters=iterations, verbose=true)

    N = parsed_args["N"]
    upper_bound = parsed_args["upper-bound"]
    p = parsed_args["p"]
    if p == 0
        name = "SL$(N)Z"
    else
        name = "SL$(N)_p"
    end

    S() = SL_generatingset(N, p)

    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
    @time PropertyT.check_property_T(name, ID(N,p), S, solver, upper_bound, tol)
end

main()