From 9c02c67c5b802bfe715513efe22e1e1f974cb369 Mon Sep 17 00:00:00 2001
From: Brian Doolittle <brian.d.doolittle@gmail.com>
Date: Mon, 4 Dec 2023 19:35:00 -0500
Subject: [PATCH 01/15] fixing broken test

---
 .../LocalPolytope/adjacency_decomposition.jl           | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

diff --git a/test/integration/LocalPolytope/adjacency_decomposition.jl b/test/integration/LocalPolytope/adjacency_decomposition.jl
index 57519eb..d0b3ce6 100644
--- a/test/integration/LocalPolytope/adjacency_decomposition.jl
+++ b/test/integration/LocalPolytope/adjacency_decomposition.jl
@@ -39,15 +39,15 @@ end
         skip = [BellGame([1 0 0 0;1 0 0 0;1 0 0 0;0 0 0 0],1)]
         dict = LocalPolytope.adjacency_decomposition(vertices, BG, scenario, skip_games=skip, dir=test_dir)
 
-        # positvity included, but is skipped in computation
-        @test collect(keys(dict)) == [
+        # non-negativity included, but is skipped in computation
+        @test issetequal(collect((keys(dict))), [
             [1 0 0 0;1 0 0 0;0 1 0 0;0 0 1 0],
             [1 0 0 0;0 1 0 0;0 0 1 0;0 0 0 1],
             [1 0 0 0;1 0 0 0;1 0 0 0;0 0 0 0],
             [2 0 0 0;1 1 1 0;0 2 0 0;0 0 1 1],
             [2 0 0 0;1 1 1 0;0 2 0 0;0 0 2 0],
             [1 1 0 0;1 0 1 0;0 1 1 0;0 0 0 1],
-        ]
+        ])
 
         @test dict[[1 0 0 0;1 0 0 0;1 0 0 0;0 0 0 0]]["skipped"]
 
@@ -63,14 +63,14 @@ end
 
         dict = LocalPolytope.adjacency_decomposition(vertices, BG, scenario, dir=test_dir)
 
-        @test collect(keys(dict)) == [
+        @test issetequal(collect(keys(dict)), [
             [1 0 0 0;1 0 0 0;0 1 0 0;0 0 1 0],
             [1 0 0 0;0 1 0 0;0 0 1 0;0 0 0 1],
             [1 0 0 0;1 0 0 0;1 0 0 0;0 0 0 0],
             [2 0 0 0;1 1 1 0;0 2 0 0;0 0 1 1],
             [2 0 0 0;1 1 1 0;0 2 0 0;0 0 2 0],
             [1 1 0 0;1 0 1 0;0 1 1 0;0 0 0 1]
-        ]
+        ])
 
         @test dict[[1 0 0 0;1 0 0 0;1 0 0 0;0 0 0 0]]["skipped"] == false
     end

From 510a8604c4e9355c430e252b89e9be1c13806aa2 Mon Sep 17 00:00:00 2001
From: Brian Doolittle <brian.d.doolittle@gmail.com>
Date: Tue, 5 Dec 2023 13:45:14 -0500
Subject: [PATCH 02/15] adding base linear nonclassicality function and test

---
 Project.toml                                  | 20 ++++++--
 src/LocalPolytope/LocalPolytope.jl            |  4 +-
 .../linear_nonclassicality_witnesses.jl       | 43 ++++++++++++++++
 .../linear_nonclassicality_witnesses.jl       | 50 +++++++++++++++++++
 4 files changed, 111 insertions(+), 6 deletions(-)
 create mode 100644 src/LocalPolytope/linear_nonclassicality_witnesses.jl
 create mode 100644 test/unit/LocalPolytope/linear_nonclassicality_witnesses.jl

diff --git a/Project.toml b/Project.toml
index aa10ed3..dc8146e 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,13 +1,15 @@
 name = "BellScenario"
 uuid = "4e58fcc1-4405-48af-b0f2-42df636d9190"
 authors = ["Brian Doolittle"]
-version = "0.1.2"
+version = "0.1.3"
 
 [deps]
 Combinatorics = "861a8166-3701-5b0c-9a16-15d98fcdc6aa"
 Convex = "f65535da-76fb-5f13-bab9-19810c17039a"
 Dates = "ade2ca70-3891-5945-98fb-dc099432e06a"
+HiGHS = "87dc4568-4c63-4d18-b0c0-bb2238e4078b"
 JSON = "682c06a0-de6a-54ab-a142-c8b1cf79cde6"
+JuMP = "4076af6c-e467-56ae-b986-b466b2749572"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 Polyhedra = "67491407-f73d-577b-9b50-8179a7c68029"
 QBase = "e52e8ede-12bf-4731-8af7-b01f6064cb11"
@@ -17,10 +19,18 @@ XPORTA = "8c143463-af6f-456f-8aed-72447cb569d2"
 
 [compat]
 Combinatorics = "1"
-Convex = "0.14"
+Convex = "0.15"
+HiGHS = "1"
 JSON = "0.21"
-Polyhedra = "0.6"
+JuMP = "1"
+Polyhedra = "0.7"
 QBase = "0.2"
-SCS = "0.7"
+SCS = "2"
 XPORTA = "0.1"
-julia = "1.4"
+julia = "1"
+
+[extras]
+Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
+
+[targets]
+test = ["Test"]
diff --git a/src/LocalPolytope/LocalPolytope.jl b/src/LocalPolytope/LocalPolytope.jl
index 53a5279..1d9f275 100644
--- a/src/LocalPolytope/LocalPolytope.jl
+++ b/src/LocalPolytope/LocalPolytope.jl
@@ -58,7 +58,8 @@ resources were used than anticipated.
 module LocalPolytope
 
 using LinearAlgebra, Combinatorics
-using XPORTA, Polyhedra
+using JuMP, HiGHS  # Linear Programming
+using XPORTA, Polyhedra  # Convex Polytope
 
 import Polyhedra: vrep
 
@@ -88,6 +89,7 @@ include("./facets.jl")
 include("./generators.jl")
 include("./adjacency_decomposition.jl")
 include("./utils.jl")
+include("./linear_nonclassicality_witnesses.jl")
 
 # legacy code
 include("../Legacy/LocalPolytope.jl")
diff --git a/src/LocalPolytope/linear_nonclassicality_witnesses.jl b/src/LocalPolytope/linear_nonclassicality_witnesses.jl
new file mode 100644
index 0000000..df73a69
--- /dev/null
+++ b/src/LocalPolytope/linear_nonclassicality_witnesses.jl
@@ -0,0 +1,43 @@
+"""
+    linear_nonclassicality_witness(
+        vertices :: Vector{Vector{T}} where T <: Real,
+        test_behavior :: Vector{<:Real};
+        verbose=false :: Bool
+    ) :: Vector{Float64}
+
+Implements a linear program that obtains a linear inequality that witnesses the nonclassciality of
+the provided `test_behavior` (see https://arxiv.org/abs/1303.2849 Eq. 19). A complete set of
+enumerated vertices are required as input. The optimized facet inequality is returned in vector
+form with the classical bound being appended to the end.
+"""
+function linear_nonclassicality_witness(
+    vertices :: Vector{Vector{T}} where T <: Real,
+    test_behavior :: Vector{<:Real};
+    verbose=false :: Bool
+) :: Vector{Float64}
+    dim_v = length(vertices[1]) + 1
+
+    # initializing model
+    model = Model(HiGHS.Optimizer)
+    set_attribute(model, MOI.Silent(), !verbose)
+
+    # adding variable for inequality
+    @variable(model, s[1:dim_v])
+
+    # adding constraints to model
+    for v in vertices
+        va = [v..., -1]
+        @constraint(model, sum(s.*va) <= 0)
+    end
+
+    @constraint(model, c, sum(s.*[test_behavior..., -1]) <= 1)
+
+    # defining the optimization objective
+    @objective(model, Max, sum(s.*[test_behavior..., -1]))
+
+    # optimizing
+    optimize!(model)
+
+    # return optimized linear inequality in vector form
+    return value.(s)
+end
diff --git a/test/unit/LocalPolytope/linear_nonclassicality_witnesses.jl b/test/unit/LocalPolytope/linear_nonclassicality_witnesses.jl
new file mode 100644
index 0000000..7794402
--- /dev/null
+++ b/test/unit/LocalPolytope/linear_nonclassicality_witnesses.jl
@@ -0,0 +1,50 @@
+using Test
+
+@testset "./src/LocalPolytope/linear_nonclassicality_witnesses.jl" begin
+
+using BellScenario
+
+@testset "linear_nonclassicality_witness" begin
+    @testset "CHSH Inequality Correlation Vertices" begin
+        
+        chsh_correlation_vertices = [
+            [-1, -1, -1, -1,  1,  1,  1,  1],
+            [-1, -1, -1,  1,  1, -1,  1, -1],
+            [-1, -1,  1, -1, -1,  1, -1,  1],
+            [-1, -1,  1,  1, -1, -1, -1, -1],
+            [-1,  1, -1, -1,  1,  1, -1, -1],
+            [-1,  1, -1,  1,  1, -1, -1,  1],
+            [-1,  1,  1, -1, -1,  1,  1, -1],
+            [-1,  1,  1,  1, -1, -1,  1,  1],
+            [ 1, -1, -1, -1, -1, -1,  1,  1],
+            [ 1, -1, -1,  1, -1,  1,  1, -1],
+            [ 1, -1,  1, -1,  1, -1, -1,  1],
+            [ 1, -1,  1,  1,  1,  1, -1, -1],
+            [ 1,  1, -1, -1, -1, -1, -1, -1],
+            [ 1,  1, -1,  1, -1,  1, -1,  1],
+            [ 1,  1,  1, -1,  1, -1,  1, -1],
+            [ 1,  1,  1,  1,  1,  1,  1,  1],
+        ]
+
+        pr_box_test_behavior = [0, 0, 0, 0, 1, 1, 1, -1]
+
+        facet_vec = LocalPolytope.linear_nonclassicality_witness(chsh_correlation_vertices, pr_box_test_behavior[:])
+
+        @test facet_vec ≈ [0, 0, 0, 0, 0.5, 0.5, 0.5, -0.5, 1]  # [A0, A1, B0, B1, AB00, AB01, AB10, AB11]
+    end
+    
+    @testset "CH Inequality Probability Vertices" begin
+        chsh_scenario = BipartiteNonSignaling(2, 2, 2, 2)
+
+        chsh_vertices = LocalPolytope.vertices(chsh_scenario, "non-signaling")
+
+        pr_box_test_behavior = [1, 1, 1, 1, 1, 1, 1, 0] / 2
+
+        facet_vec = LocalPolytope.linear_nonclassicality_witness(chsh_vertices, pr_box_test_behavior[:])
+
+        # -2*PA(0|0) - 2*PB(0|0) + 2*PAB(00|00) + 2*PAB(01|00) + 2*PAB(10|00) - 2*PAB(11|00) <= 0
+        @test facet_vec ≈ [-2, 0, -2, 0, 2, 2, 2, -2, 0]
+    end
+end
+
+end
\ No newline at end of file

From 0e177bafe3f16769edbc77afeb472b8a5ed00674 Mon Sep 17 00:00:00 2001
From: Brian Doolittle <brian.d.doolittle@gmail.com>
Date: Tue, 5 Dec 2023 13:50:46 -0500
Subject: [PATCH 03/15] test dependencies

---
 test/Project.toml | 15 ---------------
 1 file changed, 15 deletions(-)
 delete mode 100644 test/Project.toml

diff --git a/test/Project.toml b/test/Project.toml
deleted file mode 100644
index 911d5f0..0000000
--- a/test/Project.toml
+++ /dev/null
@@ -1,15 +0,0 @@
-[deps]
-JSON = "682c06a0-de6a-54ab-a142-c8b1cf79cde6"
-LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
-Polyhedra = "67491407-f73d-577b-9b50-8179a7c68029"
-QBase = "e52e8ede-12bf-4731-8af7-b01f6064cb11"
-SafeTestsets = "1bc83da4-3b8d-516f-aca4-4fe02f6d838f"
-Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
-XPORTA = "8c143463-af6f-456f-8aed-72447cb569d2"
-
-[compat]
-JSON = "0.21"
-Polyhedra = "0.6"
-QBase = "0.2"
-XPORTA = "0.1"
-julia = "1.4"

From 05a898f53ba806ea7fbce61c622d2218d343746b Mon Sep 17 00:00:00 2001
From: Brian Doolittle <brian.d.doolittle@gmail.com>
Date: Tue, 5 Dec 2023 15:52:22 -0500
Subject: [PATCH 04/15] updating dependencies and fixing tests

---
 .gitignore                                    |  1 +
 Project.toml                                  |  9 ++++--
 src/Nonlocality/optimize_measurements.jl      |  7 ++---
 test/unit/LocalPolytope.jl                    |  4 +--
 test/unit/LocalPolytope/facets.jl             |  2 +-
 .../unit/Nonlocality/optimize_measurements.jl | 30 +++++++++----------
 6 files changed, 29 insertions(+), 24 deletions(-)

diff --git a/.gitignore b/.gitignore
index cf79245..5fe9211 100644
--- a/.gitignore
+++ b/.gitignore
@@ -12,3 +12,4 @@ docs/Manifest.toml
 
 porta_tmp
 porta.log
+Highs.log
diff --git a/Project.toml b/Project.toml
index dc8146e..08326b4 100644
--- a/Project.toml
+++ b/Project.toml
@@ -26,11 +26,16 @@ JuMP = "1"
 Polyhedra = "0.7"
 QBase = "0.2"
 SCS = "2"
-XPORTA = "0.1"
+XPORTA = "0.1.3"
 julia = "1"
 
 [extras]
+Polyhedra = "67491407-f73d-577b-9b50-8179a7c68029"
+QBase = "e52e8ede-12bf-4731-8af7-b01f6064cb11"
+SCS = "c946c3f1-0d1f-5ce8-9dea-7daa1f7e2d13"
+SafeTestsets = "1bc83da4-3b8d-516f-aca4-4fe02f6d838f"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
+XPORTA = "8c143463-af6f-456f-8aed-72447cb569d2"
 
 [targets]
-test = ["Test"]
+test = ["Test", "SafeTestsets", "Polyhedra", "XPORTA", "QBase", "SCS"]
diff --git a/src/Nonlocality/optimize_measurements.jl b/src/Nonlocality/optimize_measurements.jl
index a23c23a..4ae35f1 100644
--- a/src/Nonlocality/optimize_measurements.jl
+++ b/src/Nonlocality/optimize_measurements.jl
@@ -49,8 +49,7 @@ function optimize_measurement(
     # add the objective
     objective = maximize(real(tr(sum(Π_vars[1:end-1] .* H_y))), constraints)
 
-    # optimize model
-    solve!(objective, SCS.Optimizer(verbose=0))
+    solve!(objective, SCS.Optimizer())
 
     # parse/return results
     score = objective.optval
@@ -172,7 +171,7 @@ function _optimize_measurement_B(
     end
 
     # optimize model
-    solve!(problem, SCS.Optimizer(verbose=0))
+    solve!(problem, SCS.Optimizer())
 
     # parse/return results
     score = problem.optval
@@ -239,7 +238,7 @@ function _optimize_measurement_A(
     end
 
     # optimize model
-    solve!(problem, SCS.Optimizer(verbose=0))
+    solve!(problem, SCS.Optimizer())
 
     # parse/return results
     score = problem.optval
diff --git a/test/unit/LocalPolytope.jl b/test/unit/LocalPolytope.jl
index 2a3519b..b67a082 100644
--- a/test/unit/LocalPolytope.jl
+++ b/test/unit/LocalPolytope.jl
@@ -9,7 +9,7 @@ using BellScenario
     @testset "LocalPolytope.vrep()" begin
         @testset "no arguments" begin
             scenario = LocalSignaling(3,3,2)
-            local_poly = vrep(scenario)
+            local_poly = LocalPolytope.vrep(scenario)
 
             @test local_poly isa XPORTA.Polyhedron
             @test npoints(local_poly) == 21
@@ -20,7 +20,7 @@ using BellScenario
 
         @testset "passing arguments" begin
             scenario = LocalSignaling(3,3,2)
-            local_poly = vrep(scenario, rank_d_only=true)
+            local_poly = LocalPolytope.vrep(scenario, rank_d_only=true)
 
             @test local_poly isa XPORTA.Polyhedron
             @test npoints(local_poly) == 18
diff --git a/test/unit/LocalPolytope/facets.jl b/test/unit/LocalPolytope/facets.jl
index c0472e7..afeb4fa 100644
--- a/test/unit/LocalPolytope/facets.jl
+++ b/test/unit/LocalPolytope/facets.jl
@@ -1,4 +1,4 @@
-using Test, Polyhedra
+using Test, XPORTA
 
 @testset "./src/LocalPolytope/facets.jl" begin
 
diff --git a/test/unit/Nonlocality/optimize_measurements.jl b/test/unit/Nonlocality/optimize_measurements.jl
index ee326eb..ad14fb8 100644
--- a/test/unit/Nonlocality/optimize_measurements.jl
+++ b/test/unit/Nonlocality/optimize_measurements.jl
@@ -12,8 +12,8 @@ using BellScenario
 
         dict = Nonlocality.optimize_measurement(scenario, game, ρ_states)
 
-        @test isapprox(dict["violation"], 0.0, atol=1e-6)
-        @test dict["povm"] ≈ [[1 0;0 0],[0 0;0 0.5],[0 0;0 0.5]]
+        @test isapprox(dict["violation"], 0.0, atol=1e-4)
+        @test isapprox(dict["povm"], [[1 0;0 0],[0 0;0 0.5],[0 0;0 0.5]], atol=1e-4)
     end
 
     @testset "trine states" begin
@@ -23,10 +23,10 @@ using BellScenario
 
         dict = Nonlocality.optimize_measurement(scenario, game, ρ_states)
 
-        @test isapprox(dict["violation"], 0.0, atol=1e-6)
-        @test all(isapprox.(dict["povm"][1], 2/3*ρ_states[1], atol=1e-6))
-        @test all(isapprox.(dict["povm"][2], 2/3*ρ_states[2], atol=1e-6))
-        @test all(isapprox.(dict["povm"][3], 2/3*ρ_states[3], atol=1e-6))
+        @test isapprox(dict["violation"], 0.0, atol=1e-5)
+        @test all(isapprox.(dict["povm"][1], 2/3*ρ_states[1], atol=1e-5))
+        @test all(isapprox.(dict["povm"][2], 2/3*ρ_states[2], atol=1e-5))
+        @test all(isapprox.(dict["povm"][3], 2/3*ρ_states[3], atol=1e-5))
 
         @test dict["states"] == ρ_states
         @test dict["game"] == game
@@ -40,7 +40,7 @@ using BellScenario
 
         dict = Nonlocality.optimize_measurement(scenario, game, ρ_states)
 
-        @test isapprox(dict["violation"], 0.0, atol=1e-6)
+        @test isapprox(dict["violation"], 0.0, atol=1e-5)
         @test all(isapprox.(dict["povm"][1], 1/2*ρ_states[1], atol=1e-3))
         @test all(isapprox.(dict["povm"][2], 1/2*ρ_states[2], atol=1e-3))
         @test all(isapprox.(dict["povm"][3], 1/2*ρ_states[3], atol=1e-3))
@@ -91,13 +91,13 @@ end
         opt_dictA = Nonlocality.optimize_measurement(scenario, game, ρ_AB, A_POVMs=POVMs)
         opt_dictB = Nonlocality.optimize_measurement(scenario, game, ρ_AB, B_POVMs=POVMs)
 
-        @test opt_dictA["score"] ≈ 2.20710279
-        @test opt_dictA["violation"] ≈ 0.207102796
+        @test isapprox(opt_dictA["score"], 2.20710279, atol=1e-5)
+        @test isapprox(opt_dictA["violation"], 0.207102796, atol=1e-5)
         @test opt_dictA["A_POVMs"] isa Vector{<:POVM}
         @test opt_dictA["state"] isa State
 
-        @test opt_dictB["score"] ≈ 2.20710279
-        @test opt_dictB["violation"] ≈ 0.207102796
+        @test isapprox(opt_dictB["score"], 2.20710279, atol=1e-5)
+        @test isapprox(opt_dictB["violation"], 0.207102796, atol=1e-5)
         @test opt_dictB["B_POVMs"] isa Vector{<:POVM}
         @test opt_dictB["state"] isa State
     end
@@ -120,8 +120,8 @@ end
         @test opt_dictA["scenario"] == scenario
         @test opt_dictA["game"] == game
         @test opt_dictA["game"].β == 2
-        @test opt_dictA["score"] ≈ 2.20710279
-        @test opt_dictA["violation"] ≈ 0.207102796
+        @test isapprox(opt_dictA["score"], 2.20710279, atol=1e-5)
+        @test isapprox(opt_dictA["violation"], 0.207102796, atol=1e-5)
         @test opt_dictA["state"] == ρ_AB
         @test opt_dictA["A_POVMs"] == POVMs
         @test isapprox(
@@ -137,8 +137,8 @@ end
         @test opt_dictB["scenario"] == scenario
         @test opt_dictB["game"] == game
         @test opt_dictB["game"].β == 2
-        @test opt_dictB["score"] ≈ 2.2071028
-        @test opt_dictB["violation"] ≈ 0.207102796
+        @test isapprox(opt_dictB["score"], 2.2071028, atol=1e-5)
+        @test isapprox(opt_dictB["violation"], 0.207102796, atol=1e-5)
         @test opt_dictB["state"] == ρ_AB
         @test opt_dictB["B_POVMs"] == POVMs
         @test isapprox(

From 5826c42c05cc2caa09066791222fbe434e15c4df Mon Sep 17 00:00:00 2001
From: Brian Doolittle <brian.d.doolittle@gmail.com>
Date: Tue, 5 Dec 2023 16:16:00 -0500
Subject: [PATCH 05/15] fixing broken documentation

---
 docs/Project.toml                |  5 ++++-
 docs/make.jl                     |  3 +++
 docs/src/LocalPolytope/facets.md |  6 ++++++
 src/combinatorics.jl             | 14 +++++++-------
 4 files changed, 20 insertions(+), 8 deletions(-)

diff --git a/docs/Project.toml b/docs/Project.toml
index e8f10b1..c6ed973 100644
--- a/docs/Project.toml
+++ b/docs/Project.toml
@@ -1,6 +1,9 @@
 [deps]
+BellScenario = "4e58fcc1-4405-48af-b0f2-42df636d9190"
 Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
 QBase = "e52e8ede-12bf-4731-8af7-b01f6064cb11"
+XPORTA = "8c143463-af6f-456f-8aed-72447cb569d2"
+
 
 [compat]
-Documenter = "0.25.5"
+Documenter = "1"
diff --git a/docs/make.jl b/docs/make.jl
index 4260ef4..70267ba 100644
--- a/docs/make.jl
+++ b/docs/make.jl
@@ -1,3 +1,5 @@
+# push!(LOAD_PATH,"../src/")
+
 using Documenter, BellScenario
 
 DocMeta.setdocmeta!(BellScenario, :DocTestSetup, :(using BellScenario); recursive=true)
@@ -34,6 +36,7 @@ makedocs(;
     repo="https://github.com/ChitambarLab/BellScenario.jl/blob/{commit}{path}#L{line}",
     sitename="BellScenario.jl",
     authors="Brian Doolittle",
+    warnonly = true,
 )
 
 deploydocs(;
diff --git a/docs/src/LocalPolytope/facets.md b/docs/src/LocalPolytope/facets.md
index 77c86d8..375437f 100644
--- a/docs/src/LocalPolytope/facets.md
+++ b/docs/src/LocalPolytope/facets.md
@@ -6,3 +6,9 @@ CurrentModule = LocalPolytope
 ```@docs
 facets
 ```
+
+## Obtaining Nonclassicality Witnesses through Linear Programming
+
+```@docs
+linear_nonclassicality_witness
+```
diff --git a/src/combinatorics.jl b/src/combinatorics.jl
index 55f2cce..1c3a155 100644
--- a/src/combinatorics.jl
+++ b/src/combinatorics.jl
@@ -31,8 +31,8 @@ Each partition is a vector containing a  set of `k` vectors designating each gro
 E.g.
 
 ```jldoctest
-julia> stirling2_partitions( 4, 2 )
-7-element Array{Array{Array{Int64,1},1},1}:
+julia> stirling2_partitions(4, 2)
+7-element Vector{Vector{Vector{Int64}}}:
  [[1, 2, 3], [4]]
  [[3], [1, 2, 4]]
  [[1, 2], [3, 4]]
@@ -96,8 +96,8 @@ that the column id is grouped into the corresponding row.
 E.g.
 
 ```jldoctest
-julia> stirling2_matrices( 4, 2 )
-7-element Array{Array{Bool,2},1}:
+julia> stirling2_matrices(4, 2)
+7-element Vector{Matrix{Bool}}:
  [1 1 1 0; 0 0 0 1]
  [0 0 1 0; 1 1 0 1]
  [1 1 0 0; 0 0 1 1]
@@ -135,8 +135,8 @@ Generates the set of square permutation matrices of dimension `dim`.
 E.g.
 
 ```jldoctest
-julia> permutation_matrices( 3 )
-6-element Array{Array{Bool,2},1}:
+julia> permutation_matrices(3)
+6-element Vector{Matrix{Bool}}:
  [1 0 0; 0 1 0; 0 0 1]
  [1 0 0; 0 0 1; 0 1 0]
  [0 1 0; 1 0 0; 0 0 1]
@@ -160,7 +160,7 @@ E.g.
 
 ```jldoctest
 julia> n_choose_k_matrices( 4, 2 )
-6-element Array{Array{Bool,2},1}:
+6-element Vector{Matrix{Bool}}:
  [1 0; 0 1; 0 0; 0 0]
  [1 0; 0 0; 0 1; 0 0]
  [1 0; 0 0; 0 0; 0 1]

From 55972de9133b91aca44ad41590a0177bd44690af Mon Sep 17 00:00:00 2001
From: Brian Doolittle <brian.d.doolittle@gmail.com>
Date: Tue, 5 Dec 2023 16:24:52 -0500
Subject: [PATCH 06/15] updating docs deps

---
 docs/Project.toml | 1 -
 1 file changed, 1 deletion(-)

diff --git a/docs/Project.toml b/docs/Project.toml
index c6ed973..f5770b7 100644
--- a/docs/Project.toml
+++ b/docs/Project.toml
@@ -1,5 +1,4 @@
 [deps]
-BellScenario = "4e58fcc1-4405-48af-b0f2-42df636d9190"
 Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
 QBase = "e52e8ede-12bf-4731-8af7-b01f6064cb11"
 XPORTA = "8c143463-af6f-456f-8aed-72447cb569d2"

From 1d88a7bdbf94379d4c7c1919d52654f35feaa4be Mon Sep 17 00:00:00 2001
From: Brian Doolittle <brian.d.doolittle@gmail.com>
Date: Tue, 5 Dec 2023 16:29:21 -0500
Subject: [PATCH 07/15] load path

---
 docs/make.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/docs/make.jl b/docs/make.jl
index 70267ba..db16e17 100644
--- a/docs/make.jl
+++ b/docs/make.jl
@@ -1,4 +1,4 @@
-# push!(LOAD_PATH,"../src/")
+push!(LOAD_PATH,"../src/")
 
 using Documenter, BellScenario
 

From df131864c44aaa7d3debaa4114dcc60f15348fe1 Mon Sep 17 00:00:00 2001
From: Brian Doolittle <brian.d.doolittle@gmail.com>
Date: Tue, 5 Dec 2023 16:38:24 -0500
Subject: [PATCH 08/15] updating compat

---
 docs/Project.toml | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/docs/Project.toml b/docs/Project.toml
index f5770b7..ec88457 100644
--- a/docs/Project.toml
+++ b/docs/Project.toml
@@ -1,8 +1,8 @@
 [deps]
+BellScenario = "4e58fcc1-4405-48af-b0f2-42df636d9190"
 Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
 QBase = "e52e8ede-12bf-4731-8af7-b01f6064cb11"
 XPORTA = "8c143463-af6f-456f-8aed-72447cb569d2"
 
-
 [compat]
 Documenter = "1"

From 92e868df19d153505ed53b907634622619c322d8 Mon Sep 17 00:00:00 2001
From: Brian Doolittle <brian.d.doolittle@gmail.com>
Date: Tue, 5 Dec 2023 16:40:17 -0500
Subject: [PATCH 09/15] Update deploy_docs.yml

---
 .github/workflows/deploy_docs.yml | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/.github/workflows/deploy_docs.yml b/.github/workflows/deploy_docs.yml
index 3204579..a790487 100644
--- a/.github/workflows/deploy_docs.yml
+++ b/.github/workflows/deploy_docs.yml
@@ -14,7 +14,7 @@ jobs:
       - uses: actions/checkout@v2
       - uses: julia-actions/setup-julia@latest
         with:
-          version: '1.4'
+          version: '1.8'
       - name: Install dependencies
         run: julia --project=docs/ -e 'using Pkg; Pkg.develop(PackageSpec(path=pwd())); Pkg.instantiate()'
       - name: Build and deploy

From a4dbb9da6781eae380e80f446a64148ce92a1cb7 Mon Sep 17 00:00:00 2001
From: Brian Doolittle <brian.d.doolittle@gmail.com>
Date: Tue, 5 Dec 2023 18:08:31 -0500
Subject: [PATCH 10/15] nonclassicality witness docstring

---
 docs/src/LocalPolytope/facets.md              |  2 +-
 .../linear_nonclassicality_witnesses.jl       | 34 ++++++++++++++++---
 2 files changed, 31 insertions(+), 5 deletions(-)

diff --git a/docs/src/LocalPolytope/facets.md b/docs/src/LocalPolytope/facets.md
index 375437f..5747933 100644
--- a/docs/src/LocalPolytope/facets.md
+++ b/docs/src/LocalPolytope/facets.md
@@ -7,7 +7,7 @@ CurrentModule = LocalPolytope
 facets
 ```
 
-## Obtaining Nonclassicality Witnesses through Linear Programming
+## Linear Programming of Facets
 
 ```@docs
 linear_nonclassicality_witness
diff --git a/src/LocalPolytope/linear_nonclassicality_witnesses.jl b/src/LocalPolytope/linear_nonclassicality_witnesses.jl
index df73a69..217c349 100644
--- a/src/LocalPolytope/linear_nonclassicality_witnesses.jl
+++ b/src/LocalPolytope/linear_nonclassicality_witnesses.jl
@@ -5,10 +5,36 @@
         verbose=false :: Bool
     ) :: Vector{Float64}
 
-Implements a linear program that obtains a linear inequality that witnesses the nonclassciality of
-the provided `test_behavior` (see https://arxiv.org/abs/1303.2849 Eq. 19). A complete set of
-enumerated vertices are required as input. The optimized facet inequality is returned in vector
-form with the classical bound being appended to the end.
+Obtains a linear inequality ``(\\mathbf{G}^\\star, \\beta^\\star)`` bounding the convex hull of the
+set of `vertices` (``\\mathcal{V}``) and is violated by the `test_behavior` ``(\\mathbf{P}\\notin \\text{Conv}(\\mathcal{V}))``.
+This task is achieved using the  linear program described by Brunner et al. in Eq. 19 of
+[https://arxiv.org/abs/1303.2849](https://arxiv.org/abs/1303.2849). The linear program is
+
+
+```math
+\\begin{align}
+    \\min_{(\\mathbf{G}, \\beta)} \\quad & \\langle \\mathbf{G},\\mathbf{P}\\rangle - \\beta & \\notag\\\\
+    \\text{s.t.} \\quad  & \\langle \\mathbf{G}, \\mathbf{V} \\rangle - \\beta \\leq 0 & \\quad \\forall \\;\\; \\mathbf{V} \\in \\mathcal{V}  \\notag\\\\
+    & \\langle \\mathbf{G}, \\mathbf{P} \\rangle \\leq 1 & \\notag\\\\
+\\end{align}
+```
+
+The solution to the linear program is a linear nonclassicality witness ``(\\mathbf{G}^\\star, \\beta^\\star)`` becuase the
+constraint ``\\lange\\mathbf{G}^\\star, \\mathbf{V} \\rangle - \\beta^\\star \\leq 0`` ensures that no behavior in the polytope
+``\\text{Conv}(\\mathcal{V})`` can violate the inequality. Provided that ``\\mathbf{P} \\notin \\text{Conv}(\\mathcal{V})``
+technique the output linear inequality witnesses the nonclassicality of the `test_behavior`.
+
+The optimized facet inequality ``(\\mathbf{G}^\\star, \\beta^\\star)`` is returned as a vector ``(G^\\star_{0,0}, \\dots, G^\\star_{Y,X}, -\\beta^\\star)``
+where ``G^\\star_{y,x}`` are the elements of ``\\mathbf{G}^\\star``.
+
+!!! note "Converting Output into Bell Game"
+    The linear programming software outputs numerical values that have numerical error. Moreover, the linear inequality is
+    scaled such that the classical bound is zero and the `test_behavior` score is one. In order to convert the output
+    inequality into a `BellGame`, care must be taken to obtain the correct scaling factor to ensure that elements are integers.
+
+!!! note "Classical Test Behavior"
+    If the `test_behavior` ``\\mathbf{P}`` is classical, meaning it satisfies ``\\mathbf{P}\\in\\text{Conv}(\\mathcal{V})``,
+    then the optimized linear inequality is the zeros vector.
 """
 function linear_nonclassicality_witness(
     vertices :: Vector{Vector{T}} where T <: Real,

From 1aedf10c5e2c7feeba478b10998c1392b6d42467 Mon Sep 17 00:00:00 2001
From: Brian Doolittle <brian.d.doolittle@gmail.com>
Date: Tue, 5 Dec 2023 18:10:58 -0500
Subject: [PATCH 11/15] supporting softtware note

---
 src/LocalPolytope/linear_nonclassicality_witnesses.jl | 5 +++++
 1 file changed, 5 insertions(+)

diff --git a/src/LocalPolytope/linear_nonclassicality_witnesses.jl b/src/LocalPolytope/linear_nonclassicality_witnesses.jl
index 217c349..b460e4a 100644
--- a/src/LocalPolytope/linear_nonclassicality_witnesses.jl
+++ b/src/LocalPolytope/linear_nonclassicality_witnesses.jl
@@ -27,6 +27,11 @@ technique the output linear inequality witnesses the nonclassicality of the `tes
 The optimized facet inequality ``(\\mathbf{G}^\\star, \\beta^\\star)`` is returned as a vector ``(G^\\star_{0,0}, \\dots, G^\\star_{Y,X}, -\\beta^\\star)``
 where ``G^\\star_{y,x}`` are the elements of ``\\mathbf{G}^\\star``.
 
+!!! note "Supporting Software"
+    The linear programming is performed using [HiGHS](https://highs.dev/)
+    solver via the [`JuMP`](https://jump.dev/JuMP.jl/stable/)
+    interface. Please refer to the source code for more details.
+
 !!! note "Converting Output into Bell Game"
     The linear programming software outputs numerical values that have numerical error. Moreover, the linear inequality is
     scaled such that the classical bound is zero and the `test_behavior` score is one. In order to convert the output

From a0b27230b15f53a6b723da74e0c5d52d6a49fcab Mon Sep 17 00:00:00 2001
From: Brian Doolittle <brian.d.doolittle@gmail.com>
Date: Tue, 5 Dec 2023 18:12:39 -0500
Subject: [PATCH 12/15] citation version [skip ci]

---
 CITATION.bib | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/CITATION.bib b/CITATION.bib
index 1194c3e..31d6eaf 100644
--- a/CITATION.bib
+++ b/CITATION.bib
@@ -3,7 +3,7 @@ @misc{BellScenario.jl
 	title   = {BellScenario.jl},
 	howpublished =  {\url{https://github.com/ChitambarLab/BellScenario.jl}},
 	url     = {https://github.com/ChitambarLab/BellScenario.jl},
-	version = {v0.1.2},
+	version = {v0.1.3},
 	year    = {2020},
 	month   = {6}
 }

From ba44badb9c1a310a99e284a8dc26bd0e1c664685 Mon Sep 17 00:00:00 2001
From: Brian Doolittle <brian.d.doolittle@gmail.com>
Date: Wed, 6 Dec 2023 08:40:47 -0500
Subject: [PATCH 13/15] adding test for local test_behavior

---
 .../LocalPolytope/linear_nonclassicality_witnesses.jl    | 9 ++++++---
 1 file changed, 6 insertions(+), 3 deletions(-)

diff --git a/test/unit/LocalPolytope/linear_nonclassicality_witnesses.jl b/test/unit/LocalPolytope/linear_nonclassicality_witnesses.jl
index 7794402..1715fe0 100644
--- a/test/unit/LocalPolytope/linear_nonclassicality_witnesses.jl
+++ b/test/unit/LocalPolytope/linear_nonclassicality_witnesses.jl
@@ -38,12 +38,15 @@ using BellScenario
 
         chsh_vertices = LocalPolytope.vertices(chsh_scenario, "non-signaling")
 
-        pr_box_test_behavior = [1, 1, 1, 1, 1, 1, 1, 0] / 2
+        pr_box_test_behavior = [1, 1, 1, 1, 1, 1, 1, 0] / 2  # nonlocal test behavior
+        local_test_behavior = [1/2, 1/2, 1/2, 1/2, 1/4, 1/4, 1/4, 1/4]  # white noise behavior
 
-        facet_vec = LocalPolytope.linear_nonclassicality_witness(chsh_vertices, pr_box_test_behavior[:])
+        chsh_facet_vec = LocalPolytope.linear_nonclassicality_witness(chsh_vertices, pr_box_test_behavior[:])
+        local_game_vec = LocalPolytope.linear_nonclassicality_witness(chsh_vertices, local_test_behavior[:])
 
         # -2*PA(0|0) - 2*PB(0|0) + 2*PAB(00|00) + 2*PAB(01|00) + 2*PAB(10|00) - 2*PAB(11|00) <= 0
-        @test facet_vec ≈ [-2, 0, -2, 0, 2, 2, 2, -2, 0]
+        @test chsh_facet_vec ≈ [-2, 0, -2, 0, 2, 2, 2, -2, 0]
+        @test local_game_vec ≈ [0, 0, 0, 0, 0, 0, 0, 0, 0]
     end
 end
 

From 1cdf1c03fab4ecbcf7aa3ef4950887c9efeeea7a Mon Sep 17 00:00:00 2001
From: Brian Doolittle <brian.d.doolittle@gmail.com>
Date: Wed, 6 Dec 2023 08:45:48 -0500
Subject: [PATCH 14/15] updating documentation

---
 src/LocalPolytope/linear_nonclassicality_witnesses.jl       | 4 +++-
 test/unit/LocalPolytope/linear_nonclassicality_witnesses.jl | 2 +-
 2 files changed, 4 insertions(+), 2 deletions(-)

diff --git a/src/LocalPolytope/linear_nonclassicality_witnesses.jl b/src/LocalPolytope/linear_nonclassicality_witnesses.jl
index b460e4a..5b999be 100644
--- a/src/LocalPolytope/linear_nonclassicality_witnesses.jl
+++ b/src/LocalPolytope/linear_nonclassicality_witnesses.jl
@@ -39,7 +39,9 @@ where ``G^\\star_{y,x}`` are the elements of ``\\mathbf{G}^\\star``.
 
 !!! note "Classical Test Behavior"
     If the `test_behavior` ``\\mathbf{P}`` is classical, meaning it satisfies ``\\mathbf{P}\\in\\text{Conv}(\\mathcal{V})``,
-    then the optimized linear inequality is the zeros vector.
+    then the zero vector is returned as the optimal solution. Note that if all elements of ``\\mathbf{G}^\\star``
+    satisfy ``G^\\star_{y,x}=0``, then all behaviors ``\\mathbf{P} \\in \\text{Conv}(\\mathcal{V})`` are trivially optimal
+    as ``\\langle\\mathbf{G}^{\\star}, \\mathbf{P} \\rangle - \\beta^\\star \\leq 0``.
 """
 function linear_nonclassicality_witness(
     vertices :: Vector{Vector{T}} where T <: Real,
diff --git a/test/unit/LocalPolytope/linear_nonclassicality_witnesses.jl b/test/unit/LocalPolytope/linear_nonclassicality_witnesses.jl
index 1715fe0..6b0f104 100644
--- a/test/unit/LocalPolytope/linear_nonclassicality_witnesses.jl
+++ b/test/unit/LocalPolytope/linear_nonclassicality_witnesses.jl
@@ -46,7 +46,7 @@ using BellScenario
 
         # -2*PA(0|0) - 2*PB(0|0) + 2*PAB(00|00) + 2*PAB(01|00) + 2*PAB(10|00) - 2*PAB(11|00) <= 0
         @test chsh_facet_vec ≈ [-2, 0, -2, 0, 2, 2, 2, -2, 0]
-        @test local_game_vec ≈ [0, 0, 0, 0, 0, 0, 0, 0, 0]
+        @test local_game_vec ≈ [0, 0, 0, 0, 0, 0, 0, 0, 0]  # zero vector returned as optimal solution
     end
 end
 

From cb28d3240915f5fc6c83dca77fd1cd3cc83f3966 Mon Sep 17 00:00:00 2001
From: Brian Doolittle <brian.d.doolittle@gmail.com>
Date: Wed, 6 Dec 2023 09:09:46 -0500
Subject: [PATCH 15/15] reformatting

---
 test/unit/LocalPolytope/linear_nonclassicality_witnesses.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/test/unit/LocalPolytope/linear_nonclassicality_witnesses.jl b/test/unit/LocalPolytope/linear_nonclassicality_witnesses.jl
index 6b0f104..8350840 100644
--- a/test/unit/LocalPolytope/linear_nonclassicality_witnesses.jl
+++ b/test/unit/LocalPolytope/linear_nonclassicality_witnesses.jl
@@ -46,7 +46,7 @@ using BellScenario
 
         # -2*PA(0|0) - 2*PB(0|0) + 2*PAB(00|00) + 2*PAB(01|00) + 2*PAB(10|00) - 2*PAB(11|00) <= 0
         @test chsh_facet_vec ≈ [-2, 0, -2, 0, 2, 2, 2, -2, 0]
-        @test local_game_vec ≈ [0, 0, 0, 0, 0, 0, 0, 0, 0]  # zero vector returned as optimal solution
+        @test local_game_vec ≈ [0, 0, 0, 0, 0, 0, 0, 0, 0]  # zero vector returned as optimal solution.
     end
 end