From bce3fc3ba8d9de0c3ca1d652dfa6539ecd957b9c Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Radek=20Hu=C5=A1ek?= <husek@iuuk.mff.cuni.cz>
Date: Tue, 6 Apr 2021 15:55:32 +0200
Subject: [PATCH] Flower snark experiment
---
experiments/flower-snarks.py | 88 ++++++++++++++++++++++++++++++++++++
1 file changed, 88 insertions(+)
create mode 100755 experiments/flower-snarks.py
diff --git a/experiments/flower-snarks.py b/experiments/flower-snarks.py
new file mode 100755
index 0000000..b1e5243
--- /dev/null
+++ b/experiments/flower-snarks.py
@@ -0,0 +1,88 @@
+#!/usr/bin/python
+
+DOC = """
+Calculate that the number of CDCs of Flower snark $J_k$ is $16^{(1-o(1))k}$.
+
+We calculate the step matrix $M$, show that 16 is its eigenvalue by finding
+its eigenvector. The eigenvector is guessed for the ratios of coefficients
+of $M^10 x$ and $M^9 x$ where $x$ is the initial vector. This eigenvector
+shares one non-zero coordinate with the final vector.
+"""
+
+import sys, os
+sys.path.append(os.path.dirname(__file__) + "/..")
+
+from graph_tools.all import *
+from sage.all import identity_matrix, QQ, DiGraph, matrix
+
+def test_eigenvalue(M, x):
+ H = M - x*identity_matrix(QQ, M.ncols(), sparse=True)
+ if H.is_singular():
+ print(" Value %i is an eigenvalue of M with geometric multiplicity %i" %
+ (x, H.ncols() - H.rank()))
+
+def matrix_to_rev_graph(M):
+ assert M.ncols() == M.nrows()
+ return DiGraph([range(M.ncols()), M.dict().keys()],
+ format="vertices_and_edges", loops=True)
+
+
+if __name__ == "__main__":
+ print(DOC)
+ print("Stabilizing...")
+ S = FlowerSnark.stabilize(CircuitDoubleCover, jordan=False)
+ print("Done")
+ M = S["step_matrix"]
+
+ print("\nThe rank of the step matrix M: %s" % M.rank())
+ upto = 16
+ print("Testing if integers up to %i are eigenvalues:" % upto)
+ for x in range(upto + 1):
+ test_eigenvalue(M, x)
+
+ fin = S["finalize"]
+
+ print("\nCheking correctness of the matrix")
+ N = 16
+ A = S["initial_vector"]
+ for i in range(3, N):
+ nu1 = FlowerSnark.graph(i).eval(CircuitDoubleCover)
+ A = M * A
+ nu2 = fin * A
+ assert nu1 == nu2
+ print(" ν(J_%i) = %i" % (i, nu1))
+ print(" Done")
+
+ print("\nCalculating the eigenvector for eigenvalue 16")
+ to_dict = lambda g: { b.boundary: b.value for b in g }
+ gadget_to_dict = lambda i: to_dict(FlowerSnark.gadget(i).eval_gadget(CircuitDoubleCover))
+
+ X10 = gadget_to_dict(10)
+ X9 = gadget_to_dict(9)
+ B = [ b for b in X10.keys() if b in X9 and X10[b] / X9[b] > 15 ]
+ m = min( X10[b] for b in B )
+ vec = { b: int(round(X10[b] / m)) for b in B }
+ print(" Guessed eigenvector (with %i non-zero coordinates):" % len(vec))
+ for b, v in sorted(vec.items()):
+ print(" %i at %s" % (v, b))
+
+ print(" Checking it...")
+ nex = to_dict(FlowerSnark._next_gadget(FakeGadget(6,
+ [ Boundary(b, v) for b, v in vec.items() ])).eval_gadget(CircuitDoubleCover))
+
+ assert nex.keys() == vec.keys()
+ for b in vec:
+ assert vec[b] * 16 == nex[b], "Oops at %b" % b
+
+ assert set(S["variables"]).issuperset(vec.keys())
+ index = { b: i for i, b in enumerate(S["variables"]) }
+ vec_ = matrix(QQ, M.ncols(), 1, { (index[b], 0): v for b, v in vec.items() })
+ assert M * vec_ == 16 * vec_
+
+ print(" Done")
+
+ print(" Boundaries of with the non-zero value in both the eigenvector and the final vector:")
+ for i in range(vec_.nrows()):
+ if vec_[i, 0] > 0 and fin[0, i] > 0:
+ print(" %s" % (S["variables"][i],))
+
--
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