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test.py
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test.py
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# Python Symbolic Information Theoretic Inequality Prover
# Copyright (C) 2020 Cheuk Ting Li
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <https://www.gnu.org/licenses/>.
"""
Python Symbolic Information Theoretic Inequality Prover
Examples and tests
Copyright (C) 2020 Cheuk Ting Li
References for the examples used are provided at the end.
"""
from psitip import *
import unittest
import time
class TestPsitip(unittest.TestCase):
def test_marton(self):
# Fourier-Motzkin elimination for Marton's inner bound with common message
# [Marton 1979], [Liang-Kramer 2007]
R0, R1, R2, R10, R20, Rs = real("R0", "R1", "R2", "R10", "R20", "Rs")
U0, U1, U2, X, Y1, Y2 = rv("U0", "U1", "U2", "X", "Y1", "Y2")
r = universe()
r &= R0 >= 0
r &= R1 >= 0
r &= R2 >= 0
r &= R10 >= 0
r &= R10 <= R1
r &= R20 >= 0
r &= R20 <= R2
r &= Rs >= 0
r &= R0 + R20 + R1 + Rs <= I(U0 + U1 & Y1)
r &= R1 - R10 + Rs <= I(U1 & Y1 | U0)
r &= R0 + R10 + R2 - Rs <= I(U0 + U2 & Y2) - I(U1 & U2 | U0)
r &= R0 + R10 + R2 <= I(U0 + U2 & Y2)
r &= R2 - R20 - Rs <= I(U2 & Y2 | U0) - I(U1 & U2 | U0)
r &= R2 - R20 <= I(U2 & Y2 | U0)
r &= markov(U0+U1+U2, X, Y1+Y2)
region_str = r.exists(R10+R20+Rs+U0+U1+U2).tostring(tosort = True, lhsvar = R0+R1+R2)
print(region_str)
self.assertEqual(region_str,
("{ R0 >= 0,\n"
" R1 >= 0,\n"
" R2 >= 0,\n"
" R0+R1 <= I(U0,U1;Y1),\n"
" R0+R2 <= I(U0,U2;Y2),\n"
" R0+R1+R2 <= I(U0,U1;Y1)+I(U2;Y2|U0)-I(U1;U2|U0),\n"
" R0+R1+R2 <= I(U0,U2;Y2)+I(U1;Y1|U0)-I(U1;U2|U0),\n"
" 2R0+R1+R2 <= I(U0,U1;Y1)+I(U0,U2;Y2)-I(U1;U2|U0),\n"
" I(U0,U1,U2;Y1,Y2|X) == 0,\n"
" I(U1;U2|U0) <= I(U1;Y1|U0)+I(U2;Y2|U0) } , exists U0,U1,U2")
)
def test_implication(self):
X, Y1, Y2, U, V = rv("X", "Y1", "Y2", "U", "V")
# Degraded broadcast channel [Cover 1972]
r_deg = markov(X, Y1, Y2)
# Less noisy [Korner-Marton 1975]
# Reads: For all marginal distr. of X, for all U, markov(U,X,Y1+Y2) implies I(U & Y1) >= I(U & Y2)
r_ln = (markov(U+V,X,Y1+Y2) >> (I(U & Y1 | V) >= I(U & Y2 | V))).forall(U+V)
# More capable [Korner-Marton 1975]
# Reads: For all marginal distr. of X, I(X & Y1) >= I(X & Y2)
r_mc = (markov(V,X,Y1+Y2) >> (I(X & Y1 | V) >= I(X & Y2 | V))).forall(V)
self.assertTrue(r_deg.implies(r_ln))
self.assertTrue(r_ln.implies(r_mc))
self.assertFalse(r_ln.implies(r_deg))
self.assertFalse(r_mc.implies(r_ln))
def test_commoninfo(self):
X, Y, Z, W, U = rv("X", "Y", "Z", "W", "U")
# Properties of Gacs-Korner common information [Gacs-Korner 1973],
# Wyner's common information [Wyner 1975] and
# Common entropy (one-shot exact common information) [Kumar-Li-El Gamal 2014]
# Comparisons
self.assertTrue(emin(H(X), H(Y)) >= exact_ci(X & Y))
self.assertTrue(exact_ci(X & Y) >= wyner_ci(X & Y))
self.assertTrue(wyner_ci(X & Y) >= I(X & Y))
self.assertTrue(I(X & Y) >= gacs_korner(X & Y))
self.assertFalse(emin(H(X), H(Y)) <= exact_ci(X & Y))
self.assertFalse(exact_ci(X & Y) <= wyner_ci(X & Y))
self.assertFalse(wyner_ci(X & Y) <= I(X & Y))
self.assertFalse(I(X & Y) <= gacs_korner(X & Y))
# Data processing
r = markov(X, Y, Z)
self.assertTrue(r >> (exact_ci(X & Y) >= exact_ci(X & Z)))
self.assertTrue(r >> (wyner_ci(X & Y) >= wyner_ci(X & Z)))
self.assertTrue(r >> (gacs_korner(X & Y) >= gacs_korner(X & Z)))
self.assertFalse(r >> (exact_ci(X & Y) <= exact_ci(X & Z)))
self.assertFalse(r >> (wyner_ci(X & Y) <= wyner_ci(X & Z)))
self.assertFalse(r >> (gacs_korner(X & Y) <= gacs_korner(X & Z)))
# Tensorize
r = indep(X + Y, Z + W)
self.assertTrue(r >> (exact_ci(X & Y) + exact_ci(Z & W) >= exact_ci(X + Z & Y + W)))
self.assertFalse(r >> (exact_ci(X & Y) + exact_ci(Z & W) <= exact_ci(X + Z & Y + W)))
self.assertTrue(r >> (wyner_ci(X & Y) + wyner_ci(Z & W) >= wyner_ci(X + Z & Y + W)))
self.assertTrue(r >> (wyner_ci(X & Y) + wyner_ci(Z & W) <= wyner_ci(X + Z & Y + W)))
# self.assertTrue(r >> (gacs_korner(X & Y) + gacs_korner(Z & W) >= gacs_korner(X + Z & Y + W)))
self.assertTrue(r >> (gacs_korner(X & Y) + gacs_korner(Z & W) <= gacs_korner(X + Z & Y + W)))
# Independent parts
r = indep(X, Y, Z)
self.assertTrue(r >> (exact_ci(X+Z & Y+Z) == H(Z)))
self.assertTrue(r >> (wyner_ci(X+Z & Y+Z) == H(Z)))
self.assertTrue(r >> (gacs_korner(X+Z & Y+Z) == H(Z)))
# Symmetry
self.assertTrue(exact_ci(X & Y) == exact_ci(Y & X))
self.assertTrue(wyner_ci(X & Y) == wyner_ci(Y & X))
self.assertTrue(gacs_korner(X & Y) == gacs_korner(Y & X))
# Independence
self.assertTrue((exact_ci(X & Y) == 0) == indep(X, Y))
self.assertTrue((wyner_ci(X & Y) == 0) == indep(X, Y))
self.assertFalse((gacs_korner(X & Y) == 0) == indep(X, Y))
def test_commoninfo_def(self):
X, Y, Z, W, U = rv("X", "Y", "Z", "W", "U")
# Defining Wyner's common information etc. explicitly
eci = markov(X, U, Y).minimum(H(U), U)
wci = markov(X, U, Y).minimum(I(U & X+Y), U)
gkci = ((H(U|X) == 0) & (H(U|Y) == 0)).maximum(H(U), U)
self.assertTrue(exact_ci(X & Y) == eci)
self.assertTrue(wyner_ci(X & Y) == wci)
self.assertTrue(gacs_korner(X & Y) == gkci)
self.assertTrue(gacs_korner(X & Y) == H(meet(X, Y)))
# Comparisons
self.assertTrue(emax(H(X), H(Y)) >= eci)
self.assertTrue(eci >= wci)
self.assertTrue(wci >= I(X & Y))
self.assertTrue(I(X & Y) >= gkci)
self.assertFalse(emax(H(X), H(Y)) <= eci)
self.assertFalse(eci <= wci)
self.assertFalse(wci <= I(X & Y))
self.assertFalse(I(X & Y) <= gkci)
# self.assertTrue((H(X) >= eci) & (H(Y) >= eci)) # This currently fails
def test_commoninfo_discover(self):
X, Y = rv("X", "Y")
K = gacs_korner(X&Y)
J = wyner_ci(X&Y)
G = exact_ci(X&Y)
RK, RJ, RG = real("RK", "RJ", "RG")
# Automatically discover relationship between these quantities
region_str = str(universe().discover([(RK, K), (RJ, J), (RG, G), X, Y], maxsize = 2))
print("test_commoninfo_discover")
print(region_str)
self.assertEqual(region_str,
("{ RK >= 0,\n"
" RJ <= RG,\n"
" RG <= H(X),\n"
" RG <= H(Y),\n"
" RJ >= I(X;Y),\n"
" RK <= I(X;Y) }")
)
# ("{ RK >= 0,\n"
# " RJ <= RG,\n"
# " RG <= H(X),\n"
# " RG <= H(Y),\n"
# " RK <= I(X;Y),\n"
# " RJ >= I(X;Y) }")
def test_new_rv(self):
X, Y, Z, W, U = rv("X", "Y", "Z", "W", "U")
# Gacs-Korner common part [Gacs-Korner 1973]
K = meet(X, Y)
self.assertTrue(H(K | X) == 0)
self.assertTrue(((H(U | X) == 0) & (H(U | Y) == 0)) >> (H(U | K) == 0))
self.assertFalse(H(X | K) == 0)
self.assertFalse((H(U | X) == 0) >> (H(U | K) == 0))
self.assertTrue(indep(X, Y) >> (H(K) == 0))
self.assertTrue(equiv(meet(X+Y, Y), Y))
self.assertTrue(H(meet(X+Y,Y+Z)) >= H(Y))
self.assertTrue(H(Y | meet(X + Y, Y + Z)) == 0)
self.assertTrue(H(meet(X, Y) | meet(X, Y + Z)) == 0)
self.assertTrue(indep(X, Y, Z) >> equiv(meet(X+Y,Y+Z), Y))
self.assertFalse(indep(X, Y, Z) << equiv(meet(X+Y,Y+Z), Y))
# Minimal sufficient statistic of X about Y
S = mss(X, Y)
self.assertTrue(H(S | X) == 0)
self.assertTrue(markov(X, S, Y))
self.assertTrue(((H(U | X) == 0) & markov(X, U, Y)) >> (H(S | U) == 0))
self.assertTrue(indep(X, Y) >> (H(S) == 0))
self.assertTrue(equiv(mss(X+Y, Y), Y))
self.assertTrue(equiv(mss(X, X+Y), X))
self.assertTrue((indep(Y, X + Z) | indep(Z, X + Y)).implies(equiv(mss(X + Y, X + Z), X)))
self.assertTrue(H(meet(X, Y) | mss(X, Y)) == 0)
# Strong functional representation lemma [Li-El Gamal 2018]
logg = real("logg")
F = sfrl_rv(X, Y, logg)
self.assertTrue(indep(X, F))
self.assertTrue(H(Y | X+F) == 0)
self.assertTrue(H(Y | F) <= I(X & Y) + logg)
self.assertTrue(markov(F, X+Y, Z))
def test_csie_region(self):
R = real("R")
U, X, Y, S, M = rv("U", "X", "Y", "S", "M")
logg = real("logg")
# Channel with state information at encoder, lower bound
r_op = ((R <= I(M & Y)) & indep(M,S) & markov(M, X+S, Y)
& (R >= 0)).exists(M).marginal_exists(X)
# Gelfand-Pinsker theorem [Gel'fand-Pinsker 1980]
r = ((R <= I(U & Y) - I(U & S)) & markov(U, X+S, Y)
& (R >= 0)).exists(U).marginal_exists(X)
# Strong functional representation lemma with logarithmic gap [Li-El Gamal 2018]
with sfrl(logg).assumed():
# Achievability: GP region implies operational region relaxed by log gap
self.assertTrue(r >> r_op.relaxed(R, logg * 5))
# Converse: Specify that S comes from nature, and are independent across time
aux = r.check_converse(r_op, nature = S)
print("test_csie_region")
print(iutil.list_tostr_std(aux))
self.assertEqual(str(aux[0][1]), "M,Y_1")
self.assertEqual(str(aux[1][1]), "M,S")
def test_wz_region(self):
R = real("R")
U, X, Y, Z, M = rv("U", "X", "Y", "Z", "M")
# Lossy source coding with side information at decoder, upper bound
r_op = ((R >= I(M & X)) & markov(M, X, Y) & markov(X, M+Y, Z)
).exists(M).kernel_exists(Z)
# Wyner-Ziv theorem [Wyner-Ziv 1976]
r = ((R >= I(X & U | Y)) & markov(U, X, Y) & markov(X, U+Y, Z)
).exists(U).kernel_exists(Z)
# Converse
aux = r.check_converse(r_op)
print("test_wz_region")
print(iutil.list_tostr_std(aux))
self.assertTrue(aux is not None)
#self.assertEqual(str(aux[0][1]), "M,Y_")
#self.assertEqual(str(aux[1][1]), "M,Y")
def test_dbc_region(self):
R1, R2 = real("R1", "R2")
U, X, Y1, Y2, M1, M2 = rv("U", "X", "Y1", "Y2", "M1", "M2")
# Broadcast channel operational region
r_op = ((R1 <= I(M1 & Y1)) & (R2 <= I(M2 & Y2)) & (H(X | M1+M2) == 0) & indep(M1,M2) & markov(M1+M2, X, Y1+Y2)
& (R1 >= 0) & (R2 >= 0)).exists(M1+M2).marginal_exists(X)
# Superposition coding region [Bergmans 1973], [Gallager 1974]
r = ((R2 <= I(U & Y2)) & (R1 + R2 <= I(X & Y1 | U) + I(U & Y2)) & (R1 + R2 <= I(X & Y1))
& markov(U, X, Y1+Y2) & (R1 >= 0) & (R2 >= 0)).exists(U).marginal_exists(X)
#r2 = ((R1 <= I(X & Y1 | U)) & (R2 <= I(U & Y2)) & (R1 + R2 <= I(X & Y1))
# & markov(U, X, Y1+Y2) & (R1 >= 0) & (R2 >= 0)).exists(U).marginal_exists(X)
# Degraded broadcast channel [Cover 1972]
c_dg = markov(X, Y1, Y2)
# More capable [Korner-Marton 1975]
# Reads: For all marginal distr. of X, I(X & Y1) >= I(X & Y2)
c_mc = (I(X & Y1) >= I(X & Y2)).marginal_forall(X).convexified(inp = True, forall = True)
print("test_dbc_region")
if True:
# Attempt to tensorize assuming degraded
aux = r.tensorize(chan_cond = c_dg)
# Print auxiliary RVs
print(iutil.list_tostr_std(aux))
self.assertTrue(aux is not None)
#self.assertEqual(str(aux[0][1]), "U")
#self.assertEqual(str(aux[1][1]), "U,Y1")
if True:
# Attempt to prove converse assuming degraded
aux = r.check_converse(reg_subset = r_op, chan_cond = c_dg)
# Print auxiliary RVs
print(iutil.list_tostr_std(aux))
self.assertTrue(aux is not None)
#self.assertEqual(str(aux[0][1]), "M2")
#self.assertEqual(str(aux[1][1]), "M2,Y1")
if True:
# Attempt to prove converse assuming more capable
aux = r.check_converse(reg_subset = r_op, chan_cond = c_mc)
# Print auxiliary RVs
print(iutil.list_tostr_std(aux))
self.assertTrue(aux is not None)
#self.assertEqual(str(aux[0][1]), "M1,Y1_")
#self.assertEqual(str(aux[1][1]), "M1,Y2")
#self.assertEqual(str(aux[2][1]), "M2,Y1_")
#self.assertEqual(str(aux[3][1]), "M2,Y2")
def test_gw_region(self):
R0, R1, R2 = real("R0", "R1", "R2")
X1, X2, U = rv("X1", "X2", "U")
# Gray-Wyner region [Gray-Wyner 1974]
r = ((R0 >= I(X1 + X2 & U)) & (R1 >= H(X1 | U))
& (R2 >= H(X2 | U))).exists(U)
# Attempt to tensorize
aux = r.tensorize()
print("test_gw_region")
print(iutil.list_tostr_std(aux))
self.assertTrue(aux is not None)
self.assertEqual(str(aux[0][1]), "U_2")
# self.assertEqual(str(aux[1][1]), "U_1,X1,X2")
self.assertEqual(str(aux[1][1]), "U_2,X1")
def test_nonshannon(self):
X, Y, Z, W, U = rv("X", "Y", "Z", "W", "U")
print("test_nonshannon")
# Zhang-Yeung inequality [Zhang-Yeung 1998] cannot be proved by Shannon-type inequalities
self.assertFalse(2 * I(Z & W) <= I(X & Y) + I(X & Z+W) + 3 * I(Z & W | X) + I(Z & W | Y))
# Using copy lemma [Zhang-Yeung 1998], [Dougherty-Freiling-Zeger 2011]
with copylem().assumed():
# Prove Zhang-Yeung inequality
aux = (2 * I(Z & W) <= I(X & Y) + I(X & Z+W) + 3 * I(Z & W | X) + I(Z & W | Y)).check_getaux()
self.assertTrue(aux is not None)
print(iutil.list_tostr_std(aux))
self.assertEqual(str(aux[0][1]), "X")
self.assertEqual(str(aux[1][1]), "Z")
self.assertEqual(str(aux[2][1]), "W")
def test_dblmarkov(self):
X, Y, Z, W, U = rv("X", "Y", "Z", "W", "U")
print("test_dblmarkov")
# Using double Markov property [Csiszar-Korner 2011]
with dblmarkov().assumed():
aux = ((markov(X, Y, Z) & markov(Y, X, Z))
>> (H(mss(X, Z) | mss(Y, Z)) == 0)).check_getaux()
self.assertTrue(aux is not None)
print(iutil.list_tostr_std(aux))
aux = ((markov(X, Y, Z) & markov(Y, X, Z))
>> markov(X+Y, meet(X, Y), Z)).check_getaux()
self.assertTrue(aux is not None)
print(iutil.list_tostr_std(aux))
aux = ((markov(X, Y, Z) & markov(Y, X, Z) & markov(Y, Z, X))
>> (indep(X, Y, Z).conditioned(U)
& (H(U|X)==0) & (H(U|Y)==0) & (H(U|Z)==0)).exists(U)).check_getaux()
self.assertTrue(aux is not None)
print(iutil.list_tostr_std(aux))
def test_eliminate_toreal(self):
X, Y, Z, W, U, V = rv("X", "Y", "Z", "W", "U", "V")
IX, IY, IXY = real("IX", "IY", "IXY")
self.assertEqual(str(
(markov(X, Y, Z) & indep(X+Z, Y)).exists(Y, toreal = True)
), "I(X;Z) == 0")
# Mutual information region outer bound [Li-El Gamal 2017]
self.assertEqual((
(I(X & U) == IX) & (I(Y & U) == IY) & (I(X+Y & U) == IXY)
).exists(U, toreal = True).tostring(tosort = True, lhsvar = IX+IY+IXY),
("{ IX >= 0,\n"
" IY >= 0,\n"
" IX <= IXY,\n"
" IY <= IXY,\n"
" IXY-IX <= H(Y|X),\n"
" IXY-IY <= H(X|Y),\n"
" IX+IY-IXY <= I(X;Y) }")
)
self.assertEqual(str(
(markov(X, Y, U, W) & (H(U) == 0)).exists(U, toreal = True)),
("I(W;X,Y) == 0")
)
# This currently fails
# print(((H(X | U) == 0) >> (H(Y | U) == 0)).forall(U, toreal = True))
def test_convex(self):
R1, R2 = real("R1", "R2")
X, X1, X2 = rv("X", "X1", "X2")
Y, Y1, Y2 = rv("Y", "Y1", "Y2")
U, U1, U2 = rv("U", "U1", "U2")
Q = rv("Q")
# Multiple access channel region without time sharing [Ahlswede 1971], [Liao 1972]
r = ((R1 <= I(X1 & Y | X2)) & (R2 <= I(X2 & Y | X1))
& (R1 + R2 <= I(X1 + X2 & Y)) & (I(X1 & X2) == 0)).marginal_exists(X1 + X2)
self.assertFalse(r.isconvex(inp = True))
# Multiple access channel region with time sharing
r = ((R1 <= I(X1 & Y | X2 + Q)) & (R2 <= I(X2 & Y | X1 + Q))
& (R1 + R2 <= I(X1 + X2 & Y | Q)) & (I(X1 & X2 | Q) == 0)).exists(Q).marginal_exists(X1 + X2)
self.assertTrue(r.isconvex(inp = True))
# Degraded broadcast channel region [Bergmans 1973], [Gallager 1974]
r = ((R1 <= I(X & Y1 | U)) & (R2 <= I(U & Y2))
& (R1 + R2 <= I(X & Y1))).exists(U).marginal_exists(X)
self.assertTrue(r.isconvex(inp = True))
# Berger-Tung inner bound without time sharing [Berger 1978], [Tung 1978]
r = ((R1 >= I(X1 & U1 | U2)) & (R2 >= I(X2 & U2 | U1))
& (R1 + R2 >= I(X1 + X2 & U1 + U2))
& (I(U1 & X2 + U2 | X1) == 0) & (I(U2 & X1 + U1 | X2) == 0)).kernel_exists(U1+U2)
self.assertFalse(r.isconvex(inp = True))
# Berger-Tung inner bound with time sharing
r = ((R1 >= I(X1 & U1 | U2 + Q)) & (R2 >= I(X2 & U2 | U1 + Q))
& (R1 + R2 >= I(X1 + X2 & U1 + U2 | Q))
& (I(U1 & X2 + U2 | X1 + Q) == 0) & (I(U2 & X1 + U1 | X2 + Q) == 0)).exists(Q).kernel_exists(U1+U2)
self.assertTrue(r.isconvex(inp = True))
# Berger-Tung outer bound
r = ((R1 >= I(X1 + X2 & U1 | U2)) & (R2 >= I(X1 + X2 & U2 | U1))
& (R1 + R2 >= I(X1 + X2 & U1 + U2))
& (I(U1 & X2 | X1) == 0) & (I(U2 & X1 | X2) == 0)).kernel_exists(U1+U2)
# self.assertTrue(r.isconvex()) # This test currently fails
def test_other_quantities(self):
X, Y, Z, W, U = rv("X", "Y", "Z", "W", "U")
# Necessary conditional entropy [Cuff-Permuter-Cover 2010]
self.assertTrue(H_nec(Y | X) <= H(Y | X))
# Excess functional information [Li-El Gamal 2018]
self.assertTrue(excess_fi(X, Y) <= H(Y | X))
# The following two requires functional representation lemma to prove
with sfrl().assumed():
# with PsiOpts(flatten_distribute = False):
self.assertTrue(excess_fi(X, Y) <= exact_ci(X & Y) - I(X & Y))
self.assertTrue(excess_fi(X, Y) <= H(X | Y))
# Independent parts
r = indep(X, Y, Z)
self.assertTrue(r >> (excess_fi(X+Z, Y+Z) == 0))
self.assertTrue(r >> (H_nec(X+Z | Y+Z) == 0))
# Total correlation [Watanabe 1960]
self.assertEqual(str(total_corr(X & Y & Z+W | U)),
"H(X|U)+H(Y|U)+H(Z,W|U)-H(X,Y,Z,W|U)")
self.assertTrue(total_corr(X & Y & Z) == H(X) + H(Y) + H(Z) - H(X+Y+Z))
# Dual total correlation [Han 1978]
self.assertEqual(str(dual_total_corr(X & Y & Z+W | U)),
"H(X,Y,Z,W|U)-H(X|Y,Z,W,U)-H(Y|X,Z,W,U)-H(Z,W|X,Y,U)")
self.assertTrue(dual_total_corr(X & Y & Z) == H(X+Y+Z) - H(X|Y+Z) - H(Y|X+Z) - H(Z|X+Y))
# Mutual dependence [Csiszar-Narayan 2004]
self.assertTrue(mutual_dep(X & Y & Z) == emin(I(X+Y & Z), I(X+Z & Y), I(Y+Z & X), total_corr(X & Y & Z) / 2))
self.assertTrue(mutual_dep(X & Y & Z) <= total_corr(X & Y & Z) / 2)
self.assertTrue(mutual_dep(X & Y & Z) <= dual_total_corr(X & Y & Z))
self.assertTrue(markov(X, Y, Z) >>
(mutual_dep(X & Y & Z) == emin(I(X & Y), I(Y & Z))))
# Intrinsic mutual information [Maurer-Wolf 1999]
self.assertTrue(intrinsic_mi(X & Y | Z) == markov(X+Y, Z, U).minimum(I(X & Y | U), U))
self.assertTrue(intrinsic_mi(X & Y | Z) <= I(X & Y | Z))
self.assertFalse(intrinsic_mi(X & Y | Z) >= I(X & Y | Z))
# Information bottleneck [Tishby-Pereira-Bialek 1999]
def info_bot(X, Y, t):
U = rv("U")
return (markov(U, X, Y) & (I(Y & U) >= t)).minimum(I(X & U), U)
t1, t2 = real("t1", "t2")
self.assertTrue((t1 <= t2) >> (info_bot(X, Y, t1) <= info_bot(X, Y, t2)))
def test_exists(self):
X, Y, Z, W, U, V = rv("X", "Y", "Z", "W", "U", "V")
self.assertFalse((H(X) > 0).exists(X))
self.assertTrue((H(Y)==1) >> (H(X)==1).exists(X))
self.assertFalse((H(Y)==1) >> (H(X)==2).exists(X))
# Assume there exists a random variable with entropy 1
with exists_bit().assumed():
self.assertTrue((H(X) > 0).exists(X))
self.assertTrue(((H(X) > 0) & indep(X, Y)).exists(X))
self.assertFalse(((H(X) > 0) & indep(X, X)).exists(X))
self.assertTrue((~(H(X) <= 0)).exists(X))
self.assertTrue((~(H(X) <= 0) & indep(X, Y)).exists(X))
self.assertFalse((~(H(X) <= 0) & indep(X, X)).exists(X))
self.assertTrue((~(H(X) <= 0) & ~(H(Y) <= 0)).exists(X + Y))
self.assertFalse((~(H(X) <= 0) & ~(H(Y) <= 0) & indep(X, Y)).exists(X + Y))
self.assertFalse((~(H(X) <= 0) & ~(H(Y | X) <= 0)).exists(X + Y))
# Assume there exists two random variables with entropy 1
with (exists_bit() ** 2).assumed():
self.assertTrue((~(H(X) <= 0) & ~(H(Y) <= 0) & indep(X, Y)).exists(X + Y))
self.assertTrue((~(H(X) <= 0) & ~(H(Y | X) <= 0)).exists(X + Y))
def test_sfrl(self):
X, Y, Z, W, U = rv("X", "Y", "Z", "W", "U")
with PsiOpts(sfrl = "frl"):
self.assertTrue(((I(X & U) == 0) & (H(Y | X + U) == 0)).exists(U))
def test_ratio(self):
X, Y, Z, W, U = rv("X", "Y", "Z", "W", "U")
self.assertEqual(H(X) * 2 / (I(X & Y) + H(X | Y)), 2.0)
self.assertEqual((H(X) + H(Y) - H(X+Y) - I(X&Y)) / (I(X & Y) + H(X | Y)), 0.0)
# self.assertEqual((H(X) + H(Y)) / (H(X) * 3 - H(X) * 3), None)
# self.assertEqual(H(X) * 2 / I(X & Y), None)
self.assertEqual((H(X) * 2 + H(Z) * 6) / (H(Z) * 3 + I(X & Y) + H(X | Y)), 2.0)
# self.assertEqual((H(X) * 2 + H(Z) * 6) / (H(Z) * 2 + I(X & Y) + H(X | Y)), None)
# self.assertEqual((H(X) * 2 + H(Z) * 6) / (H(Z) * 3 + I(X & Y) + H(X | Y) + H(W)), None)
# self.assertEqual((H(X) * 2 + H(Z) * 6 + H(W)) / (H(Z) * 3 + I(X & Y) + H(X | Y)), None)
def test_bayesnet(self):
X, Y, Z, W, U = rv("X", "Y", "Z", "W", "U")
if False:
print(str(((I(X&Y|Z) == 0) & (I(U&X+Z|Y) <= 0)).get_bayesnet()))
print(str(((I(X&Y|Z) == 0) & (I(U&X+Z|Y) <= 0)).get_bayesnet().get_ic()))
print(str((I((W+X) & (Y+U) | Z) == 0).get_bayesnet()))
print(str((I((W+X) & (Y+U) | Z) == 0).get_bayesnet().get_ic()))
print(str(((I(X&Y) == 0) & (I(X+Y&W|Z) <= 0)).get_bayesnet()))
print(str(((I(X&Y) == 0) & (I(X+Y&W|Z) <= 0)).get_bayesnet().get_ic()))
self.assertTrue(((I(X&Y|Z) == 0) & (I(U&X+Z|Y) <= 0)).get_bayesnet().check_ic(I(X&U|Z)))
self.assertFalse(((I(X&Y|Z) == 0) & (I(U&X+Z|Y) <= 0)).get_bayesnet().check_ic(I(X&Y|U)))
self.assertTrue((I((W+X) & (Y+U) | Z) == 0).get_bayesnet().check_ic(I(X&U|Z)))
self.assertFalse((I((W+X) & (Y+U) | Z) == 0).get_bayesnet().check_ic(I(X&U)))
self.assertTrue(((I(X&Y) == 0) & (I(X+Y&W|Z) <= 0)).get_bayesnet().check_ic(I(X&Y)))
self.assertTrue(((I(X&Y) == 0) & (I(X+Y&W|Z) <= 0)).get_bayesnet().check_ic(I(X&W|Z)))
self.assertFalse(((I(X&Y) == 0) & (I(X+Y&W|Z) <= 0)).get_bayesnet().check_ic(I(X&Y|W)))
b = BayesNet([(X, Z), (Y, Z), (Z, W, U)])
b.set_fcn(Z+W)
self.assertFalse(b.check_ic(H(U|X)))
self.assertTrue(b.check_ic(H(W|Z)))
self.assertFalse(b.check_ic(H(W|X)))
self.assertTrue(b.check_ic(H(W|X+Y)))
self.assertTrue(b.check_ic(I(X & Y)))
self.assertTrue(b.check_ic(I(W & W|X+Y)))
self.assertTrue(b.check_ic(I(W & X+W|X+Y)))
self.assertTrue(b.check_ic(I(W & W+Z|X+Y)))
self.assertTrue(b.check_ic(I(W & Z|X+Y)))
self.assertTrue(b.check_ic(I(U & Z|X+Y)))
self.assertFalse(b.check_ic(I(X & Y | Z)))
self.assertFalse(b.check_ic(I(U & U | Z)))
self.assertTrue(b.check_ic(I(X & U | W)))
def test_markov(self):
X = rv_seq("X", 0, 9)
self.assertTrue(markov(*X) >> markov(X[0], X[4], X[8]))
self.assertTrue(markov(*X) >> (I(X[0] & X[8]) <= H(X[4])))
def test_regionop(self):
X, Y, Z, W, U = rv("X", "Y", "Z", "W", "U")
print("test_regionop")
aux = (H(X | U) == 0).exists(U).forall(X).check_getaux()
self.assertTrue(aux is not None)
print(iutil.list_tostr_std(aux))
aux = (H(X | U) == 0).forall(X).exists(U).check_getaux()
self.assertFalse(aux is not None)
aux = (H(U | X) == 0).exists(U).forall(X).check_getaux()
self.assertTrue(aux is not None)
print(iutil.list_tostr_std(aux))
aux = (H(U | X) == 0).forall(X).exists(U).check_getaux()
self.assertTrue(aux is not None)
print(iutil.list_tostr_std(aux))
def test_nested_implication(self):
X, Y, Z, W, U = rv("X", "Y", "Z", "W", "U")
# If H(U|X)=0 implies H(U|Y)=0, then H(X|Y)=0
self.assertTrue(((H(U | X) == 0) >> (H(U | Y) == 0)).forall(U) >> (H(X | Y) == 0))
self.assertTrue(((H(U | X) == 0) >> (H(U | Y) == 0)).forall(U) << (H(X | Y) == 0))
self.assertFalse(((H(U | X) == 0) >> (H(U | Y) == 0)).forall(U) >> (H(Y | X) == 0))
self.assertFalse(((H(U | X) == 0) >> (H(U | Y) == 0)).forall(U) << (H(Y | X) == 0))
self.assertFalse(((H(U | X) == 0) >> (H(U | Y) == 0)).forall(U) >> (H(Z | Y) == 0))
self.assertFalse(((H(U | X) == 0) >> (H(U | Y) == 0)).forall(U) << (H(Z | Y) == 0))
self.assertTrue((H(X | U) == 0).forall(U) >> (H(X) == 0))
# self.assertTrue((H(U | X) == 0).forall(U) >> (H(X) == 0))
self.assertTrue(((H(U | X) == 0) >> (I(Y & Z | U) == 0)).forall(U)
>> ((I(Y & Z) == 0) & (I(Y & Z | X) == 0)))
self.assertTrue(((H(U | X) == 0) >> (I(Y & Z | U) == 0)).forall(U)
>> (I(Y & Z & X) == 0))
self.assertFalse(((H(U | X) == 0) >> (H(U | X) == 0)).forall(U) >> (H(Y | X) == 0))
self.assertTrue(((H(U | X) == 0) >> (I(U & Y) == 0)).forall(U) >>
(((I(U & Y) == 0) >> (H(U | Z) == 0)).forall(U) >> (H(X | Z) == 0)))
self.assertTrue(((I(U & Y) == 0) >> (H(U | Z) == 0)).forall(U) >>
(((H(U | X) == 0) >> (I(U & Y) == 0)).forall(U) >> (H(X | Z) == 0)))
def test_union(self):
X, Y, Z, W, U = rv("X", "Y", "Z", "W", "U")
if False:
print(((H(X) >= H(Z)) | (H(Y) >= H(Z))) >> (H(X+Y) >= H(Z)))
print((((H(X) >= H(Z)) | (H(Y) >= H(Z))) >> (H(X+Y) >= H(Z))).simplified())
print(((H(X) >= H(Z)) | (H(Y) >= H(Z))) >> ((H(X) >= H(Z)) | (H(Y) >= H(Z))))
print((((H(X) >= H(Z)) | (H(Y) >= H(Z))) >> ((H(X) >= H(Z)) | (H(Y) >= H(Z)))).simplified_quick())
print((((H(X) >= H(Z)) | (H(Y) >= H(Z))) >> ((H(X) >= H(Z)) | (H(Y) >= H(Z)))).simplified())
self.assertTrue(((H(X) >= H(Z)) | (H(Y) >= H(Z))) >> (H(X+Y) >= H(Z)))
self.assertTrue(((H(X) >= H(Z)) | (H(Y) >= H(Z))) >> ((H(X) >= H(Z)) | (H(Y) >= H(Z))))
self.assertTrue((H(X) >= H(Z)) >> ((H(X) >= H(Z)) | (H(Y) >= H(Z))))
self.assertTrue(emax(H(X), H(Y)) >= H(X))
self.assertFalse(emax(H(X), H(Y)) <= H(X))
self.assertTrue(emin(H(X), H(Y)) <= H(X))
self.assertFalse(emin(H(X), H(Y)) >= H(X))
self.assertTrue(emax(H(X), H(Y)) >= emin(H(X), H(Y)))
self.assertFalse(emax(H(X), H(Y)) <= emin(H(X), H(Y)))
self.assertFalse(emin(H(X), H(Y)) <= emin(H(X), H(Y), H(Z)))
self.assertTrue(emin(H(X), H(Y)) >= emin(H(X), H(Y), H(Z)))
self.assertTrue((emin(H(X), H(Y)) == 0) >> (I(X&Y) == 0))
self.assertTrue((emax(H(X), H(Y)) == 0) >> (I(X&Y) == 0))
#self.assertTrue(((emin(H(X), H(Y)) <= 0) & (emin(H(Z), H(W)) <= 0))
# >> (indep(X,Z) | indep(X,W) | indep(Y,Z) | indep(Y,W))) # This currently fails
self.assertTrue(emax(H(X), emax(H(Y),H(Z))) >= H(Y))
self.assertTrue(emax(H(X), emax(H(Y),H(Z))) >= emax(H(X), H(Y)))
self.assertFalse(emax(H(X), emax(H(Y),H(Z))) <= emax(H(X), H(Y)))
def test_regterm(self):
X, Y, Z, W, U = rv("X", "Y", "Z", "W", "U")
R = real("R")
a = ((R <= H(X)) & (R <= H(Y))).maximum(R, R, reg_outer = ((R <= H(X)+H(Z)) & (R <= H(Y))))
self.assertTrue(a <= H(X)+H(Z))
self.assertFalse(a <= H(X))
b = ((R >= H(X)) & (R >= H(Y))).minimum(R, R, reg_outer = ((R >= I(X&Z)) & (R <= H(Y))))
self.assertTrue(b >= I(X&Z))
self.assertFalse(b >= H(X))
self.assertFalse(b <= I(X&Z)+H(Y))
self.assertTrue(b <= H(X)+H(Y))
def test_reals(self):
r = universe()
a, b, c, d = real("a", "b", "c", "d")
# Set r to be the region of (a,b,c,d) satisfying a>=b, c<=b, d==c
r &= a >= b
r &= c <= b
r &= d == c
# Test always true statements
# Note that if right hand side is a number, it can only be 0
self.assertTrue(a * 0 == 0)
self.assertTrue(b - b == 0)
self.assertFalse(c * 2 == 0)
# Use the syntax "r >> (...)" to test if r implies ...
self.assertTrue(r >> (b >= c))
self.assertTrue(r >> (a >= d))
self.assertTrue(r >> (c == d))
self.assertTrue(r >> (c >= d))
self.assertTrue(r >> (c <= d))
self.assertFalse(r >> (a <= c))
self.assertFalse(r >> (a >= 0))
def test_mixed(self):
r = universe()
a = real("a")
X, Y, Z, W, U = rv("X", "Y", "Z", "W", "U")
r &= H(X) >= a
r &= a >= H(Y)
# Test always true statements (r is ignored)
self.assertTrue(H(X) >= 0)
self.assertTrue(I(X & Y) - H(X) <= 0)
self.assertTrue(I(X & Y) == H(X) + H(Y) - H(X + Y))
self.assertTrue(I(X & Y) == H(X) - H(X | Y))
self.assertFalse(H(X) >= H(Y))
self.assertFalse(a >= 0)
# Use the syntax "r >> (...)" to test if r implies ...
self.assertTrue(r >> (H(X) >= H(Y)))
self.assertTrue(r >> (a >= 0))
self.assertTrue(r >> (H(X | Y) >= H(Y | X)))
self.assertTrue(r >> (a >= I(X & Y)))
self.assertTrue(r >> (a >= I(X & Y | X)))
self.assertFalse(r >> (a <= I(X & Y)))
self.assertTrue(((H(X|Z+W)==0) & (H(Y|Z) == 0)) >> (H(X+Y|Z+W) == 0))
self.assertFalse(((H(X|Z+W)==0) & (H(Y|Z) == 0)) >> (H(X+Y|Z) == 0))
self.assertFalse(((H(X|Z+W)==0) & (H(Y|Z) == 0)) >> (H(X+Y|W) == 0))
self.assertTrue(((H(X|Y)==0) & (H(Y|Z+W) == 0)) >> (H(X+Y|Z+W) == 0))
self.assertTrue(((H(X|Y)==0) & (H(Y|Z+W) == 0)) >> (H(X|Z+W) == 0))
self.assertFalse(((H(X|Y)==0) & (H(Y|Z+W) == 0)) >> (H(X|W) == 0))
self.assertFalse(((H(X|Y)==0) & (H(Y|Z+W) == 0)) >> (H(X|U) == 0))
def test_simplify(self):
X, Y, Z, W, U = rv("X", "Y", "Z", "W", "U")
R = real("R")
self.assertEqual(str((H(X|Y) + I(X&Y)).simplified()), "H(X)")
self.assertEqual(str((H(X|Y) + H(Y)).simplified()), "H(X,Y)")
self.assertEqual(str((H(X) - I(X&Y)).simplified()), "H(X|Y)")
self.assertEqual(str((H(X) - H(X|Y)).simplified()), "I(X;Y)")
self.assertEqual(str((H(X+Y) - H(X)).simplified()), "H(Y|X)")
self.assertEqual(str((H(X+Y) - H(X|Y)).simplified()), "H(Y)")
bnet = BayesNet([(X, Y, Z)])
self.assertEqual(str((H(X|Y) + I(X&Y|Z)).simplified(bnet = bnet)), "H(X|Z)")
self.assertEqual(str((H(X|Y) + H(Y|Z)).simplified(bnet = bnet)), "H(X,Y|Z)")
self.assertEqual(str((H(X|Z) - H(X|Y)).simplified(bnet = bnet)), "I(X;Y|Z)")
self.assertEqual(str((H(X+Y|Z) - H(X|Y)).simplified(bnet = bnet)), "H(Y|Z)")
self.assertEqual(str(I(X & Y+Z & Z+Y & Z+Y+X).simplified()), "I(X;Z,Y)")
self.assertTrue(I(X & Y+Z & Z+Y & Z+Y+X) == I(X & Y+Z & Z+Y & Z+Y+X).simplified())
self.assertEqual(str(I(X & Y+Z & Z+Y | Z+X).simplified()), "0")
self.assertTrue(I(X & Y+Z & Z+Y | Z+X) == I(X & Y+Z & Z+Y | Z+X).simplified())
self.assertEqual(str((I(X & Y+Z & Z+Y) + H(X) + H(Y|Z+Y) + I(Z+Y & X) - H(X)).simplified()), "2I(X;Z,Y)")
self.assertTrue((I(X & Y+Z & Z+Y) + H(X) + H(Y|Z+Y) + I(Z+Y & X) - H(X))
== (I(X & Y+Z & Z+Y) + H(X) + H(Y|Z+Y) + I(Z+Y & X) - H(X)).simplified())
self.assertEqual(str((H(X | Y + W) - I(X & U | W) * 2 + I(Y & X | W) + H(X | W)).simplified()), "2H(X|W,U)")
self.assertTrue((H(X | Y + W) - I(X & U | W) * 2 + I(Y & X | W) + H(X | W))
== (H(X | Y + W) - I(X & U | W) * 2 + I(Y & X | W) + H(X | W)).simplified())
self.assertEqual(str((H(X|Y+Z) + I(X&Y|Z)).simplified()), "H(X|Z)")
self.assertEqual(str((H(X|Y+Z) + H(Y|Z)).simplified()), "H(X,Y|Z)")
self.assertEqual(str((H(X|Z) - I(X&Y|Z)).simplified()), "H(X|Z,Y)")
self.assertEqual(str((H(X|Z) - H(X|Y+Z)).simplified()), "I(X;Y|Z)")
self.assertEqual(str((H(X+Y|Z) - H(Y|Z)).simplified()), "H(X|Z,Y)")
self.assertEqual(str((H(X+Y|Z) - H(Y|X+Z)).simplified()), "H(X|Z)")
self.assertEqual(str((I(X&W|Y+Z) + I(X&Y&W|Z)).simplified()), "I(X;W|Z)")
self.assertEqual(str((I(X&W|Y+Z) + I(Y&W|Z)).simplified()), "I(X,Y;W|Z)")
self.assertEqual(str((I(X&W|Z) - I(X&Y&W|Z)).simplified()), "I(X;W|Z,Y)")
self.assertEqual(str((I(X&W|Z) - I(X&W|Y+Z)).simplified()), "I(X;W;Y|Z)")
self.assertEqual(str((I(X+Y&W|Z) - I(Y&W|Z)).simplified()), "I(X;W|Z,Y)")
self.assertEqual(str((I(X+Y&W|Z) - I(Y&W|X+Z)).simplified()), "I(X;W|Z)")
self.assertEqual(str((H(X+Y) - H(X) - H(Y)).simplified()), "-I(Y;X)")
self.assertTrue((H(X+Y) - H(X) - H(Y)) == (H(X+Y) - H(X) - H(Y)).simplified())
self.assertEqual(str(((H(X) >= 0) & (I(X & Y | Z) + H(W) >= 0)).simplified()), "")
self.assertEqual(str(((I(Z & W | X) + I(W & U) <= 0) & (H(X) + I(Y & U) == 0)
& (-2*H(U) - 3*I(Y & W) == 0)).simplified()),
("{ H(U) == 0,\n"
" H(X) == 0,\n"
" I(W;Y) == 0,\n"
" I(W;Z) == 0 }"))
# self.assertEqual(str(((I(Z & W | X) + I(W & U) <= 0) & (H(X) + I(Y & U) == 0)
# & (-2*H(U) - 3*I(Y & W) == 0)).simplified(zero_group = 2)),
# "{ I(W;U)+I(Y;U)+I(Y;W)+I(Z;W|X)+H(X)+H(U) <= 0 }")
self.assertEqual(str(((H(X | Y) == 0) & (I(X & Y) <= 0)).simplified()),
"H(X) == 0")
self.assertEqual(str(((H(X) == H(X|Y)) & (I(X & Z) == 0) & (I(Z & Y) == 0) & (H(X+Y+Z) == H(X+Y)+H(Z))).simplified()),
"{ I(X;Y) == 0,\n I(X,Y;Z) == 0 }")
self.assertTrue(((I(X & Y) == 0) & (I(X & Z) == 0) & (I(Z & Y) == 0) & (I(X+Y & Z) == 0))
== ((I(X & Y) == 0) & (I(X & Z) == 0) & (I(Z & Y) == 0) & (I(X+Y & Z) == 0)).simplified())
self.assertEqual(str(((H(X) >= H(Y)) & (H(Y) >= H(X))).simplified()), "H(X) == H(Y)")
self.assertEqual(str(((H(X) >= H(Y)) & (H(Y) == H(X))).simplified()), "H(Y) == H(X)")
self.assertEqual(str(((H(X) >= H(Y)) & (H(Y) >= H(X)) & (H(X) <= H(X)) & (H(X) <= H(Y))).simplified()), "H(X) == H(Y)")
self.assertEqual(str(((H(X) >= H(Y)) & (H(Y) >= H(X)) & (H(X) <= H(X)) & (H(X) <= H(Y)) & (H(Y) <= H(Z)) & (H(Y) == H(Z))).simplified()),
"{ H(X) == H(Y),\n H(Y) == H(Z) }")
self.assertEqual(str(((H(X)<=H(Y)) | (H(X)>=H(Y))).simplified()), "")
self.assertEqual(str(((H(X)<=H(Y)) | ((H(X)>=H(Y)) & (H(X)<=H(Z))) | ((H(X)>=H(Y)) & (H(X)>=H(Z)))).simplified()), "")
self.assertEqual(str((((H(X)>=H(Y)) & (H(X)<=H(Z))) | ((H(X)>=H(Y)) & (H(X)>=H(Z))) | (H(X)<=H(Y))).simplified()), "")
self.assertEqual(str(((H(X)<=H(Y)) | ((H(X)>=H(Y) + H(Z))) | ((H(X)>=H(Y) - H(Z)))).simplified()), "")
self.assertEqual(str(((H(X)<=H(Y)) | ((H(X)>=H(Y) + R)) | ((H(X)>=H(Y) - R))).simplified()), "")
self.assertEqual(str((~(H(X)<=H(Y)) | (H(X)<=H(Y+Z))).simplified()), "")
self.assertEqual(str(((H(X)<=H(Y)) | (H(X)<=H(Y+Z))).simplified()), "H(X) <= H(Y,Z)")
self.assertEqual(str((~(H(X)<=H(Y)) | (H(X)>=H(Y))).simplified()), "H(Y) <= H(X)")
self.assertEqual(str((~(H(X)<=H(Y)) & (H(X)>=H(Y))).simplified()), "{\n NOT{ H(X) <= H(Y) }\n}")
self.assertEqual(str((~(H(X)<=H(Y)) | (H(X)>=H(Y)+R) | (R >= 0)).simplified()), "{\n { R <= H(X)-H(Y) }\n OR\n { R >= 0 }\n}")
self.assertEqual(str((~(H(X)<=H(Y)) | (H(X)>=H(Y)+R) | (R <= 0)).simplified()), "{\n NOT{ H(X) <= H(Y) }\n OR\n { R <= 0 }\n}")
def test_eliminate(self):
R = real("R")
X, Y, Z, W, U = rv("X", "Y", "Z", "W", "U")
self.assertEqual(str(
(((R >= H(Y)) & (R <= H(Z)))).exists(R)),
"H(Y) <= H(Z)")
self.assertEqual(str(
(((R >= H(Y)) & (R <= H(Z)) & (R == H(W)))).exists(R)),
("{ H(W) <= H(Z),\n"
" H(Y) <= H(W) }"))
self.assertEqual(str(
((R == H(X)) >> ((R >= H(Y)) & (R <= H(Z)) & (R == H(W)))).forall(R).simplified()),
("{ H(X) <= H(Z),\n"
" H(Y) <= H(X),\n"
" H(W) == H(X) }"))
def test_eliminate_ci(self):
R = real("R")
X, Y, Z, W, U = rv("X", "Y", "Z", "W", "U")
self.assertEqual(str((R >= H(X)).exists(X, method = "ci")), "R >= 0")
self.assertEqual(str((R >= I(X + Y & W | Z)).exists(X, method = "ci")), "R >= I(W;Y|Z)")
self.assertEqual(str((R <= I(X + Y & W | Z)).exists(X, method = "ci")), "")
self.assertEqual(str((R <= I(X + Y & W | Z)).exists(Z, method = "ci")), "")
self.assertEqual(str(((R >= I(X + Y & W | Z)) & indep(X, U)).exists(Z, method = "ci")), "I(U;X) == 0")
self.assertEqual(str(((R <= H(X + Y | Z)) & indep(X, U+Z)).exists(Z, method = "ci")), "{ R <= H(X,Y),\n I(U;X) == 0 }")
# self.assertEqual(str(((R >= H(X + Y | Z)) >> indep(X, U+Z)).exists(Z, method = "ci")), "(R >= H(X,Y)\n=> I(U;X) == 0)")
# self.assertEqual(str(((R <= H(X + Y | Z)) >> indep(X, U+Z)).exists(Z, method = "ci")), "")
self.assertEqual(str((R >= I(X + Y & W & U | Z)).exists(X, method = "ci")), "")
def test_sum(self):
R, R1, R2 = real("R", "R1", "R2")
X, Y, Z, W, U = rv("X", "Y", "Z", "W", "U")
r = (R1 >= 0) & (R2 >= 0) & (R1 <= H(X)) & (R2 <= H(Y))
r2 = (R1 >= 0) & (R2 >= 0) & (R1 + R2 <= H(Z))
self.assertEqual(str(r * 2),
("{ R1 >= 0,\n"
" R2 >= 0,\n"
" R1 <= 2H(X),\n"
" R2 <= 2H(Y) }"))
self.assertEqual(str(r + r),
("{ R1 >= 0,\n"
" R1 <= 2H(X),\n"
" R2 >= 0,\n"
" R2 <= 2H(Y) }"))
self.assertEqual(str(r + r2),
("{ R1 >= 0,\n"
" R2 >= 0,\n"
" R1+R2 <= H(Z)+H(X)+H(Y),\n"
" R1 <= H(Z)+H(X),\n"
" R2 <= H(Z)+H(Y) }"))
self.assertEqual(str(r + (r | r2)),
("{\n"
" { R1 >= 0,\n"
" R1 <= 2H(X),\n"
" R2 >= 0,\n"
" R2 <= 2H(Y) }\n"
" OR\n"
" { R1 >= 0,\n"
" R2 >= 0,\n"
" R1 <= H(X)+H(Z),\n"
" R2 <= H(Y)+H(Z),\n"
" R1+R2 <= H(Y)+H(X)+H(Z) }\n"
"}"))
def test_cases(self):
R, R2 = real("R", "R2")
X, Y, Z, W, U = rv("X", "Y", "Z", "W", "U")
print("test_cases")
self.assertTrue((R >= 0) | (R <= 0))
self.assertTrue((R >= 0) | ~(R >= 0))
self.assertTrue((R == 0) | ~(R <= 0) | ~(R >= 0))
self.assertFalse(~(R <= 0) | ~(R >= 0))
self.assertFalse(~(R == 0) | ~(R >= 0))
self.assertTrue(~(R == 0) | (R >= 0))
self.assertFalse((R == 0) | ~(R >= 0))
self.assertTrue((H(X) >= H(Y)) | (H(Y) >= H(X)))
self.assertFalse((H(X) >= H(Y)) | ((H(Y) >= H(X)) & (H(Y) >= H(Z))))
with PsiOpts(auxsearch_leaveone = True):
self.assertTrue((R >= 0) | (R <= 0))
self.assertTrue((R >= 0) | ((R <= 0) & (R2 <= R)) | ((R <= 0) & (R2 >= R)))
aux = ((I(X&Y|U) >= I(X&Y|Z)) & (I(X&Y|U) >= I(X&Y))).exists(U).check_getaux()
self.assertTrue(aux is not None)
print(iutil.list_tostr_std(aux))
aux = RegionOp.inter([(I(X&Y|U) >= I(X&Y|Z)) & (I(X&Y|U) >= I(X&Y))]).exists(U).check_getaux()