# -*- coding: utf-8 -*-
"""
Boolean Node
=============
Main class for Boolean node objects.
"""
# Copyright (C) 2021 by
# Rion Brattig Correia <rionbr@gmail.com>
# Alex Gates <ajgates@gmail.com>
# Etienne Nzabarushimana <enzabaru@indiana.edu>
# All rights reserved.
# MIT license.
from __future__ import division
import math
from itertools import combinations, compress, product
from statistics import mean
import networkx as nx
import numpy as np
import pandas as pd
import cana.canalization.boolean_canalization as BCanalization
import cana.canalization.cboolean_canalization as cBCanalization
import cana.symmetry as symmetry
import cana.sensitivity as sensitivity
from cana.cutils import (
binstate_to_statenum,
flip_bit,
outputs_to_binstates_of_given_type,
statenum_to_binstate,
)
from cana.utils import input_monotone, ncr
[docs]
class BooleanNode(object):
""" """
def __init__(
self,
id=0,
name="x",
k=1,
inputs=[1],
state=False,
outputs=[0, 1],
constant=False,
network=None,
verbose=False,
*args,
**kwargs
):
self.id = id # the id of the node
self.name = name # the name of the node
self.k = k # k is the number of inputs
self.inputs = list(map(int, inputs)) # the ids of the input nodes
self.state = str(
int(state)
) # the initial state of the node. internally this is treated as a string.
self.outputs = list(map(str, outputs)) # the list of transition outputs
self.network = network # the BooleanNetwork object this nodes belongs to
self.verbose = verbose # verbose mode
# mask for inputs
if len(self.inputs) > 0:
self.mask = [(i in self.inputs) for i in range(max(self.inputs) + 1)]
else:
self.mask = []
# Consistency
if (k != 0) and (k != int(np.log2(len(outputs)))):
raise ValueError(
"Number of k (inputs) do not match the number of output transitions"
)
# If all outputs are either positive or negative, the node is treated as a constant.
if (len(set(outputs)) == 1) or (constant):
self.set_constant(constant=True, state=outputs[0])
else:
self.set_constant(constant=False)
# Canalization Variables
self._prime_implicants = (
None # A tuple of negative and positive prime implicants.
)
self._two_symbols = None # The Two Symbol (TS) Schemata
self._pi_coverage = None # The Coverage of inputs by Prime Implicants schemata
self._ts_coverage = None # The Coverage of inputs by Two Symbol schemata
def __str__(self):
if len(self.outputs) > 10:
outputs = (
"["
+ ",".join(map(str, self.outputs[:4]))
+ "..."
+ ",".join(map(str, self.outputs[-4:]))
+ "]"
)
else:
outputs = "[" + ",".join(map(str, self.outputs)) + "]"
inputs = "[" + ",".join(map(str, self.inputs)) + "]"
return "<BNode(id={id}, name='{name}', k={k}, inputs={inputs:s}, state='{state}', outputs={outputs} constant={constant})>".format(
id=self.id,
name=self.name,
k=self.k,
inputs=inputs,
state=self.state,
outputs=outputs,
constant=self.constant,
)
[docs]
@classmethod
def from_output_list(cls, outputs=list(), *args, **kwargs):
"""Instanciate a Boolean Node from a output transition list.
Args:
outputs (list) : The transition outputs of the node. Its length must be a power of two, :math:`2^k`.
Returns:
(BooleanNode) : the instanciated object. Subclasses receive an instance of the subclass.
Raises:
ValueError : if the length of ``outputs`` is not a power of two.
Example:
>>> BooleanNode.from_output_list(outputs=[0,0,0,1], name="AND")
"""
n = len(outputs)
if n == 0 or n & (n - 1):
raise ValueError(
"The length of `outputs` must be a power of two (2**k); got {:d}.".format(n)
)
k = n.bit_length() - 1
id = kwargs.pop("id") if "id" in kwargs else 0
name = kwargs.pop("name") if "name" in kwargs else "x"
inputs = (
kwargs.pop("inputs") if "inputs" in kwargs else [(x + 1) for x in range(k)]
)
state = kwargs.pop("state") if "state" in kwargs else False
return cls(
id=id,
name=name,
k=k,
inputs=inputs,
state=state,
outputs=outputs,
*args,
**kwargs
)
[docs]
def set_constant(self, constant=True, state=None):
"""Sets whether the node is to be treated as a contant
Args:
constant (Boolean) : Whether to set or unset the node as a constant.
state (str; optional) : The state value to which to set the node. Either '0' or '1'; default to current state value.
"""
if constant:
self.constant = True
self.step = self.constant_step
else:
self.constant = False
self.step = self.dynamic_step
# Set an optional value to the node
if state is not None:
self.state = str(int(state))
[docs]
def constant_step(self, input_state):
"""Treat the node as a constant variable, always returning its state."""
return self.state
[docs]
def dynamic_step(self, input_state):
"""Returns the output of the node based on a specific input.
Args:
input (str) : an input to the node (e.g.: '0111' -> 7).
Returns:
output (bool) : the output value.
"""
self.state = self.outputs[binstate_to_statenum(input_state)]
return self.state
[docs]
def edge_redundancy(self, bound="mean"):
r"""The Edge Redundancy :math:`r_{i}` is the mean number of unnecessary inputs (or ``#``) in the Prime Implicants Look Up Table (LUT) for that input.
Since there may be more than one redescription schema for each input entry, the input redundancy is bounded by an upper and lower limit.
.. math::
r_i(x_i) = \frac{ \sum_{f_{\alpha} \in F} \Phi_{\theta:f_{\alpha} \in \Theta_{\theta}} (X^{\#}_{\theta_i} ) }{ |F| }
where :math:`\Phi` is a function (:math:`min` or :math:`max`) and :math:`F` is the node LUT.
Args:
bound (string) : The bound to which compute input redundancy.
Mode "input" accepts: ["lower", "mean", "ave", upper", "tuple"].
Defaults to "mean".
Returns:
(list) : The list of :math:`r_i` for inputs.
Note:
The complete mathematical description can be found in :cite:Gates:2020`.
See also:
:func:`effective_connectivity`, :func:`input_symmetry`.
"""
self._check_compute_canalization_variables(pi_coverage=True)
redundancies = []
# Generate a per input coverage
# ex: {0: {'11': [], '10': [], '00': [], '01': []}, 1: {'11': [], '10': [], '00': [], '01': []}}
# pi_edge_coverage = { input : { binstate: [ pi[input] for pi in pis ] for binstate,pis in self._pi_coverage.items() } for input in range(self.k) }
pi_edge_coverage = cBCanalization.input_wildcard_coverage(self._pi_coverage)
# Loop ever input node
for edge, binstates2wildcard in pi_edge_coverage.items():
# {'numstate': [matches], '10': [True,False,True,...] ...}
# countslenghts = {binstate_to_statenum(binstate): ([pi=='#' for pi in pis]) for binstate,pis in binstates.items() }
# A triplet of (min, mean, max) values
if bound == "lower":
redundancy = (
sum([all(pi) for pi in binstates2wildcard.values()]) / 2**self.k
) # min(r_i)
elif bound == "mean" or bound == "avg":
redundancy = (
sum([sum(pi) / len(pi) for pi in binstates2wildcard.values()])
/ 2**self.k
) # <r_i>
elif bound == "upper":
redundancy = (
sum([any(pi) for pi in binstates2wildcard.values()]) / 2**self.k
) # max(r_i)
elif bound == "tuple":
redundancy = (
sum([all(pi) for pi in binstates2wildcard.values()]) / 2**self.k,
sum([any(pi) for pi in binstates2wildcard.values()]) / 2**self.k,
) # (min,max)
else:
raise AttributeError(
'The bound you selected does not exist. Try "upper", "mean", "lower" or "tuple".'
)
redundancies.append(redundancy)
return redundancies # r_i
[docs]
def effective_connectivity(self, operator=mean, norm=True):
r"""The Effective Connectiviy is the mean number of input nodes needed to determine the transition of the node.
.. math::
k_e(x) = k(x) - k_r(x)
Args:
operator (function) : The operator to use while computing input redundancy for the node.
Defaults to `statistics.mean`, but `min` or `max` can be also considered.
norm (bool) : Normalized between [0,1].
Use this value when comparing nodes with different input sizes. (Defaults to "True".)
:math:`k^{*}_e(x) = \frac{ k_e(x) }{ k(x) }`.
Returns:
(float) : The :math:`k_e` value.
See Also:
:func:`input_redundancy`, :func:`input_symmetry`, :func:`~cana.boolean_network.BooleanNetwork.effective_graph`.
"""
k_r = self.input_redundancy(operator=operator, norm=False)
#
k_e = self.k - k_r
if norm:
k_e = k_e / self.k
return k_e
[docs]
def edge_effectiveness(self, bound="mean"):
r"""The Edge Effectiveness is the mean number of an input's states needed to determine the transition of the node.
.. math::
e_i(x_i) = 1 - r_i(x_i)
Args:
bound (string) : The bound for the :math:`k_r` Input Redundancy
Returns:
(list) : The list of :math:`e_r` values.
See Also:
:func:`input_redundancy`, :func:`input_symmetry`, :func:`~cana.boolean_network.BooleanNetwork.effective_graph`.
"""
e_i = [1.0 - x_i for x_i in self.edge_redundancy(bound=bound)]
return e_i
[docs]
def symKernel_numDots(self, ts, sameSymbol=False):
"""compute the number of variables involved in a symmetry group.
Args:
ts ([str, [[]], [[]]]) : two symbol schema
sameSymbol (bool) : whether or not to consider same-symbol symmetry
Returns:
(int)
"""
if not sameSymbol:
groups = ts[1]
else:
groups = ts[1] + ts[2]
return len(set(i for group in groups for i in group))
def _input_symmetry(self, aggOp, kernel):
self._check_compute_canalization_variables(ts_coverage=True)
summand = []
for fAlpha, fTheta in self._ts_coverage.items():
summand.append(aggOp(list(map(kernel, fTheta))))
return math.fsum(summand) / len(summand)
# refactor ks for speed, avg op only
[docs]
def distinct_symmetry(self):
"""Compute the distinct permutation symmetry of the node LUT.
For each LUT entry, this computes the fraction of distinct input
permutations that preserve the same output, excluding the identity
permutation from both numerator and denominator.
Returns:
(float)
See also:
:func:`cana.symmetry.distinct_symmetry`
"""
return symmetry.distinct_symmetry(self.outputs, self.k)
[docs]
def raw_symmetry(self):
"""Compute the raw symmetry of the node LUT.
LUT rows are grouped by input Hamming weight. For each row, this
computes the fraction of rows in the same weight group that have the
same output, then averages across all LUT rows.
Returns:
(float)
See also:
:func:`cana.symmetry.raw_symmetry`
"""
return symmetry.raw_symmetry(self.outputs, self.k)
[docs]
def look_up_table(self):
"""Returns the Look Up Table (LUT)
Returns:
(pandas.DataFrame): the LUT
Examples:
>>> AND = BooleanNode.from_output_list([0,0,0,1])
>>> AND.look_up_table()
See also:
:func:`schemata_look_up_table`
"""
d = []
if (self.k < 2) and (self.constant is False):
k = 2
else:
k = self.k
for statenum, output in zip(range(2**k), self.outputs):
# Binary State, Transition
d.append((statenum_to_binstate(statenum, base=self.k), output))
df = pd.DataFrame(d, columns=["In:", "Out:"])
return df
[docs]
def schemata_look_up_table(
self,
type="pi",
pi_symbol="#",
ts_symbol_list=["\u030A", "\u032F", "\u0303", "\u0330", "\u0306", "\u032E"],
):
"""Returns the simplified schemata Look Up Table (LUT)
Args:
type (string) : The type of schemata to return, either Prime Implicants ``pi`` or Two-Symbol ``ts``. Defaults to 'pi'.
pi_symbol (str) : The Prime Implicant don't care symbol. Default is ``#``.
ts_symbol_list (list) : A list containing Two Symbol permutable symbols, one per permutation group of a schema (combining marks: ring above, inverted breve below, tilde, tilde below, breve, breve below). A ``ValueError`` is raised if a schema has more groups than symbols.
Returns:
(pandas.DataFrame or Latex): the schemata LUT
Examples:
>>> AND = BooleanNode.from_output_list([0,0,0,1])
>>> AND.schemata_look_up_table(type='pi')
Note:
See the full list of `combining characters <https://en.wikipedia.org/wiki/Combining_character>`_ to use other symbols as the permutation symbol.
See also:
:func:`look_up_table`
"""
r = []
# Prime Implicant LUT
if type == "pi":
self._check_compute_canalization_variables(prime_implicants=True)
pi0s = self._prime_implicants.get("0", [])
pi1s = self._prime_implicants.get("1", [])
for output, pi in zip([0, 1], [pi0s, pi1s]):
for schemata in sorted(pi):
r.append((schemata, output))
# Two Symbol LUT
elif type == "ts":
self._check_compute_canalization_variables(two_symbols=True)
ts0s, ts1s = self._two_symbols
for output, ts in zip([0, 1], [ts0s, ts1s]):
for i, (schemata, permutables, samesymbols) in enumerate(ts):
if max(len(permutables), len(samesymbols)) > len(ts_symbol_list):
raise ValueError(
"Schema %s has %d permutation groups but ts_symbol_list has only %d symbols; pass a longer ts_symbol_list."
% (schemata, max(len(permutables), len(samesymbols)), len(ts_symbol_list))
)
string = ""
if len(permutables):
string += "("
# Permutable
for j, permutable in enumerate(permutables):
#
ts_symbol_unicode = ts_symbol_list[j]
if j > 0:
string += " | "
string += "".join(
[
x
if (k not in permutable)
else "%s%s" % (x, ts_symbol_unicode)
for k, x in enumerate(schemata, start=0)
]
)
string += ")"
else:
string += "(" + schemata + ")"
if len(samesymbols):
string += " | ("
# Same Symbol
for j, samesymbol in enumerate(samesymbols):
ts_symbol_unicode = ts_symbol_list[j]
if j > 0:
string += " | "
string += "".join(
[
x
if (k not in samesymbol)
else "%s%s" % (x, ts_symbol_unicode)
for k, x in enumerate(schemata, start=0)
]
)
string += ")"
r.append((string, output))
else:
raise AttributeError(
'The schemata type could not be found. Try "PI" (Prime Implicants) or "TS" (Two-Symbol).'
)
r = [(schemata.replace("2", pi_symbol), output) for schemata, output in r]
return pd.DataFrame(r, columns=["Input", "Output"])
[docs]
def activities(self):
"""compute the activities of each incoming edge of the node, directly from the LUT.
See :func:`cana.sensitivity.activities`.
Returns:
(list of floats)
"""
return sensitivity.activities(self.outputs, self.k)
[docs]
def sensitivity(self, norm=False):
"""compute the average sensitivity of the node: the mean, over all input states,
of the number of single-input flips that change the output.
Delegates to :func:`cana.sensitivity.sensitivity`, which computes it directly from
the look-up table. Up to CANA 1.0.2 this was ``sum(self.activities())``, kept as
:func:`cana.sensitivity.sensitivity_old`; the two are bit-exactly equal, as
asserted in ``tests/test_boolean_node.py``
(``test_sensitivity_matches_original_implementation``).
Args:
norm (bool) : whether or not to normalize by the number of inputs (k)
Returns:
(float)
See Also:
:func:`cana.sensitivity.sensitivity`, :func:`activities`, :func:`c_sensitivity`.
"""
return sensitivity.sensitivity(self.outputs, self.k, norm=norm)
[docs]
def canalizing_map(self, output=None):
"""Computes the node Canalizing Map (CM).
Args:
output (int) : The output CM to return. Default is ``None``, retuning both [0,1].
Returns:
CM (networkx.DiGraph) : a directed graph representation of the CM.
See Also:
:func:`boolean_network.dynamics_canalization_map` for the DCM and :func:`drawing.draw_canalizing_map_graphviz` for plotting.
"""
self._check_compute_canalization_variables(two_symbols=True)
ts0s, ts1s = self._two_symbols
G = nx.DiGraph(name="CM: %s" % self.name)
# Outputs
if output is None or output == 0:
G.add_node(
"var-{nid:d}-out-0".format(nid=self.id),
**{
"label": self.name,
"type": "variable",
"mode": "output",
"value": 0,
"constant": self.constant,
"group": self.id,
}
)
if output is None or output == 1:
G.add_node(
"var-{nid:d}-out-1".format(nid=self.id),
**{
"label": self.name,
"type": "variable",
"mode": "output",
"value": 1,
"constant": self.constant,
"group": self.id,
}
)
tid = 0
for out, tspsss in zip([0, 1], self._two_symbols):
# Only return the requested output
if (not len(tspsss)) or ((output is not None) and (output != out)):
continue
for ts, ps, ss in tspsss:
lits = []
group0 = []
group1 = []
# group2 = []
# Tau is the threshold, counted as the sum of (0's and 1's literals; 0's in permutation group; 1's in permutation group)
nlit, ngrp0, ngrp1 = 0, 0, 0
for j in range(self.k):
# Is this input in any permutation group?
input = ts[j]
if not any([j in group for group in ps]):
if ts[j] in ["0", "1"]:
nlit += 1
source = j
lits.append(source)
else:
if ts[j] == "0":
ngrp0 += 1
group0.append(j)
elif ts[j] == "1":
ngrp1 += 1
group1.append(j)
tau = nlit + ngrp0 + ngrp1
# Threshold Node
tname = "thr-{tid:d}-var-{nid:d}-out-{out:d}".format(
tid=tid, nid=self.id, out=out
)
label = "{:d}".format(tau)
G.add_node(
tname,
**{
"label": label,
"type": "threshold",
"tau": tau,
"group": self.id,
}
)
# Add Edges from Threshold node to output
xname = "var-{nid:d}-out-{out:d}".format(nid=self.id, out=out)
if G.has_node(tname) and G.has_node(xname):
G.add_edge(tname, xname, **{"type": "out"})
else:
raise BaseException("Adding edge to node that does not exist.")
# Literal Edges
for lit in lits:
iname = "var-{nid:d}-out-{out:d}".format(
nid=self.inputs[lit], out=int(ts[lit])
)
if iname not in G.nodes():
# Can we get the name of ther nodes? Are we attached to a network?
if self.network is not None:
ilabel = self.network.get_node_name(self.inputs[lit])[0]
else:
ilabel = str(self.inputs[lit])
iout = int(ts[lit])
G.add_node(
iname,
**{
"label": ilabel,
"type": "variable",
"mode": "input",
"value": iout,
"group": self.id,
}
)
G.add_edge(iname, tname, **{"type": "literal"})
# Group0
for fusion in range(ngrp0):
fname = "fus-{fus:d}-thr-{thr:d}-var-{nid:d}-out-{out:d}".format(
fus=fusion, thr=tid, nid=self.id, out=0
)
G.add_node(fname, **{"type": "fusion", "group": self.id})
for input in ps[0]:
iname = "var-{nid:d}-out-{out:d}".format(
nid=self.inputs[input], out=0
)
if iname not in G.nodes():
# Can we get the name of ther nodes? Are we attached to a network?
if self.network is not None:
ilabel = self.network.get_node_name(self.inputs[input])[
0
]
else:
ilabel = str(self.inputs[input])
G.add_node(
iname,
**{
"label": ilabel,
"type": "variable",
"mode": "input",
"value": 0,
"group": self.id,
}
)
G.add_edge(iname, fname, **{"type": "fusing"})
G.add_edge(fname, tname, **{"type": "fused"})
# Group1
for fusion in range(ngrp1):
fname = "fus-{fus:d}-thr-{thr:d}-var-{nid:d}-out-{out:d}".format(
fus=fusion, thr=tid, nid=self.id, out=1
)
G.add_node(fname, **{"type": "fusion", "group": self.id})
for input in ps[0]:
iname = "var-{var:d}-out-{out:d}".format(
var=self.inputs[input], out=1
)
if iname not in G.nodes():
# Can we get the name of ther nodes? Are we attached to a network?
if self.network is not None:
ilabel = self.network.get_node_name(self.inputs[input])[
0
]
else:
ilabel = str(self.inputs[input])
iout = ts[input]
G.add_node(
iname,
**{
"label": ilabel,
"type": "variable",
"mode": "input",
"value": 1,
"group": self.id,
}
)
G.add_edge(iname, fname, **{"type": "fusing"})
G.add_edge(fname, tname, **{"type": "fused"})
tid += 1
return G
[docs]
def pi_coverage(self):
"""Returns the :math:`F'` (Prime Implicants) binary state coverage.
Returns:
(list)
See also:
:func:`ts_coverage`
"""
self._check_compute_canalization_variables(pi_coverage=True)
return self._pi_coverage
[docs]
def ts_coverage(self):
"""Returns the :math:`F''` (Two-Symbol schematas) binary state coverage.
Returns:
(list)
See also:
:func:`pi_coverage`
"""
self._check_compute_canalization_variables(ts_coverage=True)
return self._ts_coverage
def _check_compute_canalization_variables(self, **kwargs):
"""Recursevely check if the requested canalization variables are instantiated/computed, otherwise computes them in order.
For example: to compute `two_symbols` we need `prime_implicants` first.
Likewise, to compute `prime_implicants` we need the `transition_density_table` first.
"""
if "prime_implicants" in kwargs:
if self._prime_implicants is None:
self._prime_implicants = dict()
for output in sorted(set(self.outputs)):
output_binstates = outputs_to_binstates_of_given_type(
self.outputs, output=output, k=self.k
)
self._prime_implicants[output] = cBCanalization.find_implicants_qm(
input_binstates=output_binstates
)
elif "pi_coverage" in kwargs:
self._check_compute_canalization_variables(prime_implicants=True)
if self._pi_coverage is None:
self._pi_coverage = dict()
for output, piset in self._prime_implicants.items():
self._pi_coverage.update(cBCanalization.return_pi_coverage(piset))
# make sure every inputstate was covered by at least one prime implicant
assert len(self._pi_coverage) == 2**self.k
elif "two_symbols" in kwargs:
self._check_compute_canalization_variables(prime_implicants=True)
if self._two_symbols is None:
# this is a temporary fix until we update 'find_two_symbols' to cython.
if "0" in self._prime_implicants:
pi0 = self._prime_implicants["0"]
pi0 = set([pi.replace("#", "2") for pi in pi0])
else:
pi0 = []
if "1" in self._prime_implicants:
pi1 = self._prime_implicants["1"]
pi1 = set([pi.replace("#", "2") for pi in pi1])
else:
pi1 = []
# /end fix
self._two_symbols = (
BCanalization.find_two_symbols_v2(k=self.k, prime_implicants=pi0),
BCanalization.find_two_symbols_v2(k=self.k, prime_implicants=pi1),
)
elif "ts_coverage" in kwargs:
self._check_compute_canalization_variables(two_symbols=True)
if self._ts_coverage is None:
self._ts_coverage = BCanalization.computes_ts_coverage(
self.k, self.outputs, self._two_symbols
)
else:
raise Exception("Canalization variable name not found. %s" % kwargs)
return True
[docs]
def bias(self):
r"""The node bias. The sum of the boolean output transitions divided by the number of entries (:math:`2^k`) in the LUT.
.. math::
bias(x) = \frac{ \sum_{f_{\alpha}\in F} s_{\alpha} }{ |F| }
Returns:
(float)
See Also:
:func:`~cana.boolean_network.BooleanNetwork.network_bias`
"""
return sum(map(int, self.outputs)) / 2**self.k
[docs]
def c_sensitivity(self, c, mode="default", max_k=0):
"""Node c-sensitivity.
c-sensitivity is defined as: the mean probability that changing exactly
``c`` variables in input variables would change output value.
There is another mode "forceK", which will be used to calculate Derrida value.
In that mode, it would assume the number of input variables is specified as max_k
this methods is equivalent to Derrida value in :cite:`Kadelka:2017`, only move a normalization
coefficient from expression of Derrida value to c-sensitivity to simplify it
Args:
c (int) : the ``c`` in the definition of c-senstivity above
mode (string) : either "default" or "forceK"
max_k (int) : you must specify max_k when you set mode as 'forceK'
Returns:
(float)
See Also:
:func:`~cana.boolean_network.derrida_curve`
"""
S_c_f = 0
ic = min(c, self.k)
if mode != "forceK":
for j in product("01", repeat=self.k):
origin_config = list(j)
for mut in combinations(range(self.k), ic):
mut_config = origin_config[:]
for i_mut in mut:
mut_config[i_mut] = flip_bit(mut_config[i_mut])
if self.step("".join(origin_config)) != self.step(
"".join(mut_config)
):
S_c_f += 1
if S_c_f == 0:
return 0.0
return S_c_f / float(ncr(self.k, c)) / float(2**self.k)
else:
assert max_k >= self.k
for ic in range(max(1, c + self.k - max_k), min(c, self.k) + 1):
for j in product("01", repeat=self.k):
origin_config = list(j)
for mut in combinations(range(self.k), ic):
mut_config = origin_config[:]
for i_mut in mut:
mut_config[i_mut] = flip_bit(mut_config[i_mut])
if self.step("".join(origin_config)) != self.step(
"".join(mut_config)
):
S_c_f += ncr(max_k - self.k, c - ic)
return S_c_f / float(ncr(max_k, c)) / float(2**self.k)