utils4e.py

"""Provides some utilities widely used by other modules"""

import bisect
import collections
import collections.abc
import heapq
import operator
import os.path
import random
import math
import functools
import numpy as np
from itertools import chain, combinations
from statistics import mean
import warnings

# part1. General data structures and their functions
# ______________________________________________________________________________
# Queues: Stack, FIFOQueue, PriorityQueue
# Stack and FIFOQueue are implemented as list and collection.deque
# PriorityQueue is implemented here


class PriorityQueue:
    """A Queue in which the minimum (or maximum) element (as determined by f and
    order) is returned first.
    If order is 'min', the item with minimum f(x) is
    returned first; if order is 'max', then it is the item with maximum f(x).
    Also supports dict-like lookup."""

    def __init__(self, order='min', f=lambda x: x):
        self.heap = []

        if order == 'min':
            self.f = f
        elif order == 'max':  # now item with max f(x)
            self.f = lambda x: -f(x)  # will be popped first
        else:
            raise ValueError("order must be either 'min' or 'max'.")

    def append(self, item):
        """Insert item at its correct position."""
        heapq.heappush(self.heap, (self.f(item), item))

    def extend(self, items):
        """Insert each item in items at its correct position."""
        for item in items:
            self.append(item)

    def pop(self):
        """Pop and return the item (with min or max f(x) value)
        depending on the order."""
        if self.heap:
            return heapq.heappop(self.heap)[1]
        else:
            raise Exception('Trying to pop from empty PriorityQueue.')

    def __len__(self):
        """Return current capacity of PriorityQueue."""
        return len(self.heap)

    def __contains__(self, key):
        """Return True if the key is in PriorityQueue."""
        return any([item == key for _, item in self.heap])

    def __getitem__(self, key):
        """Returns the first value associated with key in PriorityQueue.
        Raises KeyError if key is not present."""
        for value, item in self.heap:
            if item == key:
                return value
        raise KeyError(str(key) + " is not in the priority queue")

    def __delitem__(self, key):
        """Delete the first occurrence of key."""
        try:
            del self.heap[[item == key for _, item in self.heap].index(True)]
        except ValueError:
            raise KeyError(str(key) + " is not in the priority queue")
        heapq.heapify(self.heap)

# ______________________________________________________________________________
# Functions on Sequences and Iterables


def sequence(iterable):
    """Converts iterable to sequence, if it is not already one."""
    return (iterable if isinstance(iterable, collections.abc.Sequence)
            else tuple([iterable]))


def removeall(item, seq):
    """Return a copy of seq (or string) with all occurrences of item removed."""
    if isinstance(seq, str):
        return seq.replace(item, '')
    else:
        return [x for x in seq if x != item]


def unique(seq):
    """Remove duplicate elements from seq. Assumes hashable elements."""
    return list(set(seq))


def count(seq):
    """Count the number of items in sequence that are interpreted as true."""
    return sum(map(bool, seq))


def multimap(items):
    """Given (key, val) pairs, return {key: [val, ....], ...}."""
    result = collections.defaultdict(list)
    for (key, val) in items:
        result[key].append(val)
    return dict(result)


def multimap_items(mmap):
    """Yield all (key, val) pairs stored in the multimap."""
    for (key, vals) in mmap.items():
        for val in vals:
            yield key, val


def product(numbers):
    """Return the product of the numbers, e.g. product([2, 3, 10]) == 60"""
    result = 1
    for x in numbers:
        result *= x
    return result


def first(iterable, default=None):
    """Return the first element of an iterable; or default."""
    return next(iter(iterable), default)


def is_in(elt, seq):
    """Similar to (elt in seq), but compares with 'is', not '=='."""
    return any(x is elt for x in seq)


def mode(data):
    """Return the most common data item. If there are ties, return any one of them."""
    [(item, count)] = collections.Counter(data).most_common(1)
    return item


def powerset(iterable):
    """powerset([1,2,3]) --> (1,) (2,) (3,) (1,2) (1,3) (2,3) (1,2,3)"""
    s = list(iterable)
    return list(chain.from_iterable(combinations(s, r) for r in range(len(s) + 1)))[1:]


# ______________________________________________________________________________
# argmin and argmax

identity = lambda x: x

argmin = min
argmax = max


def argmin_random_tie(seq, key=identity):
    """Return a minimum element of seq; break ties at random."""
    return argmin(shuffled(seq), key=key)


def argmax_random_tie(seq, key=identity):
    """Return an element with highest fn(seq[i]) score; break ties at random."""
    return argmax(shuffled(seq), key=key)


def shuffled(iterable):
    """Randomly shuffle a copy of iterable."""
    items = list(iterable)
    random.shuffle(items)
    return items


# part2. Mathematical and Statistical util functions
# ______________________________________________________________________________


def histogram(values, mode=0, bin_function=None):
    """Return a list of (value, count) pairs, summarizing the input values.
    Sorted by increasing value, or if mode=1, by decreasing count.
    If bin_function is given, map it over values first."""
    if bin_function:
        values = map(bin_function, values)

    bins = {}
    for val in values:
        bins[val] = bins.get(val, 0) + 1

    if mode:
        return sorted(list(bins.items()), key=lambda x: (x[1], x[0]),
                      reverse=True)
    else:
        return sorted(bins.items())


def dotproduct(X, Y):
    """Return the sum of the element-wise product of vectors X and Y."""
    return sum(x * y for x, y in zip(X, Y))


def element_wise_product_2D(X, Y):
    """Return vector as an element-wise product of vectors X and Y"""
    assert len(X) == len(Y)
    return [x * y for x, y in zip(X, Y)]


def element_wise_product(X, Y):
    if hasattr(X, '__iter__') and hasattr(Y, '__iter__'):
        assert len(X) == len(Y)
        return [element_wise_product(x,y) for x,y in zip(X,Y)]
    elif hasattr(X, '__iter__') == hasattr(Y, '__iter__'):
        return X*Y
    else:
        raise Exception("Inputs must be in the same size!")


def transpose2D(M):
    return list(map(list, zip(*M)))


def matrix_multiplication(X_M, *Y_M):
    """Return a matrix as a matrix-multiplication of X_M and arbitrary number of matrices *Y_M"""

    def _mat_mult(X_M, Y_M):
        """Return a matrix as a matrix-multiplication of two matrices X_M and Y_M
        >>> matrix_multiplication([[1, 2, 3],
                                   [2, 3, 4]],
                                   [[3, 4],
                                    [1, 2],
                                    [1, 0]])
        [[8, 8],[13, 14]]
        """
        assert len(X_M[0]) == len(Y_M)
        result = [[0 for i in range(len(Y_M[0]))] for j in range(len(X_M))]
        for i in range(len(X_M)):
            for j in range(len(Y_M[0])):
                for k in range(len(Y_M)):
                    result[i][j] += X_M[i][k] * Y_M[k][j]
        return result

    result = X_M
    for Y in Y_M:
        result = _mat_mult(result, Y)

    return result


def vector_to_diagonal(v):
    """Converts a vector to a diagonal matrix with vector elements
    as the diagonal elements of the matrix"""
    diag_matrix = [[0 for i in range(len(v))] for j in range(len(v))]
    for i in range(len(v)):
        diag_matrix[i][i] = v[i]

    return diag_matrix


def vector_add(a, b):
    """Component-wise addition of two vectors."""
    if not (a and b):
        return a or b
    if hasattr(a, '__iter__') and hasattr(b, '__iter__'):
        assert len(a) == len(b)
        return list(map(vector_add, a, b))
    else:
        try:
            return a+b
        except TypeError:
            raise Exception("Inputs must be in the same size!")


def scalar_vector_product(X, Y):
    """Return vector as a product of a scalar and a vector recursively"""
    return [scalar_vector_product(X, y) for y in Y] if hasattr(Y, '__iter__') else X*Y


def map_vector(f, X):
    """apply function f to iterable X"""
    return [map_vector(f, x) for x in X] if hasattr(X, '__iter__') else list(map(f, [X]))[0]


def scalar_matrix_product(X, Y):
    """Return matrix as a product of a scalar and a matrix"""
    return [scalar_vector_product(X, y) for y in Y]


def inverse_matrix(X):
    """Inverse a given square matrix of size 2x2"""
    assert len(X) == 2
    assert len(X[0]) == 2
    det = X[0][0] * X[1][1] - X[0][1] * X[1][0]
    assert det != 0
    inv_mat = scalar_matrix_product(1.0 / det, [[X[1][1], -X[0][1]], [-X[1][0], X[0][0]]])

    return inv_mat


def probability(p):
    """Return true with probability p."""
    return p > random.uniform(0.0, 1.0)


def weighted_sample_with_replacement(n, seq, weights):
    """Pick n samples from seq at random, with replacement, with the
    probability of each element in proportion to its corresponding
    weight."""
    sample = weighted_sampler(seq, weights)

    return [sample() for _ in range(n)]


def weighted_sampler(seq, weights):
    """Return a random-sample function that picks from seq weighted by weights."""
    totals = []
    for w in weights:
        totals.append(w + totals[-1] if totals else w)

    return lambda: seq[bisect.bisect(totals, random.uniform(0, totals[-1]))]


def weighted_choice(choices):
    """A weighted version of random.choice"""
    # NOTE: Should be replaced by random.choices if we port to Python 3.6

    total = sum(w for _, w in choices)
    r = random.uniform(0, total)
    upto = 0
    for c, w in choices:
        if upto + w >= r:
            return c, w
        upto += w


def rounder(numbers, d=4):
    """Round a single number, or sequence of numbers, to d decimal places."""
    if isinstance(numbers, (int, float)):
        return round(numbers, d)
    else:
        constructor = type(numbers)  # Can be list, set, tuple, etc.
        return constructor(rounder(n, d) for n in numbers)


def num_or_str(x): # TODO: rename as `atom`
    """The argument is a string; convert to a number if
       possible, or strip it."""
    try:
        return int(x)
    except ValueError:
        try:
            return float(x)
        except ValueError:
            return str(x).strip()


def euclidean_distance(X, Y):
    return math.sqrt(sum((x - y)**2 for x, y in zip(X, Y)))


def rms_error(X, Y):
    return math.sqrt(ms_error(X, Y))


def ms_error(X, Y):
    return mean((x - y)**2 for x, y in zip(X, Y))


def mean_error(X, Y):
    return mean(abs(x - y) for x, y in zip(X, Y))


def manhattan_distance(X, Y):
    return sum(abs(x - y) for x, y in zip(X, Y))


def mean_boolean_error(X, Y):
    return mean(int(x != y) for x, y in zip(X, Y))


def hamming_distance(X, Y):
    return sum(x != y for x, y in zip(X, Y))

# part3. Neural network util functions
# ______________________________________________________________________________


def normalize(dist):
    """Multiply each number by a constant such that the sum is 1.0"""
    if isinstance(dist, dict):
        total = sum(dist.values())
        for key in dist:
            dist[key] = dist[key] / total
            assert 0 <= dist[key] <= 1, "Probabilities must be between 0 and 1."
        return dist
    total = sum(dist)
    return [(n / total) for n in dist]


def norm(X, n=2):
    """Return the n-norm of vector X"""
    return sum([x ** n for x in X]) ** (1 / n)


def random_weights(min_value, max_value, num_weights):
    return [random.uniform(min_value, max_value) for _ in range(num_weights)]


def conv1D(X, K):
    """1D convolution. X: input vector; K: kernel vector"""
    K = K[::-1]
    res = []
    for x in range(len(X)):
        res += [sum([X[x+k]*K[k]] for k in K)]
    return res



def GaussianKernel(size=3):
    mean = (size-1)/2
    stdev = 0.1
    return [gaussian(mean, stdev, x) for x in range(size)]

def gaussian_kernel_1d(size=3, sigma=0.5):
    mean = (size-1)/2
    return [gaussian(mean, sigma, x) for x in range(size)]


def gaussian_kernel_2d(size=3, sigma=0.5):
    x, y = np.mgrid[-size//2 + 1:size//2 + 1, -size//2 + 1:size//2 + 1]
    g = np.exp(-((x ** 2 + y ** 2) / (2.0 * sigma ** 2)))
    return g / g.sum()


# ______________________________________________________________________________
# loss and activation functions


class Activation:

    def derivative(self, value):
        pass

def clip(x, lowest, highest):
    """Return x clipped to the range [lowest..highest]."""
    return max(lowest, min(x, highest))


def softmax1D(Z):
    """Return the softmax vector of input vector Z"""
    exps = [math.exp(z) for z in Z]
    sum_exps = sum(exps)
    return [exp/sum_exps for exp in exps]


class sigmoid(Activation):

    def f(self, x):
        if x>=100:
            return 1
        if x<= -100:
            return 0
        return 1 / (1 + math.exp(-x))

    def derivative(self, value):
        return value * (1 - value)


class relu(Activation):

    def f(self,x):
        return max(0, x)

    def derivative(self, value):
        if value > 0:
            return 1
        else:
            return 0


class elu(Activation):

    def f(self, x, alpha=0.01):
        if x > 0:
            return x
        else:
            return alpha * (math.exp(x) - 1)

    def derivative(self, value, alpha = 0.01):
        if value > 0:
            return 1
        else:
            return alpha * math.exp(value)


class tanh(Activation):

    def f(self, x):
        return np.tanh(x)

    def derivative(self, value):
        return (1 - (value ** 2))


class leaky_relu(Activation):

    def f(self, x, alpha = 0.01):
        if x > 0:
            return x
        else:
            return alpha * x

    def derivative(self, value, alpha=0.01):
        if value > 0:
            return 1
        else:
            return alpha


def step(x):
    """Return activation value of x with sign function"""
    return 1 if x >= 0 else 0


def gaussian(mean, st_dev, x):
    """Given the mean and standard deviation of a distribution, it returns the probability of x."""
    return 1 / (math.sqrt(2 * math.pi) * st_dev) * math.exp(-0.5 * (float(x - mean) / st_dev) ** 2)


def gaussian_2D(means, sigma, point):
    det = sigma[0][0] * sigma[1][1] - sigma[0][1] * sigma[1][0]
    inverse = inverse_matrix(sigma)
    assert det != 0
    x_u = vector_add(point, scalar_vector_product(-1, means))
    buff = matrix_multiplication(matrix_multiplication([x_u], inverse), transpose2D([x_u]))
    return 1/(math.sqrt(det)*2*math.pi) * math.exp(-0.5 * buff[0][0])


try:  # math.isclose was added in Python 3.5; but we might be in 3.4
    from math import isclose
except ImportError:
    def isclose(a, b, rel_tol=1e-09, abs_tol=0.0):
        """Return true if numbers a and b are close to each other."""
        return abs(a - b) <= max(rel_tol * max(abs(a), abs(b)), abs_tol)

# part4. Self defined data structures
# ______________________________________________________________________________
# Grid Functions


orientations = EAST, NORTH, WEST, SOUTH = [(1, 0), (0, 1), (-1, 0), (0, -1)]
turns = LEFT, RIGHT = (+1, -1)


def turn_heading(heading, inc, headings=orientations):
    return headings[(headings.index(heading) + inc) % len(headings)]


def turn_right(heading):
    return turn_heading(heading, RIGHT)


def turn_left(heading):
    return turn_heading(heading, LEFT)


def distance(a, b):
    """The distance between two (x, y) points."""
    xA, yA = a
    xB, yB = b
    return math.hypot((xA - xB), (yA - yB))


def distance_squared(a, b):
    """The square of the distance between two (x, y) points."""
    xA, yA = a
    xB, yB = b
    return (xA - xB) ** 2 + (yA - yB) ** 2


def vector_clip(vector, lowest, highest):
    """Return vector, except if any element is less than the corresponding
    value of lowest or more than the corresponding value of highest, clip to
    those values."""
    return type(vector)(map(clip, vector, lowest, highest))


# ______________________________________________________________________________
# Misc Functions

class injection():
    """Dependency injection of temporary values for global functions/classes/etc.
    E.g., `with injection(DataBase=MockDataBase): ...`"""

    def __init__(self, **kwds):
        self.new = kwds

    def __enter__(self):
        self.old = {v: globals()[v] for v in self.new}
        globals().update(self.new)

    def __exit__(self, type, value, traceback):
        globals().update(self.old)


def memoize(fn, slot=None, maxsize=32):
    """Memoize fn: make it remember the computed value for any argument list.
    If slot is specified, store result in that slot of first argument.
    If slot is false, use lru_cache for caching the values."""
    if slot:
        def memoized_fn(obj, *args):
            if hasattr(obj, slot):
                return getattr(obj, slot)
            else:
                val = fn(obj, *args)
                setattr(obj, slot, val)
                return val
    else:
        @functools.lru_cache(maxsize=maxsize)
        def memoized_fn(*args):
            return fn(*args)

    return memoized_fn


def name(obj):
    """Try to find some reasonable name for the object."""
    return (getattr(obj, 'name', 0) or getattr(obj, '__name__', 0) or
            getattr(getattr(obj, '__class__', 0), '__name__', 0) or
            str(obj))


def isnumber(x):
    """Is x a number?"""
    return hasattr(x, '__int__')


def issequence(x):
    """Is x a sequence?"""
    return isinstance(x, collections.abc.Sequence)


def print_table(table, header=None, sep='   ', numfmt='{}'):
    """Print a list of lists as a table, so that columns line up nicely.
    header, if specified, will be printed as the first row.
    numfmt is the format for all numbers; you might want e.g. '{:.2f}'.
    (If you want different formats in different columns,
    don't use print_table.) sep is the separator between columns."""
    justs = ['rjust' if isnumber(x) else 'ljust' for x in table[0]]

    if header:
        table.insert(0, header)

    table = [[numfmt.format(x) if isnumber(x) else x for x in row]
             for row in table]

    sizes = list(
        map(lambda seq: max(map(len, seq)),
            list(zip(*[map(str, row) for row in table]))))

    for row in table:
        print(sep.join(getattr(
            str(x), j)(size) for (j, size, x) in zip(justs, sizes, row)))


def open_data(name, mode='r'):
    aima_root = os.path.dirname(__file__)
    aima_file = os.path.join(aima_root, *['aima-data', name])

    return open(aima_file, mode=mode)


def failure_test(algorithm, tests):
    """Grades the given algorithm based on how many tests it passes.
    Most algorithms have arbitrary output on correct execution, which is difficult
    to check for correctness. On the other hand, a lot of algorithms output something
    particular on fail (for example, False, or None).
    tests is a list with each element in the form: (values, failure_output)."""
    from statistics import mean
    return mean(int(algorithm(x) != y) for x, y in tests)


# ______________________________________________________________________________
# Expressions

# See https://docs.python.org/3/reference/expressions.html#operator-precedence
# See https://docs.python.org/3/reference/datamodel.html#special-method-names

class Expr(object):
    """A mathematical expression with an operator and 0 or more arguments.
    op is a str like '+' or 'sin'; args are Expressions.
    Expr('x') or Symbol('x') creates a symbol (a nullary Expr).
    Expr('-', x) creates a unary; Expr('+', x, 1) creates a binary."""

    def __init__(self, op, *args):
        self.op = str(op)
        self.args = args

    # Operator overloads
    def __neg__(self):
        return Expr('-', self)

    def __pos__(self):
        return Expr('+', self)

    def __invert__(self):
        return Expr('~', self)

    def __add__(self, rhs):
        return Expr('+', self, rhs)

    def __sub__(self, rhs):
        return Expr('-', self, rhs)

    def __mul__(self, rhs):
        return Expr('*', self, rhs)

    def __pow__(self, rhs):
        return Expr('**', self, rhs)

    def __mod__(self, rhs):
        return Expr('%', self, rhs)

    def __and__(self, rhs):
        return Expr('&', self, rhs)

    def __xor__(self, rhs):
        return Expr('^', self, rhs)

    def __rshift__(self, rhs):
        return Expr('>>', self, rhs)

    def __lshift__(self, rhs):
        return Expr('<<', self, rhs)

    def __truediv__(self, rhs):
        return Expr('/', self, rhs)

    def __floordiv__(self, rhs):
        return Expr('//', self, rhs)

    def __matmul__(self, rhs):
        return Expr('@', self, rhs)

    def __or__(self, rhs):
        """Allow both P | Q, and P |'==>'| Q."""
        if isinstance(rhs, Expression):
            return Expr('|', self, rhs)
        else:
            return PartialExpr(rhs, self)

    # Reverse operator overloads
    def __radd__(self, lhs):
        return Expr('+', lhs, self)

    def __rsub__(self, lhs):
        return Expr('-', lhs, self)

    def __rmul__(self, lhs):
        return Expr('*', lhs, self)

    def __rdiv__(self, lhs):
        return Expr('/', lhs, self)

    def __rpow__(self, lhs):
        return Expr('**', lhs, self)

    def __rmod__(self, lhs):
        return Expr('%', lhs, self)

    def __rand__(self, lhs):
        return Expr('&', lhs, self)

    def __rxor__(self, lhs):
        return Expr('^', lhs, self)

    def __ror__(self, lhs):
        return Expr('|', lhs, self)

    def __rrshift__(self, lhs):
        return Expr('>>', lhs, self)

    def __rlshift__(self, lhs):
        return Expr('<<', lhs, self)

    def __rtruediv__(self, lhs):
        return Expr('/', lhs, self)

    def __rfloordiv__(self, lhs):
        return Expr('//', lhs, self)

    def __rmatmul__(self, lhs):
        return Expr('@', lhs, self)

    def __call__(self, *args):
        "Call: if 'f' is a Symbol, then f(0) == Expr('f', 0)."
        if self.args:
            raise ValueError('can only do a call for a Symbol, not an Expr')
        else:
            return Expr(self.op, *args)

    # Equality and repr
    def __eq__(self, other):
        "'x == y' evaluates to True or False; does not build an Expr."
        return (isinstance(other, Expr)
                and self.op == other.op
                and self.args == other.args)

    def __hash__(self):
        return hash(self.op) ^ hash(self.args)

    def __repr__(self):
        op = self.op
        args = [str(arg) for arg in self.args]
        if op.isidentifier():  # f(x) or f(x, y)
            return '{}({})'.format(op, ', '.join(args)) if args else op
        elif len(args) == 1:  # -x or -(x + 1)
            return op + args[0]
        else:  # (x - y)
            opp = (' ' + op + ' ')
            return '(' + opp.join(args) + ')'


# An 'Expression' is either an Expr or a Number.
# Symbol is not an explicit type; it is any Expr with 0 args.


Number = (int, float, complex)
Expression = (Expr, Number)


def Symbol(name):
    """A Symbol is just an Expr with no args."""
    return Expr(name)


def symbols(names):
    """Return a tuple of Symbols; names is a comma/whitespace delimited str."""
    return tuple(Symbol(name) for name in names.replace(',', ' ').split())


def subexpressions(x):
    """Yield the subexpressions of an Expression (including x itself)."""
    yield x
    if isinstance(x, Expr):
        for arg in x.args:
            yield from subexpressions(arg)


def arity(expression):
    """The number of sub-expressions in this expression."""
    if isinstance(expression, Expr):
        return len(expression.args)
    else:  # expression is a number
        return 0


# For operators that are not defined in Python, we allow new InfixOps:


class PartialExpr:
    """Given 'P |'==>'| Q, first form PartialExpr('==>', P), then combine with Q."""

    def __init__(self, op, lhs):
        self.op, self.lhs = op, lhs

    def __or__(self, rhs):
        return Expr(self.op, self.lhs, rhs)

    def __repr__(self):
        return "PartialExpr('{}', {})".format(self.op, self.lhs)


def expr(x):
    """Shortcut to create an Expression. x is a str in which:
    - identifiers are automatically defined as Symbols.
    - ==> is treated as an infix |'==>'|, as are <== and <=>.
    If x is already an Expression, it is returned unchanged. Example:
    >>> expr('P & Q ==> Q')
    ((P & Q) ==> Q)
    """
    if isinstance(x, str):
        return eval(expr_handle_infix_ops(x), defaultkeydict(Symbol))
    else:
        return x


infix_ops = '==> <== <=>'.split()


def expr_handle_infix_ops(x):
    """Given a str, return a new str with ==> replaced by |'==>'|, etc.
    >>> expr_handle_infix_ops('P ==> Q')
    "P |'==>'| Q"
    """
    for op in infix_ops:
        x = x.replace(op, '|' + repr(op) + '|')
    return x


class defaultkeydict(collections.defaultdict):
    """Like defaultdict, but the default_factory is a function of the key.
    >>> d = defaultkeydict(len); d['four']
    4
    """

    def __missing__(self, key):
        self[key] = result = self.default_factory(key)
        return result


class hashabledict(dict):
    """Allows hashing by representing a dictionary as tuple of key:value pairs
       May cause problems as the hash value may change during runtime
    """

    def __hash__(self):
        return 1

# ______________________________________________________________________________
# Useful Shorthands


class Bool(int):
    """Just like `bool`, except values display as 'T' and 'F' instead of 'True' and 'False'"""
    __str__ = __repr__ = lambda self: 'T' if self else 'F'


T = Bool(True)
F = Bool(False)