def compose(value, sample_args, context=None):
"""E.g., "Let f(x)=2x+1, let g(x)=3x+10. What is f(g(x))?"."""
del value # unused
if context is None:
context = composition.Context()
entropy, sample_args = sample_args.peel()
entropy_f, entropy_g = entropy * np.random.dirichlet([1, 1])
coeffs_f = polynomials.sample_coefficients([random.randint(1, 2)],
entropy_f)
coeffs_g = polynomials.sample_coefficients([random.randint(1, 2)],
entropy_g)
entity_f, entity_g = context.sample(
sample_args,
[composition.Polynomial(coeffs_f),
composition.Polynomial(coeffs_g)])
variable = sympy.var(context.pop())
poly_f = polynomials.coefficients_to_polynomial(coeffs_f, variable)
poly_g = polynomials.coefficients_to_polynomial(coeffs_g, variable)
poly_f_g = poly_f.sympy().subs(variable, poly_g.sympy()).expand()
expression = composition.FunctionHandle(entity_f, entity_g).apply(variable)
template = random.choice(_TEMPLATES)
return example.Problem(question=example.question(context,
template,
composed=expression),
answer=poly_f_g)
def coefficient_named(value, sample_args, context=None):
"""E.g., "Express x^2 + 2x in the form h * x^2 + k * x + t and give h."."""
del value # not used
if context is None:
context = composition.Context()
variable = sympy.Symbol(context.pop())
entropy, sample_args = sample_args.peel()
degree = random.randint(1, 4)
if random.choice([False, True]):
coefficients = polynomials.sample_coefficients(
degree,
entropy / 2,
min_non_zero=random.randint(degree - 1, degree))
expanded = polynomials.expand_coefficients(coefficients, entropy / 2)
expression = polynomials.coefficients_to_polynomial(expanded, variable)
else:
expression = polynomials.sample_with_brackets(variable, degree,
entropy)
coefficients = list(reversed(sympy.Poly(expression).all_coeffs()))
named_coeffs = [sympy.Symbol(context.pop()) for _ in range(degree + 1)]
canonical = polynomials.coefficients_to_polynomial(named_coeffs, variable)
if random.random() < 0.2: # only small probability of non-zero power
power = random.randint(0, degree)
else:
non_zero_powers = [
i for i in range(degree + 1) if coefficients[i] != 0
]
power = random.choice(non_zero_powers)
value = coefficients[power]
named_coeff = named_coeffs[power]
template = random.choice([
'Выразите {expression} в форме {canonical} и найдите {target}.',
'Преобразуйте {expression} в форму {canonical} и найдите {target}.',
# 'Express {expression} in the form {canonical} and give {target}.',
# 'Rearrange {expression} to the form {canonical} and give {target}.',
])
return example.Problem(question=example.question(context,
template,
expression=expression,
canonical=canonical,
target=named_coeff),
answer=value)
def testMultivariate(self):
# Test generation for: x**2 + 2*x*y + 3*y**2 - x + 5
x, y = sympy.symbols('x y')
coeffs = [[5, 0, 3], [-1, 2, 0], [1, 0, 0]]
for _ in range(10):
expanded = polynomials.expand_coefficients(coeffs, 5.0, length=10)
polynomial = polynomials.coefficients_to_polynomial(expanded, [x, y])
sympified = sympy.sympify(polynomial)
self.assertEqual(sympified, x*x + 2*x*y + 3*y*y - x + 5)
def testUnivariate(self):
# Test generation for: x**2 + 2*x + 1
x = sympy.Symbol('x')
coeffs = [1, 2, 3]
for _ in range(10):
expanded = polynomials.expand_coefficients(coeffs, 5.0)
polynomial = polynomials.coefficients_to_polynomial(expanded, [x])
sympified = sympy.sympify(polynomial)
self.assertEqual(sympified, 1 + 2*x + 3*x*x)
def collect(value, sample_args, context=None):
"""Collect terms in an unsimplified polynomial."""
is_question = context is None
if context is None:
context = composition.Context()
entropy, sample_args = sample_args.peel()
if value is None:
entropy_value, entropy = entropy * np.random.dirichlet([2, 3])
degrees = [random.randint(1, 3)]
value = composition.Polynomial(
polynomials.sample_coefficients(degrees, entropy_value))
assert isinstance(value, composition.Polynomial)
coefficients = value.coefficients
all_coefficients_are_integer = True
for coeff in coefficients.flat:
if not number.is_integer(coeff):
all_coefficients_are_integer = False
break
if all_coefficients_are_integer:
coefficients = polynomials.expand_coefficients(coefficients, entropy)
else:
# put back the unused entropy
sample_args = composition.SampleArgs(sample_args.num_modules,
sample_args.entropy + entropy)
num_variables = coefficients.ndim
variables = [sympy.Symbol(context.pop()) for _ in range(num_variables)]
unsimplified = polynomials.coefficients_to_polynomial(
coefficients, variables)
simplified = unsimplified.sympy().expand()
# Bit of a hack: handle the very rare case where no number constants appearing
if not ops.number_constants(unsimplified):
unsimplified = ops.Add(unsimplified, ops.Constant(0))
context.sample_by_replacing_constants(sample_args, unsimplified)
if is_question:
template = 'Упростите {unsimplified}.'
return example.Problem(question=example.question(
context, template, unsimplified=unsimplified),
answer=simplified)
else:
function_symbol = context.pop()
function = sympy.Function(function_symbol)(*variables)
return composition.Entity(
context=context,
value=value,
handle=composition.FunctionHandle(function_symbol),
expression=unsimplified,
polynomial_variables=variables,
description='Пусть {function} = {unsimplified}.',
function=function,
unsimplified=unsimplified)
def testPolynomialRoots(self):
variable = sympy.Symbol('x')
for _ in range(10):
roots = random.sample(list(range(-9, 10)), 3)
coeffs = algebra._polynomial_coeffs_with_roots(roots,
scale_entropy=10.0)
polynomial = polynomials.coefficients_to_polynomial(
coeffs, variable)
calc_roots = sympy.polys.polytools.real_roots(polynomial)
self.assertEqual(calc_roots, sorted(roots))
def add(value, sample_args, context=None):
"""E.g., "Let f(x)=2x+1, g(x)=3x+2. What is 5*f(x) - 7*g(x)?"."""
is_question = context is None
if context is None:
context = composition.Context()
entropy, sample_args = sample_args.peel()
if value is None:
max_degree = 3
degree = random.randint(1, max_degree)
entropy -= math.log10(max_degree)
entropy_value = entropy / 2
entropy -= entropy_value
value = polynomials.sample_coefficients(degree,
entropy=entropy_value,
min_non_zero=random.randint(
1, 3))
value = composition.Polynomial(value)
c1, c2, coeffs1, coeffs2 = polynomials.coefficients_linear_split(
value.coefficients, entropy)
coeffs1 = polynomials.trim(coeffs1)
coeffs2 = polynomials.trim(coeffs2)
c1, c2, fn1, fn2 = context.sample(sample_args, [
c1, c2,
composition.Polynomial(coeffs1),
composition.Polynomial(coeffs2)
])
var = sympy.var(context.pop())
expression = (c1.handle * fn1.handle.apply(var) +
c2.handle * fn2.handle.apply(var))
if is_question:
answer = polynomials.coefficients_to_polynomial(
value.coefficients, var)
answer = answer.sympy()
template = random.choice(_TEMPLATES)
return example.Problem(question=example.question(context,
template,
composed=expression),
answer=answer)
else:
intermediate_symbol = context.pop()
intermediate = sympy.Function(intermediate_symbol)(var)
return composition.Entity(
context=context,
value=value,
description='Пусть {intermediate} = {composed}.',
handle=composition.FunctionHandle(intermediate_symbol),
intermediate=intermediate,
composed=expression)
def _polynomial_entity(value, context):
"""Create a generic `Entity` describing a polynomial."""
assert isinstance(value, Polynomial)
coefficients = np.asarray(value.coefficients)
num_variables = coefficients.ndim
variables = [sympy.Symbol(context.pop()) for _ in range(num_variables)]
function_symbol = context.pop()
handle = FunctionHandle(function_symbol)
handle_description = sympy.Function(function_symbol)(*variables)
polynomial = polynomials.coefficients_to_polynomial(
coefficients, variables)
polynomial = polynomial.sympy()
return Entity(context=context,
value=value,
expression=polynomial,
polynomial_variables=variables,
description='Пусть {function} = {polynomial}.',
handle=handle,
function=handle_description,
polynomial=polynomial)
def testCoefficientsToPolynomial(self):
coeffs = [3, 2, 1]
x = sympy.Symbol('x')
polynomial = polynomials.coefficients_to_polynomial(coeffs, [x])
polynomial = sympy.sympify(polynomial)
self.assertEqual(polynomial, x*x + 2*x + 3)
def _differentiate_polynomial(value, sample_args, context, num_variables):
"""Generates a question for differentiating a polynomial."""
is_question = context is None
if context is None:
context = composition.Context()
if value is not None:
num_variables = value.coefficients.ndim
entropy, sample_args = sample_args.peel()
max_derivative_order = 3
derivative_order = random.randint(1, max_derivative_order)
entropy = max(0, entropy - math.log10(max_derivative_order))
derivative_axis = random.randint(0, num_variables - 1)
if value is None:
coefficients = _generate_polynomial(num_variables, entropy,
derivative_order, derivative_axis)
else:
coefficients = _sample_integrand(value.coefficients, derivative_order,
derivative_axis, entropy)
(entity, ) = context.sample(sample_args,
[composition.Polynomial(coefficients)])
value = coefficients
for _ in range(derivative_order):
value = polynomials.differentiate(value, axis=derivative_axis)
nth = display.StringOrdinal(derivative_order)
nth_fem = display.StringOrdinal_fem(derivative_order)
nth_fem_gen = display.StringOrdinal_fem_gen(derivative_order)
if entity.has_expression():
polynomial = entity.expression
variables = entity.polynomial_variables
else:
variables = [sympy.Symbol(context.pop()) for _ in range(num_variables)]
polynomial = entity.handle.apply(*variables)
variable = variables[derivative_axis]
if is_question:
template = _template(context.module_count, derivative_order,
len(variables))
answer = polynomials.coefficients_to_polynomial(value,
variables).sympy()
return example.Problem(question=example.question(
context,
template,
eq=polynomial,
var=variable,
nth=nth,
nth_fem=nth_fem,
nth_fem_gen=nth_fem_gen),
answer=answer)
else:
fn_symbol = context.pop()
variables_string = ', '.join(str(variable) for variable in variables)
assert len(
variables) == 1 # since below we don't specify var we diff wrt
return composition.Entity(
context=context,
value=composition.Polynomial(value),
description=
'Пусть {fn}({variables}) - это {nth_fem} производная функции {eq}.',
handle=composition.FunctionHandle(fn_symbol),
fn=fn_symbol,
variables=variables_string,
nth=nth,
nth_fem=nth_fem,
nth_fem_gen=nth_fem_gen,
eq=polynomial)
