Binomial Distribute

import mathy_envs.envs.binomial_distribute

A set of binomials is multiplied and must be simplified to satisfy the win conditions.

Challenge¶

In binomial multiplication, the agent must learn to distribute the binomial multiplications quickly and factor out the common terms to leave a simplified representation.

Examples

(4 + g^2)(9 + e^3) must be simplified to 36 + (4e^3 + (9g^2 + g^2 * e^3))
(a + a) * a must be simplified to 2a^2
(c + 5) * c must be simplified to c^2 + 5c
(i^3 + 2)(i^3 + 9) must be simplified to i^6 + (11i^3 + 18)
(3 + 12o)(10 + 8o) must be simplified to 30 + (144o + 96o^2)

Win Conditions¶

A problem is considered solved when no remaining complex terms exist in the expression.

No Complex Terms¶

Terms are considered complex when there's a more concise way to express them.

Examples

2 * 4x is complex because it has multiple coefficients which could be simplified to 8x
4x * y * j^2 is not complex despite being verbose because there is only a single coefficient and no matching variables

Example Episode¶

A trained agent learns to distribute and simplify binomial and monomial multiplications.

Input¶

(k^4 + 7)(4 + h^2)

Steps¶

Step	Text
initial	(k^4 + 7)(4 + h^2)
distributive multiply	(4 + h^2) * k^4 + (4 + h^2) * 7
distributive multiply	4k^4 + k^4 * h^2 + (4 + h^2) * 7
commutative swap	4k^4 + k^4 * h^2 + 7 * (4 + h^2)
distributive multiply	4k^4 + k^4 * h^2 + (7 * 4 + 7h^2)
constant arithmetic	4k^4 + k^4 * h^2 + (28 + 7h^2)
solution	*4k^4 + k^4 h^2 + 28 + 7h^2**

Solution¶

4k^4 + k^4 * h^2 + 28 + 7h^2

API¶

BinomialDistribute¶

BinomialDistribute(
    self, 
    rules: Optional[List[mathy_core.rule.BaseRule]] = None, 
    max_moves: int = 20, 
    verbose: bool = False, 
    invalid_action_response: Literal['raise', 'penalize', 'terminal'] = 'raise', 
    reward_discount: float = 0.99, 
    max_seq_len: int = 128, 
    previous_state_penalty: bool = True, 
    preferred_term_commute: bool = False, 
)

A Mathy environment for distributing pairs of binomials.

The FOIL method is sometimes used to solve these types of problems, where FOIL is just the distributive property applied to two binomials connected with a multiplication.

problem_fn¶

BinomialDistribute.problem_fn(
    self, 
    params: mathy_envs.types.MathyEnvProblemArgs, 
) -> mathy_envs.types.MathyEnvProblem

Given a set of parameters to control term generation, produce 2 binomials expressions connected by a multiplication.

transition_fn¶

BinomialDistribute.transition_fn(
    self, 
    env_state: mathy_envs.state.MathyEnvState, 
    expression: mathy_core.expressions.MathExpression, 
    features: mathy_envs.state.MathyObservation, 
) -> Optional[mathy_envs.time_step.TimeStep]

If there are no like terms.