In my last post on strategies I introduced a set of higher order functions to represent common control patterns (like top_down). We combined these with transformation rules (like flatten) to create complex functions for tree manipulation (like flatten_tree)

rule     :: expr -> expr
strategy :: parameters, rule -> rule

In my post on unification we showed how to easily create rules from patterns. At the end of this post I described that because patterns might match in multiple ways one rule might produce many different results. To avoid combinatorial blowup in the number of possible matches we solved this by yielding matches lazily.

Transformation rules produced by unify don’t return values, they yield possible solutions lazily. How do we reconcile this with our previous notion of rules and strategies? We make a new set of strategies for branching rules.

branching-rule      :: expr -> {expr}
branching-strategy  :: parameters, branching-rule -> branching-rule

In sympy.rules.branch we have implemented lazy analogs for the strategies found in sympy.rules. This allows us to apply strategies to transformations like the sincos_to_one rule created in the unification post.

## Toy Problem

Lets see branching strategies with a toy problem. Consider the following “function”

$$f(x) = \cases{ x+1 & \text{if } 5 < x < 10 \\ x-1 & \text{if } 0 < x < 5 \\ x+1 \text{ or } x-1 & \text{if } x = 5 \\ x & \text{otherwise} \\ }$$

And it’s equivalent in Python

def f(x):
if 0 < x < 5:   yield x - 1
if 5 < x < 10:  yield x + 1
if x == 5:      yield x + 1; yield x - 1


Notice that in the case where x = 5 there are two possible outcomes. Each of these is preserved by the application of branching strategies. We use the branching version of the exhaust strategy to make a new exhaustive version of this function

>>> from sympy.rules.branch import exhaust
>>> newf = exhaust(f)
>>> set(newf(6))
{10}
>>> set(newf(3))
{0}
>>> set(newf(5))
{0, 10}


## Practical Problem

We have all the machinery necessary. Lets make a sin(x)**2 + cos(x)**2 -> 1 tree-wise simplification function.

>>> from sympy.rules.branch.traverse import top_down
>>> from sympy.unify.rewrite import rewriterule
>>> from sympy.abc import a, b, c, x, y

>>> sincos_to_one = rewriterule(sin(x)**2 + cos(x)**2, S.One, wilds=[x])
>>> sincos_tree = top_down(sincos_to_one)

>>> list(sincos_tree(2 + c**(sin(a+b)**2 + cos(a+b)**2)))  # see footnote
[c**1 + 2]


Lets make a rule to simplify expressions like c**1

>>> pow_simp = rewriterule(Pow(x, 1, evaluate=False), x, wilds=[x]) # footnote 2

>>> from sympy.rules.branch.strat_pure import multiplex, exhaust
>>> simplify = exhaust(top_down(multiplex(sincos_to_one, pow_simp)))

>>> list(simplify(2 + c**(sin(a+b)**2 + cos(a+b)**2)))
[c + 2]


We see how we can easiy build up powerful simplification functions through the separate description of logic

sin(x)**2 + cos(x)**2 -> 1
x ** 1 -> x

and control

simplify = exhaust(top_down(multiplex( ... )))

Footnote 1: At the time of this writing this line should actually be

map(rebuild, sincos_tree( ... )

The rebuild function is necessary because rules don’t play well with Exprs. Expr’s need to be constructed normally in order to function properly. In particular all expressions built by rules lack the is_commutative flag which is attached onto the object at construction time. I neglected to mention this above to simplify the discussion.

Footnote 2: This also requires a slight modification due to the Expr/rules mismatch. In particular the pattern Pow(x, 1, evaluate=False) unfortunately matches to just x because x == x**1 in SymPy.