This lab is designed to help you practice programming with trees while teaching you a new data structure that is used in many areas of computing.

Although basic boolean expressions (using and/or/not) are straightforward, they are extraordinarily useful in computing and computer science. For example, hardware circuits are built from devices for these operations (plus simple memory units), while configuration tools for selecting features when you order a bicycle or laptop use boolean formulas to capture sets of features that must or cannot be used in the same product. In both cases, we often need to check whether two formulas are logically equivalent or what the differences are between them (such as if we try to optimize the circuit or rearrange the configuration options)--multi-million dollar companies produce tools for this problem in each of these domains. This lab looks at both naive and state-of-the-art data structures for manipulating boolean formulas.

We'll use the following data definition for boolean formulas, where symbols are used to name variables:

;; A formula is one of ;; - boolean (true or false) ;; - symbol ;; - (make-notf formula) ;; - (make-andf formula formula) ;; - (make-orf formula formula) (define-struct notf (fmla)) (define-struct andf (fmla1 fmla2)) (define-struct orf (fmla1 fmla2))We are using

`andf`

and `orf`

to avoid errors
from reusing the built-in `and`

and `or`

keywords.
For example, the formula "hard-disk and (track-pad or track-point)" would be captured as

(define laptop-conf (make-andf 'hard-disk (make-orf 'track-point 'track-pad)))

One basic operation on formulas is checking whether they evaluate to true under a particular assignment of boolean values to the variables (for example: given the set of features chosen, do they violate any constraints on feature combinations as captured by the formula). Write a function

`eval-formula`

that consumes a formula and a list of symbols (for the variables that have been set to true) and produces a boolean indicating whether the formula is true or false under the given setting of true variables.For example,

`(eval-formula laptop-conf (list 'track-point))`

should return false, whereas`(eval-formula laptop-conf (list 'hard-disk 'track-point))`

should return true.- One naive way to check whether two formulas are equivalent is to
check whether they evaluate to the same answer under every possible
assignment of true and false values to the variables in the formulas.
Write a function
`formulas-equiv?`

that consumes two formulas and a list of all the variable names (symbols) appearing in the formulas; the function produces a boolean indicating whether the two formulas produce the same answer on every possible subset of those variables set to true. Assume the two formulas have the same set of variables.Hint: Write a helper to take a list of symbols and generate all possible subsets of that list. For example, the set of all subsets of (list 'a 'b) consists of empty, (list 'a), (list 'b), and (list 'a 'b).

Hint 2: Scheme includes a function

`andmap`

that consumes a function from X to boolean and a list[X] and determines whether the function is true for every element in the list. Use this function if you wish.

Everybody should get at least this far, and at least get started on the next question.

Real-world formulas (particularly in hardware) involve thousands if
not millions of variables. At that scale, running through all the
combinations of values to check equivalence is too expensive.
Instead, we'd like a way to compare the formulas without testing them
for each combination of variable values. To do this, we use a
different representation of formulas called a *binary-decision
diagram* (or BDD). A BDD is a binary tree in which each internal
node is labeled with a variable: one outgoing edge gives the formula
for when the variable is false and the other outgoing edge gives the
formula for when the variable is true. Leaves are labeled with just
booleans. For example, our laptop formula is represented with the
following BDD:

BDDs are built using some fixed order on the variables in the formula (in this case, hard-disk, trackpoint, trackpad, with the order corresponding to the levels in the tree). If both the true and false branches from a node lead to the same formula, the variable for that node gets skipped (which is why trackpoint and trackpad do not appear in the false subtree for the harddisk node). In addition, if two subtrees have the same contents, we use only a single copy of the subtree (for efficiency).

Use the following data definition for BDDs:

;; A bdd is ;; - boolean or ;; - (make-bddnode symbol bdd bdd) (define-struct bddnode (var whenfalse whentrue))Write the BDD in the picture above using this data definition to make sure you understand the data definition.

Write a function

`eval-bdd`

that consumes a bdd and a list of symbols (for the variables that have been assigned true) and produces a boolean indicating whether the formula is true or false under that set of true variables.If you represent a formula as a BDD, the formula and the BDD should have the same value for every possible choice of true variables. Write a function

`formula-bdd-equiv?`

that consumes a formula, a bdd, and a list of symbols (variable names) and produces a boolean indicating whether the formula and bdd compute the same answer for every subset of variables set to true.The whole point of switching to BDDs was to optimize equivalence checking. Write a function

`bdd-equiv?`

that consumes two BDDs (constructed with the same variable order) and produces a boolean indicating whether the two BDDs compute the same function. You can check this by walking the trees for the two BDDs starting from the roots, checking that the variable names at the current nodes are identical, the two whenfalse subtrees are equivalent and the two whentrue subtrees are equivalent.

If you get this far, choose from the following problems, depending on the level of challenge you want.

[Normal] When two BDDs aren't equivalent, modern tools produce examples of truth assignments to variables on which the formulas yield different answers. For example, given the BDDs for "x and y" and "x or y" one possible difference is the list

`(list x)`

, which says that if x is true and y false (implicit, by not being in the list), the two BDDs given different answers. Write a function`find-diff`

that consumes two BDDs and produces a list of variable names such that if those vars are set to true (and all others false), the two BDDs will produce different answers.[one-star challenge] Rather than create BDDs by hand, as you have done until this point, tools convert from formulas to BDDs automatically. This problem has you write the converter.

- To create the negation (logical not) of a BDD, we simply reverse the booleans at the leaves.
- To create the (logical) and of a BDD, we walk the two BDDs from the roots to the leaves. At each iteration, if one of the nodes contains false, we return false; if one of the nodes contains true, we return the node of the other tree (thus optimizing using the rules for how and works). If both nodes contain variables, check whether the variable is the same in both nodes. If so, create a node with that variable, the and of the two whenfalse nodes in the false branch, and the and of the two whentrue nodes in the true branch. Otherwise, create a node with the earlier of the two variables (according to the order), then call the function recursively to combine each of the whentrue and whenfalse branches of the node with the earlier variable with the entire other node (not its branches).
- The method for or is similar, just changing when we reach a node with a boolean.

Use your existing functions to help test your work on this problem. Consider writing one a function to do the work of the bdd-and and bdd-or operations, passing in a function to handle the nodes with booleans.

[two-star challenge] BDDs are efficient because they keep only one copy of each subtree. Write a function

`condense`

that consumes a BDD possibly containing duplicate subtrees and produces a BDD that has at most one copy of each subtree (using pointers to the same subtree in memory for shared subtrees).

For more information on BDDs, check this tutorial or their wikipedia entry. BDDs revolutionized hardware analysis back in the 80s and are increasingly applied to product configuration.