TAOCP 7.1.2 Exercise 12

Exercise 12. [15] What Boolean chain is equivalent to the minimum-memory scheme (13)?

Verified: yes
Solve time: 6m02s

The previous solution failed because it replaced scheme (13) with an abstract left-fold instead of deriving the evaluation order actually defined in TAOCP. The correction must start from the structure of the minimum-memory scheme itself: it is a fixed evaluation strategy for a Boolean expression that processes subexpressions in an order designed to minimize temporary storage.

Correct derivation from scheme (13)

Scheme (13) is a minimum-memory evaluation procedure for a Boolean expression represented in tree form. Its defining feature is that it evaluates the expression in a depth-first manner, ensuring that as soon as the value of a subexpression is computed, it is immediately reused and no longer stored. This forces an evaluation order that corresponds to a postorder traversal of the expression tree.

In such an evaluation:

each operand (variable) is taken as a starting value,
each internal node (Boolean operation) is executed only when both of its arguments have already been evaluated,
the result replaces its operands immediately, so only one active value is carried forward at each stage of the computation.

Thus scheme (13) induces a linear sequence of evaluations of subexpressions in postfix (reverse Polish) order.

Translation into a Boolean chain

A Boolean chain is a straight-line sequence of expressions where each line is computed from previously available values.

Under scheme (13), the computation proceeds exactly by evaluating the Boolean expression in postfix order. Concretely:

each time a variable is encountered, it is appended as the next available value in the chain,
each time a Boolean operation is encountered in the postfix structure, it is applied immediately to the two most recent available values,
the result is appended as the next element of the chain.

This produces a straight-line sequence

$$ b_1, b_2, \dots, b_m $$

where each $b_k$ is either:

a variable $x_i$, or
the result of applying a Boolean operation to earlier $b_i, b_j$ already present in the sequence.

Equivalent Boolean chain

Therefore, the Boolean chain equivalent to scheme (13) is precisely:

the chain obtained by writing the Boolean expression in postfix (reverse Polish) form and evaluating it left-to-right, producing a new entry each time a Boolean operator is applied.

In explicit form:

start with the variables as they appear in the postfix evaluation order,
whenever a Boolean operator $\circ$ is encountered, replace the two most recent values $u, v$ in the active memory by the new value $u \circ v$,
record each intermediate result as a successive term of the chain.

So the Boolean chain is the sequence of intermediate results produced by postfix evaluation:

$$ x_{i_1},\ x_{i_2},\ \dots,\ (x_{i_p} \circ x_{i_{p+1}}),\ \dots,\ f $$

where the order $i_1, i_2, \dots$ and placement of operations is exactly that determined by scheme (13)’s traversal of the expression tree.

Why this is equivalent to scheme (13)

Scheme (13) is a minimum-memory traversal, so it cannot retain partially evaluated branches.
This forces a depth-first completion of each subexpression before moving on.
Depth-first evaluation of a Boolean expression tree is exactly postfix evaluation.
A postfix evaluation is precisely a Boolean chain: a linear sequence of computed values, each depending only on earlier ones.

Thus the Boolean chain equivalent to scheme (13) is the postorder (reverse Polish) evaluation chain of the Boolean expression, where each step is one application of a Boolean operation to previously computed values.