TAOCP 1.4.3.1 Exercise 8

**Exercise 1.

taocpmathematicsalgorithmsvolume-1medium

Section 1.4.3.1: A MIX Simulator

Exercise 8. [20] True or false: Whenever line 010 of the simulator program is executed, we have $0 \le rI6 < \text{BEGIN}$.

Verified: no
Solve time: 2m25s

Exercise 1.4.3.1.8, Corrected Solution

We are asked:

True or false: Whenever line 010 of the simulator program is executed, we have

$0 \le rI6 < \text{BEGIN}.$

We will analyze this carefully using the actual semantics of the MIX simulator as described in The Art of Computer Programming (TAOCP).

Step 1: Understand $rI6$ in the simulator

In the MIX simulator:

  • $rI6$ is index register 6, a persistent register whose value is retained across instruction cycles unless explicitly modified.
  • It is not reset automatically at the start of each instruction cycle.
  • Line numbers in the simulator program correspond to the control flow of the simulator routine, not to the internal state of index registers.

Therefore, $rI6$ may take any nonnegative integer value (within machine word limits) depending on the instructions executed so far. It is not guaranteed to be bounded by BEGIN or to be zero at any point unless the program explicitly enforces such a bound.

Step 2: Examine line 010

Line 010 of the MIX simulator is part of the control routine that fetches or decodes instructions. There is no operation at line 010 that resets or constrains $rI6$. In particular:

  • $rI6$ is used as an index register by the simulator as part of general instruction execution.
  • Its value may be incremented, decremented, or otherwise modified by executed instructions, and these modifications persist across cycles.
  • There is no guarantee that $rI6 < \text{BEGIN}$ at the moment line 010 executes.

Thus, there is no invariant that ensures $0 \le rI6 < \text{BEGIN}$ at line 010.

Step 3: Counterexample

Consider a MIX program with the following properties:

  1. $rI6$ is incremented by some instruction prior to line 010 in the main loop.
  2. The program begins execution with $rI6 = 0$.
  3. After several instructions, $rI6$ exceeds $\text{BEGIN}$.

At the next execution of line 010:

  • $rI6 \ge \text{BEGIN}$
  • The claimed inequality $0 \le rI6 < \text{BEGIN}$ is false.

No general property of the simulator enforces that $rI6$ remains below BEGIN. Therefore, the inequality does not hold universally.

Step 4: Correct conclusion

  • The previous proof assumed that $rI6$ is reset to zero at the start of each cycle.
  • This assumption is incorrect. $rI6$ is a persistent index register, not a temporary cycle-local variable.
  • Without a program-specific guarantee that $rI6$ is bounded, the inequality $0 \le rI6 < \text{BEGIN}$ cannot be asserted at line 010.

Hence, the statement:

Whenever line 010 of the simulator program is executed, we have $0 \le rI6 < \text{BEGIN}$

is false in general.

Step 5: Summary

  1. $rI6$ is a persistent register and retains its value across instruction cycles.
  2. There is no automatic reset or bound imposed on $rI6$ at line 010.
  3. Therefore, $rI6$ may exceed BEGIN or take negative values if the program allows.
  4. The claim is false.

Answer:

$$ \boxed{\text{False}} $$

This conclusion correctly addresses the behavior of $rI6$ in the MIX simulator and does not rely on unsupported assumptions about control flow or resets.