Implementing Subroutines and Recursion with Stacks

Key Insight

To avoid duplicating machine components and instruction sequences for repeated computations, subroutines are employed. Initially, a 'continue' register could hold a simple flag (e.g., 0 or 1) to indicate the appropriate return point after a shared subroutine completes. A more robust method involves storing the actual controller sequence label where execution should resume into the 'continue' register. This requires extending the 'assign' instruction to accept labels as values and the 'goto' instruction to branch to the entry point specified by a register's contents, enabling dynamic control flow.

Iterative processes can be implemented with a fixed set of registers and a controller loop, where the machine's state is fully determined by register contents. Recursive processes, however, present a challenge because they often require intermediate values from the parent computation after a subproblem is solved. For example, computing n! requires the old value of 'n' to multiply with (n-1)!. Simply reusing registers for subproblems would overwrite these needed values, implying a conceptually infinite nesting of machines.

The solution for implementing recursion with a finite machine is the use of a stack, a 'last in, first out' data structure. When a recursive subproblem is encountered, the current computation is suspended, and crucial register values (like 'n' and the 'continue' label for returning) are saved onto the stack. The machine then reuses its components to solve the subproblem. Upon its completion, the saved values are restored from the stack in reverse order, allowing the suspended computation to resume using the subproblem's result. This generalized strategy allows arbitrary depths of recursive calls, with the stack being the potentially unbounded, yet physically implementable, component.