Needless to say that not only initial and final expression, but also temporary expressions should be stored in memory somehow. The way of how expressions are represented has impact on hardware design.
During reduction the cycles, expression symbols are being copied one by one. When a function symbol is found, then the symbol itself along with possible other consecutive symbols are rewritten in accordance with the function definition (this is used in a broader sense, and includes definition of built-in functions).
Please note that the following discussion is related only to expression evaluation, as user-defined functions are stored in a separate memory, possible in ROM.
Separate source and destination memory
Separate fragment of memory is used for source and destination expressions. Upon each reduction cycle the role of the memories are swapped. Source memory will become destination and vice versa. This is illustrated in the figure below.
If l is the longest possible size (in memory cells including EOX) of the expression during the evaluation, then the minimum memory requirement is 2xl cells.
Shared memory, with index increasing and decreasing
A shared memory can also be used. In this case source index is increasing, whereas destination is decreasing while copying symbols. After each evaluation cycle, the role of the indexes will be swapped again. This memory model may be more econimic, since in some cases it will be suitable to reduce expressions over half of the memory size, provided that the reduced expression becomes smaller, satisfying the following equation, which in fact should hold in every cycle: e+r≤m, where e, r and m denote expression size (excluding EOX), reduced expression size and memory size respectively.
Shared memory with index following
The last model is also comparable to the previous one in econimical terms, as the same equation among expression lengths and memory size must hold. However, it has the additional benefit, namely that index roles do not need to be changed. This simplifies hardware design, as multiplexers (swapping the source and destination registers) and its relevant control signal and logic can be avoided. As source index reaches the end of input expression marked by the EOX symbol, it should just copy and step over that symbol and the continue with the next reduction cycle. If memory end is reached, then indexes will wrap around starting from a lower location.