An algorithm specifies transformations of symbols. A physical computer must implement them with a device that takes time, uses resources and produces accessible output. Neither description can simply replace the other.
Three different meanings of a computational step
A numerical calculation can take one thousand iterations to represent a second of a modelled process, while requiring a fraction of a second or several minutes on actual hardware. Iteration count, simulated duration and execution time are different quantities. Their relation depends on the numerical method and its implementation.
Even the phrase one operation is relative to a description. Multiplying two numbers may count as one instruction, many lower-level operations or a circuit containing numerous switching elements. A fundamental physical time cannot be obtained merely by deciding which of these descriptions to count.
The essay Time inside a simulation develops the numerical side. Here the question is what physical execution must retain, change or discard to realize a logical map.
A function need not erase its inputs
The AND function takes two binary inputs and returns one bit. Inputs 00, 01 and 10 all produce zero; input 11 produces one. Knowing only the output cannot reconstruct the inputs. This is a many-to-one logical map.
Now add a third bit z and perform the transformation (x,y,z) → (x,y,z ⊕ xy). If z starts at zero, its final value is the AND result, but the two inputs remain. The transformation on all eight three-bit states is a permutation: applying it twice restores the original state. Computing the result and discarding the inputs are distinct operations.
Bennett's reversible-computation construction shows how keeping and subsequently undoing intermediate work can avoid treating every logical step as an erasure. It does not mean that an arbitrary real machine runs without dissipation. Control, reliability, finite operating speed and interactions with the environment remain physical questions.
What the erasure bound actually concerns
For an ideal reset of an initially unbiased, uncorrelated bit in a symmetric memory, with the memory returned to a standard state and heat released to a bath at temperature T, the familiar Landauer bound is Qbath ≥ kBT ln 2. At 300 K this is approximately 2.87 × 10−21 joules per reset. The lower limit is approached under appropriate reversible, quasistatic conditions.
The assumptions matter. A known bit, biased probabilities, side information, unequal memory free energies or an imperfect reset require a more detailed accounting. The formula is not a universal energy price for every gate, transmitted bit or observed event. Logical entropy and thermodynamic entropy are connected through an actual physical implementation and its boundary conditions.
Experiments such as Bérut and colleagues' controlled one-bit erasure study test this connection in specified physical systems. Such tests concern thermodynamic accounting, not the claim that all of reality consists of digital symbols.
Space, time and discarded work
Consider a running total. One implementation overwrites its accumulator after each addition. Another keeps every input and every intermediate sum. A third keeps enough information to reverse selected stages. They can return the same numerical answer while differing in storage, execution time and recoverable history.
An algorithmic resource claim must specify its input size and permitted operations. A physical efficiency claim additionally needs device parameters and an error criterion. Counting logical steps alone does not predict wall-clock speed or total energy. Conversely, a fast implementation does not prove that the abstract algorithm has the best scaling.
Output accessibility adds another boundary. A correct value in a temporary register may not yet be available to the intended reader. The next essay asks when a computation becomes an output.
Inner generation is a physical proposal, not a processor cycle
SMM/DTT distinguishes inner generation from outer completed observation. Computation provides a clear analogy for distinguishing internal steps from exposed results, but a computer already presupposes matter, interactions and clocks. It cannot, just by being an analogy, explain their emergence.
A physical reconstruction must define the internal states and their update rule, the completion condition, the observable output and the calibration relating that output to laboratory quantities. If distinct internal implementations produce the same output statistics, an additional test is needed to distinguish them. See the book's discussion of order, duration and physical clocks and the Reality introduction to inner generation and outer completion.
Sources and further reading
- R. Landauer, Irreversibility and Heat Generation in the Computing Process, IBM Journal of Research and Development 5 (1961), 183–191.
- C. H. Bennett, Logical Reversibility of Computation, IBM Journal of Research and Development 17 (1973), 525–532.
- A. Bérut and colleagues, Experimental verification of Landauer's principle linking information and thermodynamics, Nature 483 (2012), 187–189.