Thermodynamics of used punched tape: A weak and a strong equivalence principle

We study the repeated use of a monotonic recording medium—such as punched tape or photographic plate—where marks can be added at any time but never erased. (For practical purposes, also the electromagnetic “ether” falls into this class.) Our emphasis is on the case where the successive users act independently and selfishly, but not maliciously; typically, the “first user” would be a blind natural process tending to degrade the recording medium, and the “second user” a human trying to make the most of whatever capacity is left. To what extent is a length of used tape “equivalent”—for information transmission purposes—to a shorter length of virgin tape? Can we characterize a piece of used tape by an appropriate “effective length” and forget all other details? We identify two equivalence principles. The weak principle is exact, but only holds for a sequence of infinitesimal usage increments. The strong principle holds for any amount of incremental usage, but is only approximate; nonetheless, it is quite accurate even in the worst case and is virtually exact over most of the range—becoming exact in the limit of heavily used tape. The fact that strong equivalence does not hold exactly, but then it does almost exactly, comes as a bit of a surprise. Keywords—Thermodynamics of write-once media, Equivalence principles for storage capacity of noisy medium Bob—a poor computer science student—has found, rummaging through Alice’s dump, a large amount of used punched tape “in good conditions”. He doesn’t care for the data that is already on the tape: he would like to reuse the tape for storing his own data. He wants to be able to use a standard tape read/punch unit, which can sense holes in the tape and punch new ones but not remove holes that are already there. Since holes already made cannot be undone, the storage density Bob can expect to achieve is less than with virgin tape, and will depend on the actual conditions of the tape. To what extent is a length of used tape “equivalent”—for information transmission purposes—to a shorter length of virgin tape? Are there any qualitative differences between tapes that have been used to different degrees, or can one characterize a piece of used tape simply by its “effective length” and forget all other details? The theme we develop is complementary to that of Rivest and Shamir[7] (also cf. [6]). They stress the informationengineering aspects of reusing a tape generated by a cooperative partner in a pre-planned context. On the other hand, we are interested in a situation where the other party, while presumed non-malicious, volunteers no cooperation and pursues independent goals (if any goals can be made out); what we typically have in mind for “the other party” is natural processes. Tommaso Toffoli ([email protected]), ECE Department, Boston University, 8 Saint Mary’s St., Boston, MA 02215. 1As humans become more proficient at exploiting physical mechanisms on a finer and finer scale for computational purposes, computation will look more and more like an attempt to encroach on a turf already jammed near capacity by heavy “native” traffic—the near-equilibrium bustle of microscopic matter (cf. Dyson[4]). The present study is part of a wider program aimed at exploring this kind The cumulative channel capacity of randomly-punched used tape was first investigated in [8] (also see references therein), some of whose results we simplify and extend. References [3] and [5] discuss coding algorithms that dynamically adjust to “stuck-at-0” faults on the tape (cells that will not punch) sensed during punching, and “stuckat-1” faults sensed during or before punching. A paper related to the present one in spirit if not in detailed substance is “Writing on dirty paper” by Costa[2], whose moral (“Do the best with what you have”) we make our own. If you have no time at all, read just §V—a self-contained, intuitive debriefing.