While thinking about different calculi for predicate logic (like natural deduction and sequent calculus), I noticed that these calculi are (often) presented in a form suitable for "human computers". A "human computer" is limited to use write once read many (WORM) memory when processing large amounts of data. Pure functional programming also seems to favor a WORM memory model. In fact, it seems to me that the WORM memory model is so natural that classifying it as unconventional computing might underestimate its importance. (Understanding the strengths and limitations of the computing resources available to humans is important.)
What is known about the relation between space and time complexity for machines with WORM memory? What are the keywords to google for available material related to these questions? Do we known whether the time complexity will remain the same, if a small (for example C*log(WORM memory)^n) amount of normal memory is added?