# Time/space trade-offs for reversible computation

## Abstract

A reversible Turing machine is one whose transition function is 1:1, so that no instantaneous description (ID) has more than one predecessor. Using a pebbling argument, this paper shows that, for any ε>0, ordinary multitape Turing machines using time T and space S can be simulated by reversible ones using time O(T1+ε) and space O(S log T) or in linear time and space O(STε). The former result implies in particular that reversible machines can simulate ordinary ones in quadratic space. These results refer to reversible machines that save their input, thereby insuring a global 1:1 relation between initial and final IDs, even when the function being computed is many-to-one. Reversible machines that instead erase their input can of course compute only 1:1 partial recursive functions and indeed provide a Godel numbering of such functions. The time/space cost of computing a 1:1 function on such a machine is equal within a small polynomial to the cost of computing the function and its inverse on an ordinary Turing machine.