Non-Recursive Trade-Offs Are “Almost Everywhere”

Abstract

We briefly summarize some of the findings on non-recursive trade-offs, which were first observed by Meyer and Fischer in their seminal paper on “Economy of Description by Automata, Grammars, and Formal Systems” in 1971. This general phenomenon is about conversion problems between different (computational) description models that cannot be solved efficiently. Indeed, they evade solvability a forteriori because the change in description size caused by such a conversion cannot be bounded above by any recursive function. Hence, a result on non-recursive trade-offs can alternatively be interpreted as a compression of the description model with arbitrary space gains. Since 1971 there has been a steadily growing list of results where this phenomenon has been observed, and it appears that non-recursive trade-offs are “almost everywhere.”