Evolving Algebras and Linear Time Hierarchy

A program Q simulates a program P in lock-step under a given representation of states of P as states of Q if there is a constant c such that if P transforms a state A to a state A in time then Q transforms (A) into (A) in time c . (Cf. the related notions of real-time computation and simulation [R,S].) The constant c is the lag factor of the simulation. We speak about time rather than the number of steps because the number of steps may be too crude a measure of time and a more honest measure may be required. The representation function may be multi-valued, in which case Q transforms any (A) into some (A). The inverse function 1 is usually single-valued. Neil Jones [J] calls a universal program U for a language L e cient if there is a constant c such that U simulates every L-program in lock-step with lag factor c. He exhibits a programming language I that admits an e cient universal program and then uses such a program and diagonalization to show that functions I-computable in linear time form an in nite hierarchy. To show the robustness of his results, Jones mentions a number of languages and machine models which admit e cient universal programs and give the same class LIN of problems solvable in linear time. Among those machines models are the storage modi cation machines of Schonhage [S] and successor RAMs, that is, random access machines whose only arithmetical operation is the operation of computing the successor of a given natural number. Jones's thought-provoking paper seems to open a possibility of proving linear time lower bounds for linear time computations. Unfortunately linear time on successor RAMs is too restrictive for many applications, e.g., computational geometry, and in general linear time is not very robust notion. Should one try to extend Jones's results to addition RAMs, or to addition and multiplication RAMs, or to addition, multiplication and division RAMs, or to something else?