# RAM with compact memory: a realistic and robust model of computation

## Abstract

An operation op of arity k on ℕ, i.e. a function op: ℕ^{k} → ℕ is *linear time Turing computable* (for short, LTTC) if it is computable in linear time on a Turing machine (for usual binary or dyadic notation of integers). Let + and *Conc* respectively denote usual addition of integers and concatenation (of their dyadic notations). A RAM which uses only arithmetical operations of a set I is called an I - RAM. An LTTC-RAM is a RAM which only uses LTTC operations.

In the present paper, we use the logarithmic criterion for time measure of RAMs. A RAM works *with polynomially (resp. strongly polynomially) compact memory* if it only uses addresses (resp. addresses and register contents) ≤ t^{O(1)} where t is the time of the computation.

Theorem 1. A deterministic LTTC-RAM R with polynomially compact memory is simulated in linear time by a deterministic {+}-RAM (resp. {*Conc*}-RAM) R' with strongly polynomially compact memory.

Theorem 2. A nondeterministic LTTC-RAM R can be simulated in linear time by a nondeterministic {+}-RAM (resp. {*Conc*}-RAM) R' with strongly poynomially compact memory.

Note that Theorem 2 holds for both weak nondeterministic RAMs and strong nondeterministic RAMs, i.e. in case the RAMs have only nondeterministic goto instructions or in case they have an instruction to guess an integer.

If moreover the RAMs R of Theorem 1–2 are *sane* (i.e. R does not use noninitialised registers) then the simulating RAMs R' are sane, too.

We also study and discuss more restrictive notions of compact memory (*linearly compact memory*).

## Keywords

Hash Function Linear Time Turing Machine Hash Table Block Length## Preview

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