RAMs and TMs

A word-RAM consists of: • A fixed set of instructions P1, . . . , Pq. Allowed instructions are: – Modular arithmetic and integer division on registers; the standard model for a RAM machine does not have negative numbers, but you can verify that allowing signed integers doesn’t chance the power of the model. – Bitwise operations on registers. – Loading a constant into a register. – Transfering values between a register and a word in main memory (as addressed by another register). – Conditional GOTO. – MALLOC – increase the size S of main memory by 1. This also increments the word size if the current word size is too small to address all of main memory. – HALT. • The program counter l ∈ {1, . . . , q}, telling us where we are in the program. Except for GOTO statements, the counter increments after each instruction is executed. • The space usage S, which starts at the size of the input. • The word size w; the initial word size is dlog2 max(n+ 1, k + 1)e, where n is the length of the input and k is the maximum constant appearing in the program or input. • A constant number of registers R[0], . . . , R[r − 1], each of which is a w-bit word. • Main memory M [0], . . . ,M [S − 1], each of which is a w-bit word. The tuple (l, S, w,R,M) gives the configuration of the word-RAM. We say that a word-RAM solves a computational problem f : Σ∗ → 2N if the machine which starts with input x halts with some output in f(x) as the set of characters in M [0], . . . ,M [R[0]− 1]. As we saw with Turing Machines, there are many adjustments we can make to a model which make it easier to reason about without significantly changing its power. Here’s one for word-RAMs: