V. Register Machine

Yuxi Fu

BASICS, Shanghai Jiao Tong University Register Machines are more abstract than Turing Machines.

Computability Theory, by Y. Fu V. Register Machine 1 / 35 Register Machine Models can be classiﬁed into three groups: I CM (Counter Machine Model)

I Unlimited Register Machine Model

I RAM (Random Access Machine Model)

I RASP (Random Access Stored Program Machine Model)

Computability Theory, by Y. Fu V. Register Machine 2 / 35 Synopsis

1. Unlimited Register Machine

2. Deﬁnability in URM

3. Simulation of TM by URM

Computability Theory, by Y. Fu V. Register Machine 3 / 35 1. Unlimited Register Machine

Computability Theory, by Y. Fu V. Register Machine 4 / 35 Unlimited Register Machine Model

Unlimited Register Machines are introduced by Shepherdson and Sturgis in

Computability and Recursive Functions. Journal of Symbolic Logic, 32:1-63, 1965.

Computability Theory, by Y. Fu V. Register Machine 5 / 35 Register

An Unlimited Register Machine (URM) has an inﬁnite number of register labeled R1, R2, R3,....

r1 r2 r3 r4 r5 r6 r7 ...

R1 R2 R3 R4 R5 R6 R7 ... Every register can hold a natural number at any moment.

The registers can be equivalently written as for example

3 4 7 ∞ [r1, r2, r3]1[r4]4[r5, r6, r7]5[0, 0, 0,...]8

or simply 3 4 7 [r1, r2, r3]1[r4]4[r5, r6, r7]5.

Computability Theory, by Y. Fu V. Register Machine 6 / 35 Program

A URM also has a program, which is a ﬁnite list of instructions.

Computability Theory, by Y. Fu V. Register Machine 7 / 35 Instruction

type instruction response of URM

Zero Z(n) Replace rn by 0. Successor S(n) Add 1 to rn. Transfer T (m, n) Copy rm to Rn. Jump J(m, n, q) If rm = rn, go to the q-th instruction; otherwise go to the next instruction.

Computability Theory, by Y. Fu V. Register Machine 8 / 35 Computation

Registers:

9 7 0 0 0 0 0 ...

R1 R2 R3 R4 R5 R6 R7 Program:

I1 : J(1, 2, 6) I2 : S(2) I3 : S(3) I4 : J(1, 2, 6) I5 : J(1, 1, 2) I6 : T (3, 1)

Computability Theory, by Y. Fu V. Register Machine 9 / 35 Conﬁguration and Computation

Conﬁguration: register contents + current instruction number.

Initial configuration, computation, final configuration.

Computability Theory, by Y. Fu V. Register Machine 10 / 35 Some Notation

Suppose P is the program of a URM and a1, a2, a3,... are the numbers stored in the registers.

I P(a1, a2, a3,...) is the initial conﬁguration.

I P(a1, a2, a3,...) ↓ means that the computation converges.

I P(a1, a2, a3,...) ↑ means that the computation diverges.

I P(a1, a2,..., am) is P(a1, a2,..., am, 0, 0,...).

Computability Theory, by Y. Fu V. Register Machine 11 / 35 Every URM uses only a ﬁxed ﬁnite number of registers, no matter how large an input number is.

Computability Theory, by Y. Fu V. Register Machine 12 / 35 2. Deﬁnability in URM

Computability Theory, by Y. Fu V. Register Machine 13 / 35 URM-Computable Function

Let f (ex) be an n-ary partial function. What does it mean that a URM computes f (ex)?

Computability Theory, by Y. Fu V. Register Machine 14 / 35 URM-Computable Function

Suppose P is the program of a URM and a1,..., an, b ∈ ω.

The computation P(a1,..., an) converges to b if P(a1,..., an) ↓ and r1 = b in the ﬁnal conﬁguration.

In this case we write P(a1,..., an) ↓ b.

P URM-computes f if, for all a1,..., an, b ∈ ω, P(a1,..., an) ↓ b iﬀ f (a1,..., an) = b. The function f is URM-deﬁnable if there is a program that URM-computes f .

Computability Theory, by Y. Fu V. Register Machine 15 / 35 I1 : J(3, 2, 5) I2 : S(1) I3 : S(3) I4 : J(1, 1, 1)

Example of URM

Construct a URM that computes x + y.

Computability Theory, by Y. Fu V. Register Machine 16 / 35 Example of URM

Construct a URM that computes x + y.

I1 : J(3, 2, 5) I2 : S(1) I3 : S(3) I4 : J(1, 1, 1)

Computability Theory, by Y. Fu V. Register Machine 16 / 35 I1 : J(1, 4, 8) I2 : S(3) I3 : J(1, 3, 7) I4 : S(2) I5 : S(3) I6 : J(1, 1, 3) I7 : T (2, 1)

Example of URM

x − 1, if x > 0, Construct a URM that computes x−˙ 1 = 0, if x = 0.

Computability Theory, by Y. Fu V. Register Machine 17 / 35 Example of URM

x − 1, if x > 0, Construct a URM that computes x−˙ 1 = 0, if x = 0.

I1 : J(1, 4, 8) I2 : S(3) I3 : J(1, 3, 7) I4 : S(2) I5 : S(3) I6 : J(1, 1, 3) I7 : T (2, 1)

Computability Theory, by Y. Fu V. Register Machine 17 / 35 I1 : J(1, 2, 6) I2 : S(3) I3 : S(2) I4 : S(2) I5 : J(1, 1, 1) I6 : T (3, 1)

Example of URM

Construct a URM that computes

x/2, if x is even, x ÷ 2 = undeﬁned, if x is odd.

Computability Theory, by Y. Fu V. Register Machine 18 / 35 Example of URM

Construct a URM that computes

x/2, if x is even, x ÷ 2 = undeﬁned, if x is odd.

I1 : J(1, 2, 6) I2 : S(3) I3 : S(2) I4 : S(2) I5 : J(1, 1, 1) I6 : T (3, 1)

Computability Theory, by Y. Fu V. Register Machine 18 / 35 Function Deﬁned by Program

n b, if P(a1,..., an) ↓ b, fP (a1,..., an) = undeﬁned, if P(a1,..., an) ↑ .

Computability Theory, by Y. Fu V. Register Machine 19 / 35 Program in Standard Form

A program P = I1,..., Is is in standard form if, for every jump instruction J(m, n, q) we have q ≤ s + 1.

For every program there is a program in standard form that computes the same function.

We will focus exclusively on programs in standard form.

Computability Theory, by Y. Fu V. Register Machine 20 / 35 Program Composition

Given Programs P and Q, how do we construct the sequential composition P; Q?

The jump instructions of P and Q must be modiﬁed.

Computability Theory, by Y. Fu V. Register Machine 21 / 35 Some Notations

Suppose the program P computes f .

Let ρ(P) be the least number i such that the register Ri is not used by the program P.

Computability Theory, by Y. Fu V. Register Machine 22 / 35 Some Notations

The notation P[l1,..., ln → l] stands for the following program

I1 : T (l1, 1) . .

In : T (ln, n)

In+1 : Z(n + 1) . .

Iρ(P) : Z(ρ(P)) : P : T (1, l)

Computability Theory, by Y. Fu V. Register Machine 23 / 35 Deﬁnability of Initial Function

Fact. The initial functions are URM-deﬁnable.

Computability Theory, by Y. Fu V. Register Machine 24 / 35 Deﬁnability of Composition

Fact. If f (y1,..., yk ) and g1(ex),..., gk (ex) are URM-deﬁnable, then the composition function h(ex) given by

h(ex) ' f (g1(ex),..., gk (ex)) is URM-deﬁnable.

Computability Theory, by Y. Fu V. Register Machine 25 / 35 Deﬁnability of Composition

Let F , G1,..., Gk be programs that compute f , g1,..., gk .

Let m be max{n, k, ρ(F ), ρ(G1), . . . , ρ(Gk )}. Registers:

m m+n m+n+1 m+n+k [...]1 [ex]m+1[g1(ex)]m+n+1 ... [gk (ex)]m+n+k

Computability Theory, by Y. Fu V. Register Machine 26 / 35 Deﬁnability of Composition

The program for h:

I1 : T (1, m + 1) . .

In : T (n, m + n)

In+1 : G1[m + 1, m + 2,..., m + n → m + n + 1] . .

In+k : Gk [m + 1, m + 2,..., m + n → m + n + k]

In+k+1 : F [m + n + 1 ..., m + n + k → 1]

Computability Theory, by Y. Fu V. Register Machine 27 / 35 Deﬁnability of Recursion

Fact. Suppose f (ex) and g(ex, y, z) are URM-deﬁnable. The recursion function h(ex, y) deﬁned by the following recursion

h(ex, 0) ' f (ex), h(ex, y + 1) ' g(ex, y, h(ex, y)) is URM-deﬁnable.

Computability Theory, by Y. Fu V. Register Machine 28 / 35 Deﬁnability of Recursion Let F compute f and G compute g. Let m be max{n, ρ(F ), ρ(G)}. m m+n m+n+1 m+n+2 m+n+3 Registers: [...]1 [ex]m+1[y]m+n+1[k]m+n+2[h(ex, k)]m+n+3. Program:

I1 : T (1, m + 1) . .

In+1 : T (n + 1, m + n + 1)

In+2 : F [1, 2,..., n → m + n + 3]

In+3 : J(m + n + 2, m + n + 1, n + 7)

In+4 : G[m + 1,..., m + n, m + n + 2, m + n + 3 → m + n + 3]

In+5 : S(m + n + 2)

In+6 : J(1, 1, n + 3)

In+7 : T (m + n + 3, 1)

Computability Theory, by Y. Fu V. Register Machine 29 / 35 Deﬁnability of Minimization

Fact. If f (ex, y) is URM-deﬁnable, then the minimization function µy(f (ex, y) = 0) is URM-deﬁnable.

Computability Theory, by Y. Fu V. Register Machine 30 / 35 Deﬁnability of Minimization

Suppose F computes f (ex, y). Let m be max{n + 1, ρ(F )}. m m+n m+n+1 m+n+2 Registers: [...]1 [ex]m+1[k]m+n+1[0]m+n+2. Program:

I1 : T (1, m + 1) . .

In : T (n, m + n)

In+1 : F [m + 1, m + 2,..., m + n + 1 → 1]

In+2 : J(1, m + n + 2, n + 5)

In+3 : S(m + n + 1)

In+4 : J(1, 1, n + 1)

In+5 : T (m + n + 1, 1)

Computability Theory, by Y. Fu V. Register Machine 31 / 35 Main Result

Theorem. All recursive functions are URM-deﬁnable.

Computability Theory, by Y. Fu V. Register Machine 32 / 35 3. Simulation of TM by URM

Computability Theory, by Y. Fu V. Register Machine 33 / 35 Simulating TM by URM

Suppose M is a 3-tape TM with the alphabet {0, 1, , B}. The URM that simulates M can be designed as follows: I Suppose that Rm is the right most register that is used by a program calculating x−˙ 1.

I The head positions are stored in Rm+1, Rm+2, Rm+3. I The three binary strings in the tapes are stored respectively in Rm+4, Rm+7, Rm+10,..., Rm+5, Rm+8, Rm+11,..., Rm+6, Rm+9, Rm+12,.... I The states of M are encoded by the states of the URM. I The transition function of M can be easily simulated by the program of the URM.

Computability Theory, by Y. Fu V. Register Machine 34 / 35 Exercise. Describe an algorithm that transforms a TM to a URM.

Computability Theory, by Y. Fu V. Register Machine 35 / 35