Unlimited Register Machine (URM) Example

The goal of this URM is to calculate the function: i + (2*j) + k and place the result in R₄, where for example i = 2, j = 3 and k = 7. So the function is now 2 + (2*3) + 7

URM Program:

Z (4)

1. J (3, 4, 6)
2. S (1)
3. S (2)
4. S (4)
5. J (1, 1, 1)
6. J (2, 4, 10)
7. S (1)
8. S (4)
9. J (1, 1, 6) 
10. J (1, 3, 14)
11. S (3)
12. S (4)
13. J (1, 1, 10)

The following points are observation about the above program.

• The program uses registers R₁, R₂, R₃ and R₄
• If r₃ = r₄, the first instruction involves a jump to instruction 6
• If r₂ = r₄, the third instruction involves a jump to instruction 10
• If r₁ = r₃, the fifth instruction involves a jump to instruction 14
• The instructions 5 involves jumping back to the first instruction if r₁ = r₁ and so, as this is always true, the effect of this instruction is the unconditional jump “go to instruction number 1”. This also applies for instructions 9 and 13.
• The overall effect of the first and third, the fourth and sixth and the seventh and tenth instructions is that the program includes 3 ‘do while r₃ ≠ r_4, r₂ ≠ r₄ and r₁ ≠ r₃’ loops.

Illustration of URM transaction:

Instructions	R1	R2	R3	R4
	2	3	7	0
1	2	3	7	0
2	3	3	7	0
3	3	4	7	0
4	3	4	7	1
5	3	4	7	1
1	3	4	7	1
2	4	4	7	1
3	4	5	7	1
4	4	5	7	2
5	4	5	7	2
1	4	5	7	2
2	5	5	7	2
3	5	6	7	2
4	5	6	7	3
5	5	6	7	3
1	5	6	7	3
2	6	6	7	3
3	6	7	7	3
4	6	7	7	4
5	6	7	7	4
1	6	7	7	4
2	7	7	7	4
3	7	8	7	4
4	7	8	7	5
5	7	8	7	5
1	7	8	7	5
2	8	8	7	5
3	8	9	7	5
4	8	9	7	6
5	8	9	7	6
1	8	9	7	6
2	9	9	7	6
3	9	10	7	6
4	9	10	7	7
5	9	10	7	7
6	9	10	7	7
7	10	10	7	7
8	10	10	7	8
9	10	10	7	8
6	10	10	7	8
7	11	10	7	8
8	11	10	7	9
9	11	10	7	9
6	11	10	7	9
7	12	10	7	9
8	12	10	7	10
9	12	10	7	10
10	12	10	7	10
11	12	10	8	10
12	12	10	8	11
13	12	10	8	11
10	12	10	8	11
11	12	10	9	11
12	12	10	9	12
13	12	10	9	12
10	12	10	9	12
11	12	10	10	12
12	12	10	10	13
13	12	10	10	13
10	12	10	10	13
11	12	10	11	14
12	12	10	11	14
13	12	10	11	14
10	12	10	11	14
11	12	10	12	14
12	12	10	12	15
13	12	10	12	15
STOP	12	10	12	15

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Poster Design by Michael Lashley

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Using this theorem, below the NFSA is used to define a RE. 

by Michael Lashley