In a unit-cost random access machine, the value of a register $$r_i$$ can be set to the product of other two registers $$r_j$$ and $$r_n$$ using basic $$L O O P$$ commands as well as the algorithm for $$r_i:=0$$ , and the algorithm for $$r_i:=r_j\pm r_n$$ .

### Implementing a unit-cost random access machine capable to represent negative numbers

Although the unit-cost random access machine is not able to store negative numbers in its registers, it is not much more complicated to calculate the difference of two registers $$r_j$$ and $$r_n$$, because negative integers can be represented by pairs of natural numbers, as shown in the definition of integers. Thus, it is possible to build a unit-cost random access machine, which is able to store negative numbers using more auxiliary registers representing integers as pairs of natural numbers and implementing an appropriate subtraction operation on these pairs.

# Algorithm: Multiplication of Two Registers

class UCRAM(): # unit-cost random access machine with 3 registers r_i = 0 r_j = 0 r_n = 0

  def set_ri_to_0(self):
while True:
if self.r_i &gt; 0:
self.r_i = self.r_i - 1 #  LOOP register i
self.NOP()  #  DO nothing
else:
break

def set_rn_to_11(self):
# reset register n to 0
while True:
if self.r_n &gt; 0:
self.r_n = self.r_n - 1 #  LOOP register n
self.NOP()  #  DO nothing
else:
break
self.r_n = self.r_n + 1  # set register n to the value 11
self.r_n = self.r_n + 1
self.r_n = self.r_n + 1
self.r_n = self.r_n + 1
self.r_n = self.r_n + 1
self.r_n = self.r_n + 1
self.r_n = self.r_n + 1
self.r_n = self.r_n + 1
self.r_n = self.r_n + 1
self.r_n = self.r_n + 1
self.r_n = self.r_n + 1

def set_rj_to_7(self):
# reset register j to 0
while True:
if self.r_j &gt; 0:
self.r_j = self.r_j - 1 #  LOOP register n
self.NOP()  #  DO nothing
else:
break
self.r_j = self.r_j + 1  # set register j to the value 7
self.r_j = self.r_j + 1
self.r_j = self.r_j + 1
self.r_j = self.r_j + 1
self.r_j = self.r_j + 1
self.r_j = self.r_j + 1
self.r_j = self.r_j + 1

def LOOP_rj_DO_increment_ri(self):
while True:
if self.r_j &gt; 0:
self.r_j = self.r_j - 1 #  LOOP register j
self.r_i = self.r_i + 1 #  DO increment register i
else:
break

self.LOOP_rj_DO_increment_ri()

def NOP(self):
self.r_i = self.r_i + 1 #  LOOP register i
self.r_i = self.r_i - 1  # LOOP register i

def multiply_rj_times_rn_and_store_result_in_ri(self):
self.set_ri_to_0()
while True:
if self.r_n &gt; 0:
self.r_n = self.r_n - 1  # LOOP register n
self.set_rj_to_7()
else:
break

1. Usage for adding the r_i = r_j * r_n
2. creating a unit-cost access machine with registers initially set ucram = UCRAM() ucram.set_rj_to_7() ucram.set_rn_to_11() print(ucram.r_i, ucram.r_j, ucram.r_n) ucram.multiply_rj_times_rn_and_store_result_in_ri() print(ucram.r_i, ucram.r_j, ucram.r_n)

3. will output

4. 0 7 11
5. 0 7 11

Examples: 1

Github: ### References

#### Bibliography

1. Erk, Katrin; Priese, Lutz: "Theoretische Informatik", Springer Verlag, 2000, 2nd Edition