Editing Random-access machine (section)

=== Unbounded indirection and the partial recursive functions ===
For ''unbounded'' indirection we require a "hardware" change in our machine model. Once we make this change the model is no longer a counter machine, but rather a random-access machine.

Now when e.g. INC  is specified, the finite-state machine's instruction will have to specify ''where'' the address of the register of interest will come from. This ''where'' can be either (i) the state machine's instruction that provides an ''explicit label'', or (ii) the ''pointer-register'' whose ''contents'' is the address of interest. Whenever an instruction specifies a register address it now will ''also'' need to specify an additional parameter "i/d"{{spaced ndash}}"indirect/direct". In a sense this new "i/d" parameter is a "switch" that flips one way to get the direct address as specified in the instruction or the other way to get the indirect address from the pointer register (which pointer register{{spaced ndash}}in some models every register can be a pointer register{{spaced ndash}}is specified by the instruction). This "mutually exclusive but exhaustive choice" is yet another example of "definition by cases", and the arithmetic equivalent shown in the example below is derived from the definition in Kleene (1952) p. 229.

:Example: CPY ( indirect<sub>source</sub>, r<sub>source</sub>, direct<sub>destination</sub>, r<sub>destination</sub> )

:Assign a code to specify direct addressing as d="0" and indirect addressing as i="1". Then our machine can determine the source address as follows:
:: i*[r<sub>s</sub>] + (1-i)*r<sub>s</sub>
:For example, suppose the contents of register 3 are "5" (i.e. [3]=5 ) and the contents of register 4 are "2" (i.e. [4]=2 ):

:: Example: CPY ( 1, 3, 0, 4 ) = CPY ( indirect, reg 3, direct, reg 4 )
::: 1*[3] + 0*3 = [3] = source-register address 5
::: 0*[4] + 1*4 = 4 = destination-register address 4

:: Example: CPY ( 0, 3, 0, 4 )
::: 0*[3] + 1*3 = 3 = source-register address 3
::: 0*[4] + 1*4 = 4 = destination-register address 4

:: Example: CPY ( 0, 3, 1, 4 )
::: 0*[3] + 1*3 = 3 = source-register address 3
::: 1*[4] + 0*4 = [4] = destination-register address 2