Universal Register Machine implemented at the type-level of Haskell
This is just a proof of concept!
Available on Hackage
The machine consists of a set of registers, a contiguous instruction pool (starting at index 0) and supports the following 3 instructions:
Inc r l- increments register
rby 1, and then jumps to label
l(that is the instruction located at index
Dec r l1 l2- if the value of register
ris 0, jumps to
l2, otherwise decrements
rand jumps to
Halt- halts the machine.
This formulation is identical to the Lambek machine, with the addition of an explicit
Halt instruction, for convenience. This means that that Haskell's type system is Turing complete (with
If the execution of a given program terminates, it will result in the type
Halted ip rs, where
ip is the
index of the instruction that halted the machine (there might be multiple
Halt instructions in the code)
rs is a list representing the resulting state of the registers.
Since the machine is implemented at the type-level of Haskell, the instructions are executed during the type-checking phase. This means that a program that doesn't terminate will hang the type-checker.
(Be careful with using an on-the-fly type checker editor plugin, as checking a non-terminating (or even a relatively complex) program will consume a lot of RAM!)
Initialises R1 to 5, then raises 2 to the power of the value of R1, leaving the result (32) in R0. Uses R2 as a scratch register, thus the machine is initialised with 3 registers.
pow2 :: ('Halted a (r ': rs) ~ Run '[ -- Instr | label index -- set R1 to 5 Inc (R 1) (L 1) -- 0 , Inc (R 1) (L 2) -- 1 , Inc (R 1) (L 3) -- 2 , Inc (R 1) (L 4) -- 3 , Inc (R 1) (L 5) -- 4 -- set R0 to 1 , Inc (R 0) (L 6) -- 5 -- R0 = 2^R1 , Dec (R 1) (L 7) (L 12) -- 6 -- R2 = R0 , Dec (R 0) (L 8) (L 9) -- 7 , Inc (R 2) (L 7) -- 8 -- R0 = 2*R2 , Dec (R 2) (L 10) (L 6) -- 9 , Inc (R 0) (L 11) -- 10 , Inc (R 0) (L 9) -- 11 , Halt -- 12 ]) => Proxy r pow2 = Proxy
- composable gadgets (that is reusable components, for these to work, their used registers and labels need to be separated)
- more examples
- formally prove correctness