P′′

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P′′.png (hereafter written P′′) is a primitive programming language created by Corrado Böhm 1,2 in 1964 to describe a family of Turing machines.

Syntax

  1. R and λ are words in P′′.
  2. If p and q are words in P′′, then pq is a word in P′′.
  3. If q is a word in P′′, then (q) is a word in P′′.
  4. Only words derivable from the previous three rules are words in P′′.

In BNF: <word> ::= R|λ|<word><word>|(<word>)

Semantics

  • {a0, a1, ..., an}(n ≥ 1) is the tape-alphabet of a Turing machine with left-infinite tape, a0 being the blank symbol.
  • R means move the tape-head rightward one cell (if any).
  • λ means replace the current symbol ai by ai+1 (taking an+1 = a0), and then move the tape-head leftward one cell.
  • (q) means iterate q in a while-loop, with condition that the current symbol is not a0.
  • A program is a word in P′′. Execution of a program proceeds left-to-right, executing R, λ, and (q) as they are encountered, until there is nothing more to execute.

Relation to other programming languages

Brainfuck (apart from its I/O instructions) is a simple informal variation of Böhm's P′′. Böhm1 gives explicit P′′ programs for each of a set of basic functions sufficient to compute any partial recursive function -- and these programs are constructed entirely from six P′′ words precisely equivalent to the respective Brainfuck commands +, -, <, >, [, and ].

P′′ was the first "goto-less" or "structured programming" language proved2 to be functionally equivalent to languages that use gotos.

Implementation

A Haskell implementation:

{-# LANGUAGE TypeOperators #-}

module P'' where

import Control.Category ((>>>))
import Control.Lens hiding (Tape)
import Control.Monad

data Term = R | L | Seq Term Term | Loop Term
type Tape = Top :>> [Int] :>> Int

-- Run a P'' term over a tape, in an alphabet of size n.
-- Returns Nothing if the program moves off the tape. Note
-- that directions are flipped; the left-infinite tape is
-- implemented as a right-infinite list zipper.
runTerm :: Int -> Term -> Tape -> Maybe Tape
runTerm n R         = leftward
runTerm n L         = focus %~ (\x -> succ x `mod` n) >>> rightward
runTerm n (Seq p q) = runTerm n p >=> runTerm n q
runTerm n (Loop p)  = \tape ->
    case tape ^. focus of
        0 -> Just tape
        _ -> runTerm n p >=> runTerm n (Loop p) $ tape

-- Run a P'' program over an empty tape.
runP'' :: Int -> Term -> Maybe Tape
runP'' n p = runTerm n p blankTape
    where blankTape = zipper (repeat 0) & fromWithin traverse

Example usage: define a program that calculates n-1, then run it and print the first 10 tape elements.

prog :: Term
prog = let s = foldr1 Seq in s [L, R, Loop $ s [L, L, R, R]]

main :: IO ()
main = print $ runP'' 256 prog <&> rezip <&> take 10

References

  1. Böhm, C.: "On a family of Turing machines and the related programming language", ICC Bull. 3, (July 1964), 187-194.
  2. Böhm, C. and Jacopini, G.: "Flow diagrams, Turing machines and languages with only two formation rules", CACM 9(5), 1966. (Note: This is the seminal paper on the structured program theorem.)

External resources

  • P′′ at Wikipedia describes an explicit example program from Böhm.1