P′′

${\mathcal {P}}^{\prime \prime }$ (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, ..., a_n}(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 a_i by a_i+1 (taking a_n+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
• [P′′online interpreter] demonstrating '99 bottles of beer' in 337568 instructions.