P′′
(hereafter written P′′) is a primitive programming language created by Corrado Böhm ^{1}^{,}^{2} in 1964 to describe a family of Turing machines.
Contents
Syntax
- R and λ are words in P′′.
- If p and q are words in P′′, then pq is a word in P′′.
- If q is a word in P′′, then (q) is a word in P′′.
- Only words derivable from the previous three rules are words in P′′.
In BNF:
<word> ::= R|λ|<word><word>|(<word>)
Semantics
- {a_{0}, a_{1}, ..., a_{n}}(n ≥ 1) is the tape-alphabet of a Turing machine with left-infinite tape, a_{0} 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} = a_{0}), 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 a_{0}.
- 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′′. Böhm^{1} 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 proved^{2} 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
- Böhm, C.: "On a family of Turing machines and the related programming language", ICC Bull. 3, (July 1964), 187-194.
- 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}