# P′′

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P′′
Paradigm imperative, structured
Designed by Corrado Böhm
First appeared 1964
untyped

P′′ is a primitive computer programming language created by Corrado Böhm[1][2] in 1964 to describe a family of Turing machines.

## Definition

$\mathcal{P}^{\prime\prime}$ (hereafter written P′′) is formally defined as a set of words on the four-instruction alphabet {R, λ, (, )}, as follows:

### 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′′.

### 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 a(i+1) mod (n+1), 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

• The brainfuck language (apart from its I/O commands) is a minor informal variation of P′′. Böhm[1] gives explicit P′′ programs for each of a set of basic functions sufficient to compute any computable function, using only (, ) and the four words r ≡ λR, r′ ≡ rn (rn means rrrrr...rr [n times]), L ≡ r′λ, R. These are the equivalents of the six respective brainfuck commands [, ], +, -, <, >. Note that since an+1 = a0, performing r ("increment" symbol in current cell) n times will wrap around so that the result is to "decrement" the symbol in the current cell by one (r′).

## Example program

Böhm[1] gives the following program to compute the predecessor (x-1) of an integer x > 0:

R ( R ) L ( r' ( L ( L ) ) r' L ) R r


which translates directly to the equivalent brainfuck program

> [ > ] < [ −  [ < [ < ] ] −  < ] > +


The program expects an integer to be represented in bijective base-n notation, with a1, ..., an coding the digits 1,...,n, respectively, and to have an a0 before and after the digit-string. (E.g. in bijective base-2, the number eight would be encoded as a0a1a1a2a0, because 8 = 1*22 + 1*21 + 2*20.) At the beginning and end of the computation, the tape-head is on the a0 preceding the digit-string.

## References

1. ^ a b c d Böhm, C.: "On a family of Turing machines and the related programming language", ICC Bull. 3, 185-194, July 1964.
2. ^ a b Böhm, C. and Jacopini, G.: "Flow diagrams, Turing machines and languages with only two formation rules", CACM 9(5), 1966. (Note: This is the most-cited paper on the structured program theorem.)