P′′

Definition

P ′ ′ (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, ..., 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) 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

P′′ was the first "GOTO-less" imperative structured programming language to be proven Turing-complete

The brainfuck language (apart from its I/O commands) is a minor informal variation of P′′

Example program

Böhm 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, ..., a_n 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.