P′′

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

Definition

(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^[1][2] Turing-complete.
• 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 a_n+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, ..., 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.

References