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Paradigm Imperative, structured
Designed by Corrado Böhm
First appeared 1964
Typing discipline untyped

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


(hereinafter written P′′) is formally defined as a set of words on the four-instruction alphabet , as follows:


  1. and are words in P′′.
  2. If and are words in P′′, then is a word in P′′.
  3. If is a word in P′′, then is a word in P′′.
  4. Only words derivable from the previous three rules are words in P′′.


  • is the tape-alphabet of a Turing machine with left-infinite tape, being the blank symbol, equivalent to .
  • All instructions in P′′ are permutations of the set of all possible tape configurations; that is, all possible configurations of both the contents of the tape and the position of the tape-head.
  • is a predicate saying that the current symbol is not . It is not an instruction and is not used in programs, but is instead used to help define the language.
  • means move the tape-head rightward one cell (if possible).
  • means replace the current symbol with , and then move the tape-head leftward one cell.
  • means the function composition . In other words, the instruction is performed before .
  • means iterate in a while loop, with the condition .

Relation to other programming languages[edit]

  • P′′ was the first "GOTO-less" imperative structured programming language to be proven Turing-complete[1][2]
  • The brainfuck language (apart from its I/O commands) is a minor informal variation of P′′. Böhm gives explicit P′′ programs for each of a set of basic functions sufficient to compute any computable function, using only , and the four words where is the th iterate of , These are the equivalents of the six respective brainfuck commands [, ], +, -, <, >. Note that since , incrementing the current symbol times will wrap around so that the result is to "decrement" the symbol in the current cell by one ().

Example program[edit]

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

which translates directly to the equivalent brainfuck program:


The program expects an integer to be represented in bijective base-k notation, with encoding the digits respectively, and to have before and after the digit-string. (E.g., in bijective base-2, the number eight would be encoded as , because 8 in bijective base-2 is 112.) At the beginning and end of the computation, the tape-head is on the preceding the digit-string.


  1. ^ a b c 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.)