Suvarna Garge (Editor)

P′′

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Paradigm
  
Imperative, structured

First appeared
  
1964

Designed by
  
Corrado Böhm

Typing discipline
  
untyped

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

Contents

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, ..., 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

  • 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, ..., 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

    P′′ Wikipedia
    (Text) CC BY-SA


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