Translation to and from Polish Notation

Abstract

The question of efficient translation between an orthodox mathematical notation of the kind ordinarily used in writing algebraic formulae (and copied as closely as is practicable in FORTRAN and ALGOL) and notation has come to prominence as result of the use of what is in effect Polish notation as the basic instruction language of two recent computers.* Polish notation is so-called because of its extensive use in Polish logical writings since its invention by -Lukasiewicz (1921, 1929). -Lukasiewicz demonstrated that if operators are written always in front of their operands, instead of (as in the case of the diadic operators of arithmetic, + , —, x and so on) between them, there is never any need for brackets to indicate association of terms. Thus if in place of a -fb we write + b, and so on, the brackets in an expression such as (a + b) X c may be dispensed with in translation, since X + b c indicates unambiguously the result of operating with x on -\-ab and c: for a + (b X c) we should instead write + X b c. The resulting notation, in the case of long formulae, is little harder to read, since brackets aid the eye, but it has some other advantages. In particular Reverse Polish—the notation which results if operators are placed after operands, as in a b +—has the property that the operators appear in the order in which they are required in computation. Reverse Polish is hence in some sense natural notation for an instruction language, each symbol being interpretable as an instruction. (Number variables are fetch instructions.) The absence of brackets further makes Polish notation (— either Forward or Reverse, but probably preferably Forward—) useful in mechanized algebra, since it eliminates continual source of complication in algebraic manipulations.