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Concept
Polish notation
Summary
Polish notation (PN), also known as normal Polish notation (NPN), Łukasiewicz notation, Warsaw notation, Polish prefix notation or simply prefix notation, is a mathematical notation in which operators precede their operands, in contrast to the more common infix notation, in which operators are placed between operands, as well as reverse Polish notation (RPN), in which operators follow their operands. It does not need any parentheses as long as each operator has a fixed number of operands. The description "Polish" refers to the nationality of logician Jan Łukasiewicz, who invented Polish notation in 1924.
The term Polish notation is sometimes taken (as the opposite of infix notation) to also include reverse Polish notation.
When Polish notation is used as a syntax for mathematical expressions by programming language interpreters, it is readily parsed into abstract syntax trees and can, in fact, define a one-to-one representation for the same. Because of this, Lisp (see below) and related programming languages define their entire syntax in prefix notation (and others use postfix notation).
A quotation from a paper by Jan Łukasiewicz in 1931 states how the notation was invented:
I came upon the idea of a parenthesis-free notation in 1924. I used that notation for the first time in my article Łukasiewicz (1), p. 610, footnote.
The reference cited by Łukasiewicz, i.e., Łukasiewicz (1), is apparently a lithographed report in Polish. The referring paper by Łukasiewicz was reviewed by Henry A. Pogorzelski in the Journal of Symbolic Logic in 1965. Heinrich Behmann, editor in 1924 of the article of Moses Schönfinkel, already had the idea of eliminating parentheses in logic formulas. In one of his papers Łukasiewicz stated that his notation is the most compact and the first linearly written parentheses-free notation, but not the first one as Gottlob Frege proposed his parentheses-free Begriffsschrift notation in 1879 already.
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Arity
In logic, mathematics, and computer science, arity (ˈærᵻti) is the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank, but this word can have many other meanings. In logic and philosophy, arity may also be called adicity and degree. In linguistics, it is usually named valency. In general, functions or operators with a given arity follow the naming conventions of n-based numeral systems, such as binary and hexadecimal.
Exclusive or
Exclusive or or exclusive disjunction or exclusive alternation, also known as non-equivalence which is the negation of equivalence, is a logical operation that is true if and only if its arguments differ (one is true, the other is false). It is symbolized by the prefix operator and by the infix operators XOR (ˌɛks_ˈɔ:r, ˌɛks_ˈɔ:, 'ksɔ:r or 'ksɔ:), EOR, EXOR, , , , ⩛, , and . It gains the name "exclusive or" because the meaning of "or" is ambiguous when both operands are true; the exclusive or operator excludes that case.
Infix notation
Infix notation is the notation commonly used in arithmetical and logical formulae and statements. It is characterized by the placement of operators between operands—"infixed operators"—such as the plus sign in 2 + 2. Binary relations are often denoted by an infix symbol such as set membership a ∈ A when the set A has a for an element. In geometry, perpendicular lines a and b are denoted and in projective geometry two points b and c are in perspective when while they are connected by a projectivity when Infix notation is more difficult to parse by computers than prefix notation (e.