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Polish notation (PN), also known as normal Polish notation (NPN), Łukasiewicz notation, Warsaw notation, Polish prefix notation, Eastern Notation or simply prefix notation, is a mathematical notation in which operators ''precede'' their
operand In mathematics, an operand is the object of a mathematical operation, i.e., it is the object or quantity that is operated on. Unknown operands in equalities of expressions can be found by equation solving. Example The following arithmetic expres ...
s, in contrast to the more common
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 . Usage Binary relations are ...
, 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 Nationality is the legal status of belonging to a particular nation, defined as a group of people organized in one country, under one legal jurisdiction, or as a group of people who are united on the basis of culture. In international law, n ...
of logician
Jan Łukasiewicz Jan Łukasiewicz (; 21 December 1878 – 13 February 1956) was a Polish logician and philosopher who is best known for Polish notation and Łukasiewicz logic. His work centred on philosophical logic, mathematical logic and history of logi ...
, 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 A programming language is a system of notation for writing computer programs. Programming languages are described in terms of their Syntax (programming languages), syntax (form) and semantics (computer science), semantics (meaning), usually def ...
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 Lisp (historically LISP, an abbreviation of "list processing") is a family of programming languages with a long history and a distinctive, fully parenthesized Polish notation#Explanation, prefix notation. Originally specified in the late 1950s, ...
( see below) and related programming languages define their entire syntax in prefix notation (and others use postfix notation).


History

A quotation from a paper by
Jan Łukasiewicz Jan Łukasiewicz (; 21 December 1878 – 13 February 1956) was a Polish logician and philosopher who is best known for Polish notation and Łukasiewicz logic. His work centred on philosophical logic, mathematical logic and history of logi ...
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 Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philos ...
proposed his parentheses-free
Begriffsschrift ''Begriffsschrift'' (German for, roughly, "concept-writing") is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book. ''Begriffsschrift'' is usually translated as ''concept writing'' or ''concept notati ...
notation in 1879 already.
Alonzo Church Alonzo Church (June 14, 1903 – August 11, 1995) was an American computer scientist, mathematician, logician, and philosopher who made major contributions to mathematical logic and the foundations of theoretical computer science. He is bes ...
mentions this notation in his classic book on
mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
as worthy of remark in notational systems even contrasted to Alfred Whitehead and
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, and public intellectual. He had influence on mathematics, logic, set theory, and various areas of analytic ...
's logical notational exposition and work in
Principia Mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by the mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1 ...
. In Łukasiewicz's 1951 book, ''Aristotle's Syllogistic from the Standpoint of Modern Formal Logic'', he mentions that the principle of his notation was to write the
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
s before the
argument An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persu ...
s to avoid brackets and that he had employed his notation in his logical papers since 1929. He then goes on to cite, as an example, a 1930 paper he wrote with
Alfred Tarski Alfred Tarski (; ; born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician ...
on the sentential calculus. While no longer used much in logic, Polish notation has since found a place in
computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...
.


Explanation

The expression for adding the numbers 1 and 2 is written in Polish notation as (prefix), rather than as (infix). In more complex expressions, the operators still precede their operands, but the operands may themselves be expressions including again operators and their operands. For instance, the expression that would be written in conventional infix notation as can be written in Polish notation as Assuming a given
arity In logic, mathematics, and computer science, arity () 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 ...
of all involved operators (here the "−" denotes the binary operation of subtraction, not the unary function of sign-change), any well-formed prefix representation is unambiguous, and brackets within the prefix expression are unnecessary. As such, the above expression can be further simplified to The processing of the product is deferred until its two operands are available (i.e., 5 minus 6, and 7). As with ''any'' notation, the innermost expressions are evaluated first, but in Polish notation this "innermost-ness" can be conveyed by the sequence of operators and operands rather than by bracketing. In the conventional infix notation, parentheses are required to override the standard precedence rules, since, referring to the above example, moving them or removing them changes the meaning and the result of the expression. This version is written in Polish notation as When dealing with non-commutative operations, like division or subtraction, it is necessary to coordinate the sequential arrangement of the operands with the definition of how the operator takes its arguments, i.e., from left to right. For example, , with 10 to the left of 5, has the meaning of 10 ÷ 5 (read as "divide 10 by 5"), or , with 7 left to 6, has the meaning of 7 − 6 (read as "subtract from 7 the operand 6").


Evaluation algorithm

Prefix/postfix notation is especially popular for its innate ability to express the intended order of operations without the need for parentheses and other precedence rules, as are usually employed with
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 . Usage Binary relations are ...
. Instead, the notation uniquely indicates which operator to evaluate first. The operators are assumed to have a fixed
arity In logic, mathematics, and computer science, arity () 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 ...
each, and all necessary operands are assumed to be explicitly given. A valid prefix expression always starts with an operator and ends with an operand. Evaluation can either proceed from left to right, or in the opposite direction. Starting at the left, the input string, consisting of tokens denoting operators or operands, is pushed token for token on a stack, until the top entries of the stack contain the number of operands that fits to the top most operator (immediately beneath). This group of tokens at the stacktop (the last stacked operator and the according number of operands) is replaced by the result of executing the operator on these/this operand(s). Then the processing of the input continues in this manner. The rightmost operand in a valid prefix expression thus empties the stack, except for the result of evaluating the whole expression. When starting at the right, the pushing of tokens is performed similarly, just the evaluation is triggered by an operator, finding the appropriate number of operands that fits its arity already at the stacktop. Now the leftmost token of a valid prefix expression must be an operator, fitting to the number of operands in the stack, which again yields the result. As can be seen from the description, a push-down store with no capability of arbitrary stack inspection suffices to implement this
parsing Parsing, syntax analysis, or syntactic analysis is a process of analyzing a String (computer science), string of Symbol (formal), symbols, either in natural language, computer languages or data structures, conforming to the rules of a formal gramm ...
. The above sketched stack manipulation works—with mirrored input—also for expressions in reverse Polish notation.


Polish notation for logic

The table below shows the core of
Jan Łukasiewicz Jan Łukasiewicz (; 21 December 1878 – 13 February 1956) was a Polish logician and philosopher who is best known for Polish notation and Łukasiewicz logic. His work centred on philosophical logic, mathematical logic and history of logi ...
's notation in modern logic. Some letters in the Polish notation table stand for particular words in Polish, as shown: The quantifiers ranged over propositional values in Łukasiewicz's work on many-valued logics. Bocheński introduced a system of Polish notation that names all 16 binary connectives of classical
propositional logic The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...
. For classical propositional logic, it is a compatible extension of the notation of Łukasiewicz. But the notations are incompatible in the sense that Bocheński uses L and M (for nonimplication and converse nonimplication) in propositional logic and Łukasiewicz uses L and M in modal logic.


Implementations

Prefix notation has seen wide application in
Lisp Lisp (historically LISP, an abbreviation of "list processing") is a family of programming languages with a long history and a distinctive, fully parenthesized Polish notation#Explanation, prefix notation. Originally specified in the late 1950s, ...
S-expressions, where the parentheses are required since the operators in the language are themselves data (
first-class function In computer science, a programming language is said to have first-class functions if it treats function (programming), functions as first-class citizens. This means the language supports passing functions as arguments to other functions, returning ...
s). Lisp functions may also be variadic. The Tcl programming language, much like Lisp also uses Polish notation through the mathop library. The Ambi programming language uses Polish notation for arithmetic operations and program construction. LDAP filter syntax uses Polish prefix notation. Postfix notation is used in many stack-oriented programming languages like
PostScript PostScript (PS) is a page description language and dynamically typed, stack-based programming language. It is most commonly used in the electronic publishing and desktop publishing realm, but as a Turing complete programming language, it c ...
and Forth.
CoffeeScript CoffeeScript is a programming language that compiles to JavaScript. It adds syntactic sugar inspired by Ruby, Python, and Haskell in an effort to enhance JavaScript's brevity and readability. Some added features include list comprehension an ...
syntax also allows functions to be called using prefix notation, while still supporting the unary postfix syntax common in other languages. The number of return values of an expression equals the difference between the number of operands in an expression and the total arity of the operators minus the total number of return values of the operators. Polish notation, usually in postfix form, is the chosen notation of certain
calculator An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics. The first solid-state electronic calculator was created in the early 1960s. Pocket-si ...
s, notably from
Hewlett-Packard The Hewlett-Packard Company, commonly shortened to Hewlett-Packard ( ) or HP, was an American multinational information technology company. It was founded by Bill Hewlett and David Packard in 1939 in a one-car garage in Palo Alto, California ...
. At a lower level, postfix operators are used by some stack machines such as the Burroughs large systems.


See also

* Reverse Polish notation (RPN) *
Function application In mathematics, function application is the act of applying a function to an argument from its domain so as to obtain the corresponding value from its range. In this sense, function application can be thought of as the opposite of function abs ...
*
Lambda calculus In mathematical logic, the lambda calculus (also written as ''λ''-calculus) is a formal system for expressing computability, computation based on function Abstraction (computer science), abstraction and function application, application using var ...
*
Currying In mathematics and computer science, currying is the technique of translating a function that takes multiple arguments into a sequence of families of functions, each taking a single argument. In the prototypical example, one begins with a functi ...
*
Lisp (programming language) Lisp (historically LISP, an abbreviation of "list processing") is a family of programming languages with a long history and a distinctive, fully parenthesized prefix notation. Originally specified in the late 1950s, it is the second-oldest hig ...
**
S-expression In computer programming, an S-expression (or symbolic expression, abbreviated as sexpr or sexp) is an expression in a like-named notation for nested List (computing), list (Tree (data structure), tree-structured) data. S-expressions were invented ...
* Polish School of Mathematics *
Hungarian notation Hungarian notation is an identifier naming convention in computer programming in which the name of a variable or function indicates its intention or kind, or in some dialects, its type. The original Hungarian notation uses only intention or kin ...
* Verb–subject–object (VSO) * Verb–object–subject (VOS) * Head-directionality parameter * WFF 'N PROOF


References


Further reading

*

(77 pages) * (11 pages) (NB. Published in .) * (11 pages)


External links

* {{DEFAULTSORT:Polish Notation Mathematical notation Polish inventions Science and technology in Poland Operators (programming) Logical expressions