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Polish notation, also known as Polish prefix notation or simply prefix notation, is a form of notation for logic, arithmetic, and algebra. Its distinguishing feature is that it places operators to the left of their operands. If the arity of the operators is fixed, the result is a syntax lacking parentheses or other brackets that can still be parsed without ambiguity. The Polish logician Jan Łukasiewicz invented this notation in 1924 in order to simplify sentential logic. The term ''Polish notation'' is sometimes taken (as the opposite of ''infix notation'') to also include Polish ''postfix'' notation, or reverse Polish notation, in which the operator is placed ''after'' the operands. 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 terms of prefix notation (and others use postfix notation). Here is a quotation from a paper by Jan Łukasiewicz, ''Remarks on Nicod's Axiom and on "Generalizing Deduction"'', page 180. 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 Jan Łukasiewicz above is apparently a lithographed report in Polish. The referring paper by Łukasiewicz ''Remarks on Nicod's Axiom and on "Generalizing Deduction"'' was reviewed by H. A. Pogorzelski in the ''Journal of Symbolic Logic'' in 1965.〔Pogorzelski, H. A., ("Reviewed work(s): Remarks on Nicod's Axiom and on "Generalizing Deduction" by Jan Łukasiewicz; Jerzy Słupecki; Państwowe Wydawnictwo Naukowe" ), ''The Journal of Symbolic Logic'', Vol. 30, No. 3 (Sep. 1965), pp. 376–377. The original paper by Jan Łukasiewicz was published in Warsaw in 1961 in a volume edited by Jerzy Słupecki.〕 Alonzo Church mentions this notation in his classic book on mathematical logic as worthy of remark in notational systems even contrasted to Whitehead and Russell's logical notational exposition and work in Principia Mathematica.〔 – p. 38: "Worthy of remark is the parenthesis-free notation of Jan Łukasiewicz. In this the letters N, A, C, E, K are used in the roles of negation, disjunction, implication, equivalence, conjunction respectively. ..."〕 In Łukasiewicz 1951 book, ''Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic'', he mentions that the principle of his notation was to write the functors before the arguments to avoid brackets and that he had employed his notation in his logical papers since 1929.〔Cf. Łukasiewicz, (1951) ''Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic'', Chapter IV "Aristotle's System in Symbolic Form" (section on "Explanation of the Symbolism"), p. 78 and on.〕 He then goes on to cite, as an example, a 1930 paper he wrote with Alfred Tarski on the sentential calculus.〔Łukasiewicz, Jan; Tarski, Alfred, "Untersuchungen über den Aussagenkalkül" (into the sentential calculus" ), ''Comptes Rendus des séances de la Société des Sciences et des Lettres de Varsovie'', Vol, 23 (1930) Cl. III, pp. 31–32.〕 While no longer used much in logic,〔, p. 166: ("Polish or prefix notation has come to disuse given the difficulty that using it implies." )〕 Polish notation has since found a place in computer science. ==Arithmetic== The expression for adding the numbers 1 and 2 is, in prefix notation, written "+ 1 2" rather than "1 + 2". In more complex expressions, the operators still precede their operands, but the operands may themselves be nontrivial expressions including operators of their own. For instance, the expression that would be written in conventional infix notation as : (5 − 6) × 7 can be written in prefix as : × (− 5 6) 7 Since the simple arithmetic operators are all binary (at least, in arithmetic contexts), any prefix representation thereof is unambiguous, and bracketing the prefix expression is unnecessary. As such, the previous expression can be further simplified to : × − 5 6 7 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 prefix notation this "innermost-ness" can be conveyed by order rather than bracketing. In the classical notation, the parentheses in the infix version were required, since moving them : 5 − (6 × 7) or simply removing them : 5 − 6 × 7 would change the meaning and result of the overall expression, due to the precedence rule. Similarly : 5 − (6 × 7) can be written in Polish notation as : − 5 × 6 7 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Polish notation」の詳細全文を読む スポンサード リンク
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