View source for Polish notation

{{Short description|Mathematics notation with operators preceding operands}}
{{Use list-defined references|date=July 2022}}

'''Polish notation''' ('''PN'''), also known as '''normal Polish notation''' ('''NPN'''),<ref name="Jorke_1989"/> '''Łukasiewicz notation''', '''Warsaw notation''', '''Polish prefix notation''' or simply '''prefix notation''', is a mathematical notation in which [[Operation (mathematics)|operators]] ''precede'' their [[Operand|operand]]s, in contrast to the more common [[Infix notation|infix notation]], in which operators are placed ''between'' operands, as well as [[Reverse Polish notation|reverse Polish notation]] (RPN), in which operators ''follow'' their operands. It does not need any parentheses as long as each operator has a fixed [[Arity|number of operands]]. The description "Polish" refers to the nationality of logician [[Biography:Jan Łukasiewicz|Jan Łukasiewicz]],<ref name="Łukasiewicz_1929"/>{{rp|page=24}}<ref name="Łukasiewicz_1957"/>{{rp|page=78}}<ref name="Kennedy_1982"/> who invented Polish notation in 1924.<ref name="Łukasiewicz_1931"/>{{rp|page=367, Footnote 3}}<ref name="Łukasiewicz_1970"/>{{rp|page=180, Footnote 3}}

The term ''Polish notation'' is sometimes taken (as the opposite of ''infix notation'') to also include reverse Polish notation.<ref name="Main_2006"/>

When Polish notation is used as a syntax for mathematical expressions by [[Programming language|programming language]] [[Interpreter (computing)|interpreter]]s, it is readily parsed into [[Abstract syntax tree|abstract syntax tree]]s and can, in fact, define a [[Bijection|one-to-one representation]] for the same. Because of this, [[Lisp (programming language)|Lisp]] (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 [[Biography:Jan Łukasiewicz|Jan Łukasiewicz]] in 1931<ref name="Łukasiewicz_1931"/>{{rp|page=367, Footnote 3}}<ref name="Łukasiewicz_1970"/>{{rp|page=180, Footnote 3}} states how the notation was invented:

<blockquote>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.</blockquote>

The reference cited by Łukasiewicz, i.e., Łukasiewicz (1),<ref name="Łukasiewicz_1929a"/> is apparently a lithographed report in [[Social:Polish language|Polish]]. The referring paper<ref name="Łukasiewicz_1931"/> by Łukasiewicz was reviewed by Henry A. Pogorzelski in the ''Journal of Symbolic Logic'' in 1965.<ref name="Pogorzelski_1965"/> Heinrich Behmann, editor in 1924 of the article of [[Biography:Moses Schönfinkel|Moses Schönfinkel]],<ref name="Mengelberg"/> 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 [[Biography:Gottlob Frege|Gottlob Frege]] proposed his parentheses-free [[Begriffsschrift]] notation in 1879 already.<ref name="Gottschall_2005"/>

Alonzo Church mentions this notation in his classic book on [[Mathematical logic|mathematical logic]] as worthy of remark in notational systems even contrasted to [[Biography:Alfred North Whitehead|Alfred Whitehead]] and [[Biography:Bertrand Russell|Bertrand Russell]]'s logical notational exposition and work in [[Principia Mathematica]].<ref name="Church_1944"/>

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 functors before the [[Argument of a function|argument]]s to avoid brackets and that he had employed his notation in his logical papers since 1929.<ref name="Łukasiewicz_1957"/>{{rp|page=78}} He then goes on to cite, as an example, a 1930 paper he wrote with [[Biography:Alfred Tarski|Alfred Tarski]] on the [[Propositional calculus|sentential calculus]].<ref name="Łukasiewicz_1930"/>

While no longer used much in logic,<ref name="Martínez_2011"/> Polish notation has since found a place in [[Computer science|computer science]].

==Explanation==
The expression for adding the numbers 1 and 2 is written in Polish notation as {{nowrap|+ 1 2}} (prefix), rather than as {{nowrap|1 + 2}} (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
{{block indent|(5 − 6) × 7}}
can be written in Polish notation as
{{block indent|× (− 5 6) 7}}
Assuming a given [[Arity|arity]] 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
{{block indent|× − 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 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 [[Order of operations|precedence rules]], since, referring to the above example, moving them
{{block indent|5 − (6 × 7)}}
or removing them
{{block indent|5 − 6 × 7}}
changes the meaning and the result of the expression. This version is written in Polish notation as
{{block indent|− 5 × 6 7.}}

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, {{nowrap|÷ 10 5}}, with 10 to  the left of 5, has the meaning of 10 ÷ 5 (read as "divide 10 by 5"), or {{nowrap|− 7 6}}, 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]]. Instead, the notation uniquely indicates which operator to evaluate first. The operators are assumed to have a fixed [[Arity|arity]] 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 (abstract data type)|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 [[Deterministic pushdown automaton|push-down store]] with no capability of arbitrary stack inspection suffices to implement this [[Parsing|parsing]].

The above sketched stack manipulation works—with mirrored input—also for expressions in [[Reverse Polish notation|reverse Polish notation]].

==Polish notation for logic==
The table below shows the core of [[Biography:Jan Łukasiewicz|Jan Łukasiewicz]]'s notation in modern logic. Some letters in the Polish notation table stand for particular words in [[Social:Polish language|Polish]], as shown:

{| class=wikitable
|-
!Concept!!Conventional<br/> notation!!Polish<br/> notation!!Polish<br/>term
|-
|- style="border-top:3px solid #999;"
|-
|[[Negation]]||<math>\neg\phi</math>||<math> N\phi</math><ref name="Łukasiewicz_1929"/>{{rp|pages=27–28}}||{{lang|pl|negacja}}
|-
|[[Material conditional]]||<math>\phi\to\psi</math>||<math>C\phi\psi</math><ref name="Łukasiewicz_1929"/>{{rp|pages=28–31}}||{{lang|pl|implikacja}}
|-
|Disjunction||<math>\phi\lor\psi</math>||<math>A\phi\psi</math><ref name="Łukasiewicz_1929"/>{{rp|pages=34–35}}||{{lang|pl|alternatywa}}
|-
|[[Logical conjunction|Conjunction]]||<math>\phi\land\psi</math>||<math>K\phi\psi</math><ref name="Łukasiewicz_1929"/>{{rp|pages=35–36}}||{{lang|pl|koniunkcja}}
|-
|[[Sheffer stroke|Non-conjunction]]||<math>\phi\mid\psi </math>||<math>D\phi\psi</math><ref name="Łukasiewicz_1929"/>{{rp|page=36}}||{{lang|pl|dysjunkcja}}
|-
|Biconditional||<math>\phi\leftrightarrow\psi</math>||<math>E\phi\psi</math><ref name="Łukasiewicz_1929"/>{{rp|page=37}} or <math>Q\phi\psi</math><ref name="Łukasiewicz_1957"/>{{rp|page=108}}||{{lang|pl|ekwiwalencja}}
|-
|[[Universal quantification|Universal quantifier]]||<math>\forall p\,\phi</math>||<math>\varPi p\,\phi</math><ref name="Łukasiewicz_1929"/>{{rp|pages=154–156}}||{{lang|pl|kwantyfikator ogólny}}
|-
|[[Existential quantification|Existential quantifier]]||<math>\exists p\,\phi</math>||<math>\varSigma p\,\phi</math><ref name="Łukasiewicz_1929"/>{{rp|page=157}}||{{lang|pl|kwantyfikator szczegółowy}}
|-
|Verum||<math>\top</math>||<math>V</math><ref name="Łukasiewicz_1939"/>{{rp|page=275}}||{{lang|pl|prawdziwy}}
|-
|[[Modal logic|Possibility]]||<math>\Diamond\phi</math>||<math>M\phi</math><ref name="Łukasiewicz_1930x"/>{{rp|page=52}}<ref name="Łukasiewicz_1957"/>{{rp|page=134}} or <math>\varDelta\phi</math><ref name="Łukasiewicz_1953"/>{{rp|page=111}}||{{lang|pl|możliwość}}
|-
|[[Modal logic|Necessity]]||<math>\Box\phi</math>||<math>L\phi</math><ref name="Łukasiewicz_1957"/>{{rp|page=134}} or <math>\varGamma\phi</math><ref name="Łukasiewicz_1953"/>{{rp|page=111}}||{{lang|pl|konieczność}}
|}

Note that the [[Philosophy:Quantifier (logic)|quantifiers]] ranged over propositional values in Łukasiewicz's work on many-valued logics.

[[Biography:Józef Maria Bocheński|Bocheński]] introduced a system of Polish notation that names all 16 binary [[Logical connective|connectives]] of classical propositional logic.<ref name="Bochenski_1949"/>{{rp|page=16}} 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 <math>L</math> and <math>M</math> (for nonimplication and converse nonimplication) in propositional logic and Łukasiewicz uses <math>L</math> and <math>M</math> in modal logic.

==Implementations==
Prefix notation has seen wide application in [[Lisp (programming language)|Lisp]] S-expressions, where the brackets are required since the operators in the language are themselves data ([[First-class function|first-class function]]s). Lisp functions may also be [[Variadic function|variadic]]. The [[Tcl]] programming language, much like Lisp also uses Polish notation through the mathop library. The Ambi<ref name="Ambi"/> programming language uses Polish notation for arithmetic operations and program construction. LDAP filter syntax uses Polish prefix notation.<ref name="LDAPSyntax"/>

Postfix notation is used in many stack-oriented programming languages like [[PostScript]] and [[Forth (programming language)|Forth]]. [[CoffeeScript]] 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 [[Engineering:Calculator|calculator]]s, notably from Hewlett-Packard.<ref name="RPN_HP35S"/> At a lower level, postfix operators are used by some stack machines such as the [[Burroughs large systems]].

==See also==
{{cols|colwidth=26em}}
* [[Reverse Polish notation]] (RPN)
* [[Function application]]
* [[Lambda calculus]]
* [[Currying]]
* [[Lisp (programming language)]]
** [[S-expression]]
* [[Organization:Polish School of Mathematics|Polish School of Mathematics]]
* [[Hungarian notation]]
* [[Social:Verb–subject–object|Verb–subject–object]] (VSO)
* Verb–object–subject (VOS)
* Head-directionality parameter
* [[WFF 'N PROOF]]
{{colend}}

==References==
{{reflist|refs=
<ref name="Łukasiewicz_1929a">{{cite journal |author-last=Łukasiewicz |author-first=Jan |title=O znaczeniu i potrzebach logiki matematycznej |journal=Nauka Polska |date=1929 |volume=10 |pages=604–620 |language=pl |url=https://www.sbc.org.pl/dlibra/publication/22955/edition/20205/content?format_id=2}}</ref>
<ref name="Łukasiewicz_1929">{{cite book |title=Elementy logiki matematycznej |language=pl |author-last=Łukasiewicz |author-first=Jan |location=Warsaw, Poland |edition=1 |publisher=Państwowe Wydawnictwo Naukowe |date=1929 |postscript=none}}; {{cite book |title=Elements of Mathematical Logic |author-last=Łukasiewicz |author-first=Jan |author-link=Jan Łukasiewicz |translator-first=Olgierd Adrian |translator-last=Wojtasiewicz |translator-link=:pl:Olgierd Adrian Wojtasiewicz |date=1963 |publication-place=New York, USA |publisher=The MacMillan Company}}</ref>
<ref name="Łukasiewicz_1930">{{cite journal |title=Untersuchungen über den Aussagenkalküls |trans-title=Investigations into the Sentential Calculus |url=https://rcin.org.pl/impan/dlibra/publication/4839/edition/50601/content |language=de |author-last1=Łukasiewicz |author-first1=Jan |author-last2=Tarski |author-first2=Alfred |author-link2=Alfred Tarski |journal=Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie |volume=23 |number=Cl. III |pages=30–50 |date=1930}}</ref>
<ref name="Łukasiewicz_1930x">{{cite journal |title=Untersuchungen über den Aussagenkalküls |trans-title=Investigations into the Sentential Calculus |url=https://rcin.org.pl/impan/dlibra/publication/4839/edition/50601/content |language=de |author-last=Łukasiewicz |author-first=Jan |journal=Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie |volume=23 |number=Cl. III |pages=51–77 |date=1930}}</ref>
<ref name="Łukasiewicz_1931">{{cite encyclopedia |url=https://sbc.org.pl/dlibra/docmetadata?showContent=true&id=18864 |title=Uwagi o aksjomacie Nicoda i 'dedukcji uogólniającej' |trans-title=Comments on Nicod's Axiom and on 'Generalizing Deduction' |encyclopedia=Księga pamiątkowa Polskiego Towarzystwa Filozoficznego We Lwowie, 12. II. 1904–1912. II. 1929 |date=1931 |author-last=Łukasiewicz |author-first=Jan |publisher=Wydawnictwo Polskie Towarzystwo Filozoficzne |location=Lwów |pages=366–383 |language=pl}}</ref>
<ref name="Łukasiewicz_1939">{{cite journal |author-last=Łukasiewicz |author-first=Jan |title=Der Äquivalenzenkalkül |journal=Collectanea Logica |date=1939 |volume=1 |pages=145–169 |language=de}}</ref>
<ref name="Łukasiewicz_1957">{{cite book |title=Aristotle's Syllogistic from the Standpoint of Modern Formal Logic |author-last=Łukasiewicz |author-first=Jan |publisher=[[Organization:Oxford University Press|Oxford University Press]] |orig-date=1951 |date=1957 |edition=2}} (Reprinted by Garland Publishing in 1987, {{isbn|0-8240-6924-2}}.)</ref>
<ref name="Łukasiewicz_1953">{{cite journal |author-last=Łukasiewicz |author-first=Jan |title=A System of Modal Logic |journal=The Journal of Computing Systems |date=1953 |volume=3 |issue=1 |pages=111–149}}</ref>
<ref name="Łukasiewicz_1970">{{cite encyclopedia |title=Comments on Nicod's Axiom and on 'Generalizing Deduction' |encyclopedia=Selected Works |date=1970 |author-last=Łukasiewicz |author-first=Jan |editor-last=Borkowski |editor-first=L. |publisher=North-Holland Publishing Company/Polish Scientific Publishers |location=Amsterdam and London/Warszawa |pages=179–196}}</ref>
<ref name="Kennedy_1982">{{cite journal |title=RPN Perspective |author-first=John |author-last=Kennedy |location=Mathematics Department, Santa Monica College, Santa Monica, California, USA |journal=PPC Calculator Journal |volume=9 |number=5 |date=August 1982 |pages=26–29 |citeseerx=10.1.1.90.6448 |url=http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.90.6448&rep=rep1&type=pdf |access-date=2022-07-02 |url-status=live |archive-url=https://web.archive.org/web/20220701223543/http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.90.6448&rep=rep1&type=pdf |archive-date=2022-07-01}} (12 pages)</ref>
<ref name="Jorke_1989">{{cite book |title=Arithmetische Algorithmen der Mikrorechentechnik |trans-title=Arithmetic algorithms in microcomputers |author-first1=Günter |author-last1=Jorke |author-first2=Bernhard |author-last2=Lampe |author-first3=Norbert |author-last3=Wengel |edition=1 |publisher=VEB Verlag Technik |location=Berlin, Germany |date=1989 |language=de |isbn=3-34100515-3 |id={{EAN|978-3-34100515-6}}. MPN 5539165. License 201.370/4/89 |url=https://books.google.com/books?id=DqYWAQAAMAAJ |access-date=2015-12-01}}</ref>
<ref name="Main_2006">{{cite book |title=Data structures and other objects using Java |edition=3 |author-first=Michael |author-last=Main |publisher=Pearson PLC Addison-Wesley |date=2006 |isbn=978-0-321-37525-4 |page=334 |url=https://books.google.com/books?id=Tok_AQAAIAAJ}}</ref>
<ref name="Pogorzelski_1965">{{cite journal |title=Reviewed work(s): Remarks on Nicod's Axiom and on "Generalizing Deduction" by Jan Łukasiewicz, Jerzy Słupecki, Państwowe Wydawnictwo Naukowe |author-last=Pogorzelski |author-first=Henry Andrew |url=https://www.jstor.org/stable/2269644 |type=Review |journal=The Journal of Symbolic Logic |publisher=[[Organization:Association for Symbolic Logic|Association for Symbolic Logic]] |volume=30 |number=3 |date=September 1965 |pages=376–377 |jstor=2269644}} (NB. The original 1931 paper "Uwagi o aksjomacie Nicoda i 'dedukcji uogólniającej" by [[Biography:Jan Łukasiewicz|Jan Łukasiewicz]] was re-published at Państwowe Wydawnictwo Naukowe (National Scientific Publishers), [[Place:Warsaw|Warsaw]], Poland in 1961 in a volume edited by Jerzy Słupecki.)</ref>
<ref name="Mengelberg">{{cite journal |title=Über die Bausteine der mathematischen Logik |language=de |trans-title=On the building blocks of mathematical logic |author-last=Schönfinkel |author-first=Moses |journal=[[Mathematische Annalen]] |date=1924 |volume=92 |pages=305–316 |doi=10.1007/BF01448013 |s2cid=118507515 |issue=3–4 |postscript=none}}; {{cite book |chapter=On the building blocks of mathematical logic |translator-first=Stefan |translator-last=Bauer-Mengelberg |translator-link=:nl:Stefan Bauer-Mengelberg |editor-first=Jean |editor-last=van Heijenoort |date=1967 |title=A Source Book in Mathematical Logic, 1879–1931 |publisher=Harvard University Press |pages=355–366}}</ref>
<ref name="Church_1944">{{cite book |author-first=Alonzo |author-last=Church |title=Introduction to Mathematical Logic |location=Princeton, New Jersey, USA |publisher=Princeton University Press |date=1944 |page=38 |quote=[…] 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. […]}}</ref>
<ref name="Martínez_2011">{{citation |contribution=Mhy bib I fail logic? Dyslexia in the teaching of logic |author-first=Xóchitl |author-last=Martínez Nava |pages=162–169 |title=Tools for Teaching Logic: Third International Congress, TICTTL 2011, Salamanca, Spain, 1–4 June 2011, Proceedings |volume=6680 |series=Lecture Notes in Artificial Intelligence |publisher=[[Company:Springer Nature|Springer Nature]] |date=2011-06-01 |isbn=978-3-64221349-6 |editor-first1=Patrick |editor-last1=Blackburn |editor-first2=Hans |editor-last2=van Ditmarsch |editor-first3=Maria |editor-last3=Manzano |editor-first4=Fernando |editor-last4=Soler-Toscano |doi=10.1007/978-3-642-21350-2_19 |url=https://books.google.com/books?id=be-pTR5TmZIC&pg=PA166 |quote=[…] Polish or prefix notation has come to disuse given the difficulty that using it implies. […]}}</ref>
<ref name="Ambi">{{cite web |title=Google Code Archive - Long-term storage for Google Code Project Hosting |url=https://code.google.com/p/ambi/ |access-date=2022-11-14 |url-status=live |archive-url=https://web.archive.org/web/20170928175932/https://code.google.com/archive/p/ambi/ |archive-date=2017-09-28}}</ref>
<ref name="RPN_HP35S">{{cite web |title=HP calculators - HP 35s RPN Mode |publisher=[[Company:Hewlett-Packard|Hewlett-Packard]] |url=http://h20331.www2.hp.com/Hpsub/downloads/35_02_RPN_Mode.pdf |access-date=2022-11-14 |url-status=live |archive-url=https://web.archive.org/web/20220121192354/http://h20331.www2.hp.com/Hpsub/downloads/35_02_RPN_Mode.pdf |archive-date=2022-01-21}}</ref>
<ref name="LDAPSyntax">{{cite web |title=LDAP Filter Syntax |url=http://www.ldapexplorer.com/en/manual/109010000-ldap-filter-syntax.htm |access-date=2022-11-14 |url-status=live |archive-url=https://web.archive.org/web/20221014105027/https://www.ldapexplorer.com/en/manual/109010000-ldap-filter-syntax.htm |archive-date=2022-10-14}}</ref>
<ref name="Gottschall_2005">{{cite book |title=Logische Notationen und deren Verarbeitung auf elektronischen Rechenanlagen aus theoretischer, praktischer und historischer Sicht. |language=de |trans-title= |author-first=Christian |author-last=Gottschall |type=Diploma thesis |location=Vienna, Austria |date=2005 |quote-page=88 |quote=Die ältesten Texte in den 'Selected Works', in denen Łukasiewicz polnische Notation verwendet, datieren relativ spät, sind aber Präsentationen vorangehender Arbeiten, die 'in the course of the years 1920–1930' (S. 131) stattgefunden haben, also auch keine genauere Zeitangabe geben.}}</ref>
<ref name="Bochenski_1949">{{cite book |author-last=Bocheński |author-first=Józef Maria |title=Précis de logique mathématique |date=1949 |publication-place=Bussum, Pays-Bas, Netherlands |location=Fribourg |publisher=F. G. Kroonder |language=fr |series=Collection Synthese |volume=2 |url=https://burjcdigital.urjc.es/bitstream/handle/10115/1425/PRECIS_DE_LOGIQUE_MATHEMATIQUE.pdf?sequence=1&isAllowed=y |access-date=2023-11-12 |url-status=live |archive-url=https://web.archive.org/web/20230803130610/https://burjcdigital.urjc.es/bitstream/handle/10115/1425/PRECIS_DE_LOGIQUE_MATHEMATIQUE.pdf?sequence=1&isAllowed=y |archive-date=2023-08-03}} Translated as {{cite book |author-last=Bocheński |author-first=Józef Maria |author-link=Józef Maria Bocheński |translator-last=Bird |translator-first=Otto A. |translator-link=:d:Q94507730 |title=A Precis of Mathematical Logic |date=1959 |publisher=D. Reidel Publishing Company |location=Dordrecht, Netherlands}}</ref>
<!--
<ref name="Hamblin_1962">{{cite journal |author-first=Charles Leonard |author-last=Hamblin |date=1962 |title=Translation to and from Polish notation |journal=Computer Journal |volume=5 |issue=3 |pages=210–213 |doi=10.1093/comjnl/5.3.210 |doi-access=free}}</ref>
-->
}}

==Further reading==
* {{cite book |title=Keller, Stack und automatisches Gedächtnis – eine Struktur mit Potenzial |language=de |trans-title=Cellar, stack and automatic memory - a structure with potential |editor-first1=Michael |editor-last1=Fothe |editor-first2=Thomas |editor-last2=Wilke |type=Tagungsband zum Kolloquium 14. November 2014 in Jena |location=Jena, Germany |volume=T-7 |series=GI Series: Lecture Notes in Informatics (LNI) – Thematics |publisher=[[Organization:Gesellschaft für Informatik|Gesellschaft für Informatik]] (GI) / Köllen Druck + Verlag GmbH |isbn=978-3-88579-426-4 |issn=1614-3213 |date=2015 |orig-date=2014-11-14 |publication-place=Bonn, Germany |url=https://dl.gi.de/bitstream/handle/20.500.12116/4381/lni-t-7.pdf?sequence=1&isAllowed=y |access-date=2020-04-12 |url-status=live |archive-url=https://web.archive.org/web/20200412122706/https://dl.gi.de/bitstream/handle/20.500.12116/4381/lni-t-7.pdf?sequence=1&isAllowed=y |archive-date=2020-04-12}} [https://web.archive.org/web/20221210100112/https://dl.gi.de/handle/20.500.12116/4374/browse?type=title&sort_by=4] (77 pages)
* {{cite book |title=Friedrich L. Bauers und Klaus Samelsons Arbeiten in den 1950er-Jahren zur Einführung der Begriffe Kellerprinzip und Kellerautomat |language=de |trans-title=Friedrich L. Bauer's and Klaus Samelson's works in the 1950s on the introduction of the terms cellar principle and cellar automaton |author-first=Hans |author-last=Langmaack |location=Kiel, Germany |publisher=Institut für Informatik, Christian-Albrechts-Universität zu Kiel |publication-place=Jena, Germany |date=2015 |orig-date=2014-11-14 |pages=19–29 |url=https://dl.gi.de/bitstream/handle/20.500.12116/33413/19.pdf?sequence=1&isAllowed=y |access-date=2022-11-14 |url-status=live |archive-url=https://web.archive.org/web/20221114205159/https://dl.gi.de/bitstream/handle/20.500.12116/33413/19.pdf?sequence=1&isAllowed=y |archive-date=2022-11-14}} (11 pages) (NB. Published in {{citeref|Fothe|Wilke|2015|Fothe & Wilke|style=plain}}.)
* {{cite book |title=Geschichte der deutschsprachigen Informatik - Programmiersprachen und Übersetzerbau |language=de |trans-title=History of informatics in German-speaking countries - Programming languages and compiler design |author-first=Gerhard |author-last=Goos |publisher=Fakultät für Informatik, Karlsruhe Institute of Technology (KIT) |publication-place=Karlsruhe, Germany |date=2017-08-07 |url=http://www.kps2017.uni-jena.de/proceedings/kps2017_submission_1.pdf |access-date=2022-11-14 |url-status=live |archive-url=https://web.archive.org/web/20220519131116/http://www.kps2017.uni-jena.de/proceedings/kps2017_submission_1.pdf |archive-date=2022-05-19}} (11 pages)

==External links==

{{DEFAULTSORT:Polish Notation}}
[[Category:Mathematical notation]]
[[Category:Operators (programming)]]
[[Category:Logical expressions]]

{{Sourceattribution|Polish notation}}