Wikipedia

Reverse Polish notation

"Operational stack" redirects here. For the English Channel lorry parking procedure, seeOperation Stack.

Reverse Polish notation(RPN), also known asreverse Łukasiewicz notation,Polish postfix notationor simplypostfix notation,is a mathematical notation in whichoperatorsfollowtheiroperands,in contrast toprefixorPolish notation(PN), in which operatorsprecedetheir operands. The notation does not need any parentheses for as long as each operator has a fixednumber of operands.

Video: Keys pressed for calculating eight times six on aHP-32SII(employing RPN) from 1991

The termpostfix notationdescribes the general scheme in mathematics and computer sciences, whereas the termreverse Polish notationtypically refers specifically to the method used to enter calculations into hardware or software calculators, which often have additionalside effectsand implications depending on the actual implementation involving astack.The description "Polish" refers to thenationalityoflogicianJan Łukasiewicz,[1][2]who invented Polish notation in 1924.[3][4][5][6]

The first computer to use postfix notation, though it long remained essentially unknown outside of Germany, wasKonrad Zuse'sZ3in 1941[7][8]as well as hisZ4in 1945. The reverse Polish scheme was again proposed in 1954 byArthur Burks,Don Warren, and Jesse Wright[9]and was independently reinvented byFriedrich L. BauerandEdsger W. Dijkstrain the early 1960s to reducecomputer memoryaccess and use thestackto evaluateexpressions.Thealgorithmsand notation for this scheme were extended by the philosopher and computer scientistCharles L. Hamblinin the mid-1950s.[10][11][12][13][14][15][excessive citations]

During the 1970s and 1980s,Hewlett-Packardused RPN in all of their desktop and hand-held calculators, and has continued to use it in some models into the 2020s.[16][17]Incomputer science,reverse Polish notation is used instack-oriented programming languagessuch asForth,dc,Factor,STOIC,PostScript,RPL,andJoy.

Explanation

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In reverse Polish notation, theoperatorsfollow theiroperands.For example, to add 3 and 4 together, the expression is3 4 +rather than3 + 4.The conventional notation expression3 − 4 + 5becomes3 4 − 5 +in reverse Polish notation: 4 is first subtracted from 3, then 5 is added to it.

The concept of astack,a last-in/first-out construct, is integral to the left-to-right evaluation of RPN. In the example3 4 −,first the 3 is put onto the stack, then the 4; the 4 is now on top and the 3 below it. The subtraction operator removes the top two items from the stack, performs3 − 4,and puts the result of −1 onto the stack.

The common terminology is that added items arepushedon the stack and removed items arepopped.

The advantage of reverse Polish notation is that it removes the need for order of operations and parentheses that are required byinfix notationand can be evaluated linearly, left-to-right. For example, the infix expression(3 + 4) × (5 + 6)becomes3 4 + 5 6 + ×in reverse Polish notation.

Practical implications

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Reverse Polish notation has been compared to how one had to work through problems with aslide rule.[18]

In comparison testing of reverse Polish notation with algebraic notation, reverse Polish has been found to lead to faster calculations, for two reasons. The first reason is that reverse Polish calculators do not need expressions to be parenthesized, so fewer operations need to be entered to perform typical calculations. Additionally, users of reverse Polish calculators made fewer mistakes than for other types of calculators.[19][20]Later research clarified that the increased speed from reverse Polish notation may be attributed to the smaller number of keystrokes needed to enter this notation, rather than to a smaller cognitive load on its users.[21]However, anecdotal evidence suggests that reverse Polish notation is more difficult for users who previously learned algebraic notation.[20]

Converting from infix notation

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Main article:Shunting-yard algorithm

Edsger W. Dijkstrainvented theshunting-yard algorithmto convert infix expressions to postfix expressions (reverse Polish notation), so named because its operation resembles that of arailroad shunting yard.

There are other ways of producing postfix expressions from infix expressions. Mostoperator-precedence parserscan be modified to produce postfix expressions; in particular, once anabstract syntax treehas been constructed, the corresponding postfix expression is given by a simplepost-order traversalof that tree.

Implementations

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Hardware calculators

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Early history

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The first computer implementing a form of reverse Polish notation (but without the name and also without astack), wasKonrad Zuse'sZ3,which he started to construct in 1938 and demonstrated publicly on 12 May 1941.[22][23][24][25]In dialog mode, it allowed operators to enter two operands followed by the desired operation.[z3 1]It was destroyed on 21 December 1943 in a bombing raid.[23]With Zuse's help a first replica was built in 1961.[23]The 1945Z4also added a 2-levelstack.[31][32]

Other early computers to implement architectures enabling reverse Polish notation were theEnglish Electric Company'sKDF9machine, which was announced in 1960 and commercially available in 1963,[33]and theBurroughs B5000,announced in 1961 and also delivered in 1963:

Presumably, the KDF9 designers drew ideas from Hamblin'sGEORGE(General Order Generator),[10][11][13][34][35][32]anautocodeprogramming system written for aDEUCEcomputer installed at theUniversity of Sydney,Australia, in 1957.[10][11][13][33]

One of the designers of the B5000,Robert S. Barton,later wrote that he developed reverse Polish notation independently of Hamblin sometime in 1958 after reading a 1954 textbook on symbolic logic byIrving Copi,[36][37][38]where he found a reference to Polish notation,[38]which made him read the works of Jan Łukasiewicz as well,[38]and before he was aware of Hamblin's work.

Fridenintroduced reverse Polish notation to the desktop calculator market with theEC-130,designed byRobert "Bob" Appleby Ragen,[39]supporting a four-level stack[5]in June 1963.[40]The successorEC-132added a square root function in April 1965.[41]Around 1966, theMonroe Epiccalculator supported an unnamed input scheme resembling RPN as well.[5]

Hewlett-Packard

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Main article:HP calculators
A promotional Hewlett-Packard "No Equals" hat from the 1980s – both a boast and a reference to RPN

Hewlett-Packardengineers designed the9100A Desktop Calculatorin 1968 with reverse Polish notation[16]with only three stack levels with working registersX( "keyboard" ),Y( "accumulate" ) and visible storage registerZ( "temporary" ),[42][43]a reverse Polish notation variant later referred to asthree-level RPN.[44]This calculator popularized reverse Polish notation among the scientific and engineering communities. TheHP-35,the world's first handheld scientificcalculator,[16]introduced the classicalfour-level RPNwith its specific ruleset of the so-calledoperational (memory) stack[45][nb 1](later also calledautomatic memory stack[46][47][nb 1]) in 1972.[48]In this scheme, theEnter↑key duplicates values into Y under certain conditions (automatic stack liftwithtemporary stack lift disable), and the top registerT( "top" ) gets duplicated on drops (top copy on popakatop stack level repetition) in order to ease some calculations and to save keystrokes.[47]HP used reverse Polish notation on every handheld calculator it sold, whether scientific, financial, or programmable, until it introduced theHP-10adding machinecalculator in 1977. By this time, HP was the leading manufacturer of calculators for professionals, including engineers and accountants.

Later calculators with LCDs in the early 1980s, such as theHP-10C,HP-11C,HP-15C,HP-16C,and the financialHP-12Ccalculator also used reverse Polish notation. In 1988, Hewlett-Packard introduced a business calculator, theHP-19B,without reverse Polish notation, but its 1990 successor, theHP-19BII,gave users the option of using algebraic or reverse Polish notation again.

In 1986,[49][50]HP introducedRPL,an object-oriented successor to reverse Polish notation. It deviates from classical reverse Polish notation by using a dynamic stack only limited by the amount of available memory (instead of three or four fixed levels) and which could hold all kinds of data objects (including symbols, strings, lists, matrices, graphics, programs, etc.) instead of just numbers. The system would display an error message when running out of memory instead of just dropping values off the stack on overflow as with fixed-sized stacks.[51]It also changed the behaviour of the stack to no longer duplicate the top register on drops (since in an unlimited stack there is no longer a top register) and the behaviour of theEnter↑key so that it no longer duplicated values into Y, which had shown to sometimes cause confusion among users not familiar with the specific properties of theautomatic memory stack.From 1990 to 2003, HP manufactured theHP-48 seriesof graphing RPL calculators, followed by theHP-49 seriesbetween 1999 and 2008. The last RPL calculator was namedHP 50g,introduced in 2006 and discontinued in 2015. However, there are several community efforts likenewRPLorDB48Xto recreate RPL on modern calculators.

As of 2011, Hewlett-Packard was offering the calculator models 12C, 12C Platinum,17bII+,20b,30b,33s,35s,48gII(RPL) and 50g (RPL) which support reverse Polish notation.[52]

While calculators emulating classical models continued to support classical reverse Polish notation, new reverse Polish notation models feature a variant of reverse Polish notation, where theEnter↑key behaves as in RPL. This latter variant is sometimes known asentry RPN.[53]

In 2013, theHP Primeintroduced a128-levelform of entry RPN calledadvanced RPN.In contrast to RPL with its dynamic stack, it just drops values off the stack on overflow like other fixed-sized stacks do.[51]However, like RPL, it does not emulate the behaviour of a classical operational RPN stack to duplicate the top register on drops.

In late 2017, the list of active models supporting reverse Polish notation included only the 12C, 12C Platinum, 17bii+, 35s, and Prime. By July 2023, only the 12C, 12C Platinum, the freshly releasedHP 15C Collector's Edition,and the Prime remain active models supporting RPN.

See also:HP-related community-developed calculators

Sinclair Radionics

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In Britain,Clive Sinclair'sSinclair Scientific(1974) andScientific Programmable(1975) models used reverse Polish notation.[54][55]

Commodore

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In 1974,Commodoreproduced theMinuteman *6(MM6) without anEnter↑key and theMinuteman *6X(MM6X) with anEnter↑key, both implementing a form oftwo-level RPN.TheSR4921 RPNcame with a variant offour-level RPNwith stack levels named X, Y, Z, and W (rather than T) and anEntkey (for "entry" ). In contrast to Hewlett-Packard's reverse Polish notation implementation, W filled with 0 instead of its contents being duplicated on stack drops.[56]

Prinztronic

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PrinzandPrinztronicwere own-brand trade names of the BritishDixonsphotographic and electronic goods stores retail chain, later rebranded asCurrys Digitalstores, and became part of DSG International. A variety of calculator models was sold in the 1970s under the Prinztronic brand, all made for them by other companies.

Among these was the PROGRAM[57]Programmable Scientific Calculator which featured reverse Polish notation.

Heathkit

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TheAircraft Navigation ComputerHeathkit OC-1401/OCW-1401usedfive-level RPNin 1978.

Soviet Union / Semico

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Sovietprogrammable calculators (MK-52,MK-61,B3-34and earlierB3-21[58]models) used reverse Polish notation for both automatic mode and programming. Modern Russian calculatorsMK-161[59]andMK-152,[60]designed and manufactured inNovosibirsksince 2007 and offered bySemico,[61]are backwards compatible with them. Their extended architecture is also based on reverse Polish notation.

Others

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Community-developed hardware-based calculators

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See also:Hewlett-Packard RPN calculators

An eight-level stack was suggested by John A. Ball in 1978.[5]

The community-developed calculatorsWP 34S(2011),WP 31S(2014) andWP 34C(2015), which are based on theHP 20b/HP 30bhardware platform, support classical Hewlett-Packard-style reverse Polish notation supporting automatic stack lift behaviour of theEnter↑key and top register copies on pops, but switchable between a four- and an eight-level operational stack.

In addition to the optional support for an eight-level stack, the newerSwissMicros DM42-basedWP 43Sas well as theWP 43C(2019) /C43(2022) /C47(2023) derivatives support data types for stack objects (real numbers, infinite integers, finite integers, complex numbers, strings, matrices, dates and times). The latter three variants can also be switched betweenclassicalandentry RPNbehaviour of theEnter↑key, a feature often requested by the community.[66]They also support a rarely seensignificant figuresmode, which had already been available as a compile-time option for the WP 34S and WP 31S.[67][68]

Since 2021, theHP-42SsimulatorFree42version 3 can be enabled to support a dynamic RPN stack only limited by the amount of available memory instead of the classical 4-level stack. This feature was incorporated as a selectable function into the DM42 since firmware DMCP-3.21 / DM42-3.18.[69][70]

Software calculators

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Software calculators:

  • Atari Calculator
  • Mac OS X Calculator
  • Several AppleiPhoneapplications e.g. "reverse polish notation calculator"
  • SeveralAndroidapplications e.g. "RealCalc"
  • SeveralWindows 10 Mobileapplications e.g. "RPN9"
  • Unix systemcalculator programdc
  • Emacslisp library package calc
  • Xorgcalculator (xcalc)
  • ARPCalc,[71]a powerful scientific/engineering RPN calculator for Windows, Linux and Android that also has a web-browser based version
  • grpn[72]scientific/engineering calculator using the GIMP Toolkit (GTK+)
  • F-Correlatives inMultiValuedictionary items
  • RRDtool,a widely used tabulating and graphing software
  • grdmath, a program for algebraic operations onNetCDFgrids, part ofGeneric Mapping Tools(GMT) suite
  • galculator,[73]a GTK desktop calculator
  • Mouseless Stack-Calculator[74]scientific/engineering calculator including complex numbers
  • rpCalc,[75]a simple reverse polish notation calculator written inPythonfor Linux and MS Windows and published under theGNU GPLv2license
  • orpie, RPN calculator for the terminal for real or complex numbers or matrices
  • Qalculate!,a powerful and versatile cross-platform desktop calculator
  • WRPN Calculator

Programming languages

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Existing implementations using reverse Polish notation include:

See also

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Notes

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  1. ^abHewlett-Packard,in the 1970s, called their special RPN stack implementation anoperational (memory) stackorautomatic memory stack.Interestingly,Klaus SamelsonandFriedrich L. Bauer,the inventors of the stack principle, called their stackOperationskeller(Engl. "operational cellar") in 1955, andparallel discovererWilhelm Kämmerer[de]called his stack conceptAutomatisches Gedächtnis(Engl. "automatic memory") in 1958.
  1. ^In reference to "In dialog mode, it allowed operators to enter two operands followed by the desired operation" for the Z3 computer, the following citations are relevant:[7][26][8][27][23][28][25][29][30]

References

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  1. ^Łukasiewicz, Jan(1951). "Chapter IV. Aristotle's System in Symbolic Form (section on" Explanation of the Symbolism ")".Aristotle's Syllogistic from the Standpoint of Modern Formal Logic(1 ed.). p. 78.
  2. ^Łukasiewicz, Jan(1957).Aristotle's Syllogistic from the Standpoint of Modern Formal Logic(2 ed.).Oxford University Press.(Reprinted by Garland Publishing in 1987ISBN0-8240-6924-2.)
  3. ^Łukasiewicz, Jan(February 1929).Elementy logiki matematycznej(in Polish) (1 ed.). Warsaw, Poland:Państwowe Wydawnictwo Naukowe;Łukasiewicz, Jan(1963).Elements of mathematical logic.Translated byWojtasiewicz, Olgierd Adrian[in Polish].New York, USA:The MacMillan Company.p. 24.
  4. ^Hamblin, Charles Leonard(1962-11-01)."Translation to and from Polish notation"(PDF).Computer Journal.5(3): 210–213.doi:10.1093/comjnl/5.3.210.Archived fromthe original(PDF)on 2022-10-20.(4 pages)
  5. ^abcdBall, John A. (1978).Algorithms for RPN calculators(1 ed.). Cambridge, Massachusetts, USA:Wiley-Interscience,John Wiley & Sons, Inc.ISBN0-471-03070-8.LCCN77-14977.p. 2:[…] In their advertisements and also in a letter to me,Hewlett-PackardCompany (HP), the best known manufacturer of RPN calculators, says that RPN is based on a suggestion byJan Łukasiewicz(1878–1956), and that RPN was invented and is patented by HP. Aside from the apparent contradiction in these two statements, I do not think that either of them is quite true. My first experience with RPN involved a nice oldFriden EC-130desktop electronic calculator, circa 1964. The EC-130 has RPN with a push-down stack of four registers, all visible simultaneously on a cathode ray tube display. Furthermore, they are shown upside down, that is, the last-in-first-out register is at the bottom. […] Around 1966, theMonroe Epiccalculator offered RPN with a stack of four, a printer, and either 14 or 42 step programmability. The instruction booklets with these two calculators make no mention of RPN orJan Łukasiewicz.[…]
  6. ^Kennedy, John (August 1982)."RPN Perspective".PPC Calculator Journal.9(5). Mathematics Department, Santa Monica College, Santa Monica, California, USA: 26–29.CiteSeerX10.1.1.90.6448.Archivedfrom the original on 2022-07-01.Retrieved2022-07-02.(12 pages)
  7. ^abCeruzzi, Paul E.(April 1980)."1941 RPN Computer?".PPC Calculator Journal.7(3): 25.Archivedfrom the original on 2022-07-01.Retrieved2022-07-01.p. 25:The interesting aspect of the programming of theZ-3was that this code was very similar to that of, say, anHP-25.To perform an operation on two numbers, commands would first be given to recall the numbers from appropriate locations in the memory, followed by the command for the operation. Numbers were automatically positioned in registers in the Arithmetic Unit of the machine so that operations like division and subtraction would proceed in the right order. Results were left in a register in the AU so that long sequences of operations could be carried out. Thus, the Z-3 used a version of RPN that was nearly identical to that used by HP! I have obtained copies of early programs that Zuse had written for the evaluation of a 5 × 5 determinant, and it is possible to run these programs on anHP-41Cwith almost no modification whatsoever (once the numbers have been placed in the storage registers beforehand). The AU of the Z-3 contained 3 registers, although Zuse never referred to them as a stack, of course. These registers were labelled "f", "a", and "b". All entrance and exit to and from the AU was through the "f" register. This is sort of like the display register of the 41C, which is distinct from the stack. Arithmetic operations were performed on numbers in the a and b registers, so these may be thought of as corresponding to the x and y registers of HP's. Unlike modern computer practice, the actual numbers themselves were moved around the registers, not just a pointer.
  8. ^abRojas, Raúl(April–June 1997)."Konrad Zuse's Legacy: The Architecture of the Z1 and Z3"(PDF).IEEE Annals of the History of Computing.19(2): 5–16 [7–8].doi:10.1109/85.586067.Archived(PDF)from the original on 2022-07-03.Retrieved2022-07-03.(12 pages)
  9. ^Burks, Arthur Walter;Warren, Don W.; Wright, Jesse B. (1954). "An Analysis of a Logical Machine Using Parenthesis-Free Notation".Mathematical Tables and Other Aids to Computation.8(46): 53–57.doi:10.2307/2001990.JSTOR2001990.
  10. ^abcHamblin, Charles Leonard(May 1957).An Addressless Coding Scheme based on Mathematical Notation(Typescript).New South Wales University of Technology.
  11. ^abcHamblin, Charles Leonard(June 1957). "An addressless coding scheme based on mathematical notation".Proceedings of the First Australian Conference on Computing and Data Processing.Salisbury, South Australia:Weapons Research Establishment.
  12. ^Hamblin, Charles Leonard(1957). "Computer Languages".The Australian Journal of Science(20?): 135–139;Hamblin, Charles Leonard(November 1985). "Computer Languages".The Australian Computer Journal(Reprint).17(4): 195–198.
  13. ^abcHamblin, Charles Leonard(1958).GEORGE IA and II: A semi-translation programming scheme for DEUCE: Programming and Operation Manual(PDF).School of Humanities, University of New South Wales, Kensington, New South Wales.Archived(PDF)from the original on 2020-04-04.Retrieved2020-07-27.
  14. ^McBurney, Peter (2008-12-06)."Charles L. Hamblin and his work".Archived fromthe originalon 2008-12-06.
  15. ^McBurney, Peter (2008-07-27)."Charles L. Hamblin: Computer Pioneer".Archived fromthe originalon 2008-12-07.[…]Hamblinsoon became aware of the problems of (a) computing mathematical formulae containing brackets, and (b) the memory overhead in having dealing with memory stores each of which had its own name. One solution to the first problem wasJan Łukasiewicz's Polish notation, which enables a writer of mathematical notation to instruct a reader the order in which to execute the operations (e.g. addition, multiplication, etc) without using brackets. Polish notation achieves this by having an operator (+, ×, etc) precede the operands to which it applies, e.g., +ab, instead of the usual, a+b. Hamblin, with his training in formal logic, knew of Lukasiewicz's work. […]
  16. ^abcOsborne, Thomas E.(2010) [1994]."Tom Osborne's Story in His Own Words".Steve Leibson.Archivedfrom the original on 2022-04-04.Retrieved2016-01-01.[…] I changed the architecture to use RPN (Reverse Polish Notation), which is the ideal notation for programming environment in which coding efficiency is critical. In the beginning, that change was not well received... […]
  17. ^Peterson, Kristina (2011-05-04)."Wall Street's Cult Calculator Turns 30".The Wall Street Journal.Archived fromthe originalon 2015-03-16.Retrieved2015-12-06.
  18. ^Williams, Al (2023-06-21)."In Praise Of RPN (with Python Or C)".Hackaday.Archivedfrom the original on 2023-09-23.Retrieved2023-09-23.
  19. ^Kasprzyk, Dennis Michael; Drury, Colin G.; Bialas, Wayne F. (1979) [1978-09-25]. "Human behaviour and performance in calculator use with Algebraic and Reverse Polish Notation".Ergonomics.22(9). Department of Industrial Engineering,State University of New York at Buffalo,Amherst, New York, USA:Taylor & Francis:1011–1019.doi:10.1080/00140137908924675.eISSN1366-5847.ISSN0014-0139.S2CID62692402.(9 pages)
  20. ^abAgate, Seb J.; Drury, Colin G. (March 1980)."Electronic calculators: which notation is the better?"(PDF).Applied Ergonomics.11(1). Department of Industrial Engineering, University at Buffalo, State University of New York, USA:IPC Business Press:2–6.doi:10.1016/0003-6870(80)90114-3.eISSN1872-9126.ISSN0003-6870.PMID15676368.0003-6870/80/01 0002-05.Archived(PDF)from the original on 2023-09-23.Retrieved2018-09-22.p. 6:In terms of practical choice between calculators, it would appear that RPN is faster and more accurate overall but particularly for more complex problems.(5 pages)
  21. ^Hoffman, Errol; Ma, Patrick; See, Jason; Yong, Chee Kee; Brand, Jason; Poulton, Matthew (1994). "Calculator logic: when and why is RPN superior to algebraic?".Applied Ergonomics.25(5).Elsevier Science Ltd.:327–333.doi:10.1016/0003-6870(94)90048-5.eISSN1872-9126.ISSN0003-6870.
  22. ^"Rechenhilfe für Ingenieure".Alumni-Magazin der Technischen Universität Berlin(in German). Vol. 2, no. 3.Technische Universität Berlin.December 2000. Archived fromthe originalon 2009-02-13.
  23. ^abcdZuse, Horst,ed. (2008-02-22)."Z3 im Detail"[Z3 in details].Professor Dr.-Ing. habil. Horst Zuse(in German).Archivedfrom the original on 2022-07-01.Retrieved2022-07-01.Die Z3 konnte in zwei Betriebsmodi betrieben werden, und zwar in dem Programm- und Dialogmodus. Das Rechnen im Dialog erfolgt wie mit einem Taschenrechner in der umgekehrten polnischen Notation.[1]
  24. ^"An einem 12. Mai"(in German).Deutsches Historisches Museum(German Historical Museum). Archived fromthe originalon 2013-05-30.
  25. ^abBundesmann, Jan (June 2016)."Zum 75. Geburtstag von Konrad Zuses Z3: Ratterkasten".Report / Jubiläum.iX(in German). Vol. 2016, no. 6.Heise Verlag.p. 94.Archivedfrom the original on 2022-07-01.Retrieved2022-07-01.p. 94:Zum Eingeben der Zahlen stand eine Tastatur bereit (Dezimalzahlen, Gleitkommadarstellung). Anweisungen gaben Nutzer in umgekehrter polnischer Notation: zuerst die Argumente, um Register zu befüllen, dann der auszuführende Operator.
  26. ^Ceruzzi, Paul E.(1983)."2. Computers in Germany".Reckoners - The prehistory of the digital computer, from relays to the stored program concept, 1935–1945.Contributions to the study of computer science. Vol. 1 (1 ed.). Westport, Connecticut, USA:Greenwood Press,Congressional Information Service, Inc. p. 0010.ISBN0-313-23382-9.ISSN0734-757X.LCCN82-20980.Archivedfrom the original on 2022-07-01.Retrieved2022-07-02.
  27. ^Zuse, Horst."2. Dialogfähigkeit der Maschine Z3". Written at Berlin, Germany. In Cremers, Armin B.; Manthey, Rainer; Martini, Peter; Steinhage, Volker (eds.).Die ergonomischen Erfindungen der Zuse-Maschinen(PDF).INFORMATIK 2005 Informatik LIVE! Band 1, Beiträge der 35. Jahrestagung der Gesellschaft für Informatik e.V. (GI), 19. bis 22. September 2005 in Bonn. Lecture Notes in Informatics (in German). Bonn, Germany:Gesellschaft für Informatik(GI). pp. 200–204 [200–201].Archived(PDF)from the original on 2022-07-01.Retrieved2022-07-02.p. 201:Dazu stehen die beiden Register R1 und R2 als Kurzspeicher für die Operanden der arithmetischen Operationen zur Verfügung. Gerechnet wird in der umgekehrten polnischen Notation, wie z.B. beim TaschenrechnerHP 45(1972) oderHP11(1998).(5 pages)
  28. ^Bonten, Jo H. M. (2009-05-28) [2009-03-08]."Fast Calculators: Konrad Zuse's Z1 and Z3".Geldrop, Netherlands.Archivedfrom the original on 2022-07-01.Retrieved2022-07-02.The computer can be used as a simple hand-held calculator. In this mode besides entering the numeric values the user must enter the instructions and the addresses by pressing their keys. He has to enter the numbers and operators in the reverse Polish notation.
  29. ^"Die Computerwelt von Konrad Zuse - Auf den Spuren eines EDV-Genies"(PDF).Die Welt der technischen Museen.Welt der Fertigung[de](in German). Vol. 2018, no. 2. 2018. pp. 32–35.ISSN2194-9239.Archived(PDF)from the original on 2019-10-17.Retrieved2022-07-02.pp. 32–33:Er hat wohl auch als erster die vom polnischen MathematikerJan Lukasiewiczentwickelte ›polnische Notation‹ weiterentwickelt und daraus die ›umgekehrte polnische Notation‹ (UPN) ersonnen, da diese in seinen Rechnern verwendet wird: zunächst werden die Werte eingegeben, danach die gewünschte Rechenoperation ausgelöst. Klammern werden auf diese Weise vermieden.(4 pages)
  30. ^Tremmel, Sylvester (2021-11-21)."Computergeschichte: Zuse Z3" im Test "".c't magazin.Heise Verlag.Archivedfrom the original on 2022-03-01.Retrieved2022-07-01.Über die I/O-Einheit kann man dieZ3als reine Rechenmaschine einsetzen, Operationen nimmt sie dann in der praktischen – wenn auch gewöhnungsbedürftigen – umgekehrten polnischen Notation entgegen. Werte im Speicher ablegen (oder von dort laden) kann man so allerdings nicht.
  31. ^Blaauw, Gerrit Anne;Brooks, Jr., Frederick Phillips(1997).Computer architecture: Concepts and evolution.Boston, Massachusetts, USA:Addison-Wesley Longman Publishing Co., Inc.
  32. ^abLaForest, Charles Eric (April 2007). "2.1 Lukasiewicz and the First Generation: 2.1.2 Germany: Konrad Zuse (1910–1995); 2.2 The First Generation of Stack Computers: 2.2.1 Zuse Z4".Second-Generation Stack Computer Architecture(PDF)(thesis). Waterloo, Canada:University of Waterloo.pp. 8, 11.Archived(PDF)from the original on 2022-01-20.Retrieved2022-07-02.(178 pages)
  33. ^abBeard, Bob (Autumn 1997) [1996-10-01]."The KDF9 Computer — 30 Years On"(PDF).Resurrection- The Bulletin of the Computer Conservation Society.No. 18.Computer Conservation Society(CCS). pp. 7–15.ISSN0958-7403.Archived(PDF)from the original on 2020-07-27.Retrieved2020-07-27.p. 8:[…] TheKDF9is remarkable because it is the believed to be the first zero-address instruction format computer to have been announced (in 1960). It was first delivered at about the same time (early 1963) as the other famous zero-address computer, theBurroughs B5000in America. Like many modern pocket calculators, a zero-address machine allows the use of Reverse Polish arithmetic; this offers certain advantages to compiler writers. It is believed that the attention of the English Electric team was first drawn to the zero-address concept through contact withGeorge(General Order Generator), an autocode programming system written for aDeucecomputer by theUniversity of Sydney,Australia, in the latter half of the 1950s. George used Reversed Polish, and the KDF9 team were attracted to this convention for the pragmatic reason of wishing to enhance performance by minimising accesses to main store. This may be contrasted with the more "theoretical" line taken independently byBurroughs.Besides a hardwarenesting storeor stack - the basic mechanism of a zero-address computer - the KDF9 had other groups of central registers for improving performance which gave it an interesting internal structure. […][2](NB. This is an edited version of a talk given to North West Group of the Society at the Museum of Science and Industry, Manchester, UK on 1996-10-01.)
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  45. ^HP35 User's Manual.Hewlett-Packard.p. i. p. i:[…] The operational stack and reverse Polish (Łukasiewicz) notation used in the HP-35 are the most efficient way known to computer science for evaluating mathematical expressions. […]
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Further reading

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