Reverse Polish notation
Reverse Polish notation (RPN), also known as Polish postfix notation or simply postfix notation, is a mathematical notation in which operators follow their operands, in contrast to Polish notation (PN), in which operators precede 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 reverse Polish scheme was proposed in 1954 by Arthur Burks, Don Warren, and Jesse Wright and was independently reinvented by Friedrich L. Bauer and Edsger W. Dijkstra in the early 1960s to reduce computer memory access and utilize the stack to evaluate expressions. The algorithms and notation for this scheme were extended by Australian philosopher and computer scientist Charles L. Hamblin in the mid-1950s.
During the 1970s and 1980s, Hewlett-Packard used RPN in all of their desktop and hand-held calculators, and continued to use it in some models into the 2010s. In computer science, reverse Polish notation is used in stack-oriented programming languages such as Forth and PostScript.
In reverse Polish notation, the operators follow their operands; for instance, to add 3 and 4, one would write 3 4 + rather than 3 + 4. If there are multiple operations, operators are given immediately after their second operands; so the expression written 3 − 4 + 5 in conventional notation would be written 3 4 − 5 + in reverse Polish notation: 4 is first subtracted from 3, then 5 is added to it. An advantage of reverse Polish notation is that it removes the need for parentheses that are required by infix notation. While 3 − 4 × 5 can also be written 3 − (4 × 5), that means something quite different from (3 − 4) × 5. In reverse Polish notation, the former could be written 3 4 5 × −, which unambiguously means 3 (4 5 ×) − which reduces to 3 20 −; the latter could be written 3 4 − 5 × (or 5 3 4 − ×, if keeping similar formatting), which unambiguously means (3 4 −) 5 ×.
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, fewer operations need to be entered to perform typical calculations. Additionally, users of reverse Polish calculators made fewer mistakes than for other types of calculator. 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. However, anecdotal evidence suggests that reverse Polish notation is more difficult for users to learn than algebraic notation.
Postfix evaluation algorithmEdit
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The following algorithm evaluates postfix expressions using a stack, with the expression processed from left to right:
for each token in the postfix expression: if token is an operator: operand_2 ← pop from the stack operand_1 ← pop from the stack result ← evaluate token with operand_1 and operand_2 push result back onto the stack else if token is an operand: push token onto the stack result ← pop from the stack
The following algorithm produces the same results of the previous one, but the expression is processed from right to left:
for each token in the reversed postfix expression: if token is an operator: push token onto the operator stack pending_operand ← False else if token is an operand: operand ← token if pending_operand is True: while the operand stack is not empty: operand_1 ← pop from the operand stack operator ← pop from the operator stack operand ← evaluate operator with operand_1 and operand push operand onto the operand stack pending_operand ← True result ← pop from the operand stack
A significant difference between the two algorithms is that the first one holds only numbers on a stack, and so it is possible to write a long calculation that uses minimal stack space by putting operators as early as possible in the expression. For example, 1 2 + 3 + 4 + 5 + will accumulate the subtotal on the stack and the stack never has more than two items.
The infix expression ((15 ÷ (7 − (1 + 1))) × 3) − (2 + (1 + 1)) can be written like this in reverse Polish notation:
- 15 7 1 1 + − ÷ 3 × 2 1 1 + + −
- Evaluating this postfix expression with the above left-to-right algorithm yields (red items are the stack contents, bold is the current token):
15 7 1 1 + − ÷ 3 × 2 1 1 + + − = 15 7 1 1 + − ÷ 3 × 2 1 1 + + − = 15 7 1 1 + − ÷ 3 × 2 1 1 + + − = 15 7 1 1 + − ÷ 3 × 2 1 1 + + − = 15 7 1 1 + − ÷ 3 × 2 1 1 + + − = 15 7 2 − ÷ 3 × 2 1 1 + + − = 15 5 ÷ 3 × 2 1 1 + + − = 3 3 × 2 1 1 + + − = 3 3 × 2 1 1 + + − = 9 2 1 1 + + − = 9 2 1 1 + + − = 9 2 1 1 + + − = 9 2 1 1 + + − = 9 2 2 + − = 9 4 − = 5 = 5
- Evaluating this postfix expression with the above right-to-left algorithm yields:
15 7 1 1 + − ÷ 3 × 2 1 1 + + − = 15 7 1 1 + − ÷ 3 × 2 2 + − = 15 7 1 1 + − ÷ 3 × 4 − = 15 7 2 − ÷ 3 × 4 − = 15 5 ÷ 3 × 4 − = 3 3 × 4 − = 9 4 − = 5
The following table shows the state of the operand stack at each stage of the above left-to-right algorithm:
|15||Operand||15||Push onto stack.|
|7||Operand||15 7||Push onto stack.|
|1||Operand||15 7 1||Push onto stack.|
|1||Operand||15 7 1 1||Push onto stack.|
|+||Operator||15 7 2||Pop from stack twice (1, 1), calculate (1 + 1 = 2) and push onto stack.|
|−||Operator||15 5||Pop from stack twice (7, 2), calculate (7 − 2 = 5) and push onto stack.|
|÷||Operator||3||Pop from stack twice (15, 5), calculate (15 ÷ 5 = 3) and push onto stack.|
|3||Operand||3 3||Push onto stack.|
|×||Operator||9||Pop from stack twice (3, 3), calculate (3 × 3 = 9) and push onto stack.|
|2||Operand||9 2||Push onto stack.|
|1||Operand||9 2 1||Push onto stack.|
|1||Operand||9 2 1 1||Push onto stack.|
|+||Operator||9 2 2||Pop from stack twice (1, 1), calculate (1 + 1 = 2) and push onto stack.|
|+||Operator||9 4||Pop from stack twice (2, 2), calculate (2 + 2 = 4) and push onto stack.|
|−||Operator||5||Pop from stack twice (9, 4), calculate (9 − 4 = 5) and push onto stack.|
As was demonstrated in the Algebraic mode, it is usually easier (fewer keystrokes) in working a problem like this to begin with the arithmetic operations inside the parentheses first.
- 1 2 + 4 × 5 + 3 −
Converting from infix notationEdit
Edsger Dijkstra invented the shunting-yard algorithm to convert infix expressions to postfix expressions (reverse Polish notation), so named because its operation resembles that of a railroad shunting yard.
There are other ways of producing postfix expressions from infix expressions. Most operator-precedence parsers can be modified to produce postfix expressions; in particular, once an abstract syntax tree has been constructed, the corresponding postfix expression is given by a simple post-order traversal of that tree.
The first computers to implement architectures enabling reverse Polish notation were the English Electric Company's KDF9 machine, which was announced in 1960 and delivered (i.e. made available commercially) in 1963, and the American Burroughs B5000, announced in 1961 and also delivered in 1963. 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 by Irving Copi, where he found a reference to Polish notation, which made him read the works of Jan Łukasiewicz as well, and before he was aware of Hamblin's work. Designed by Robert "Bob" Appleby Ragen, Friden introduced reverse Polish notation to the desktop calculator market with the EC-130 supporting a four-level stack in June 1963. The successor EC-132 added a square root function in April 1965. Around 1966, the Monroe Epic calculator supported an unnamed input scheme resembling RPN as well.
Hewlett-Packard engineers designed the 9100A Desktop Calculator in 1968 with reverse Polish notation with only three stack levels, a reverse Polish notation variant later referred to as three-level RPN. This calculator popularized reverse Polish notation among the scientific and engineering communities. The HP-35, the world's first handheld scientific calculator, introduced the classical four-level RPN in 1972. HP used reverse Polish notation on every handheld calculator it sold, whether scientific, financial, or programmable, until it introduced the HP-10 adding machine calculator in 1977. By this time, HP was the leading manufacturer of calculators for professionals, including engineers and accountants.
Later LCD-based calculators in the early 1980s such as the HP-10C, HP-11C, HP-15C, HP-16C, and the financial calculator, the HP-12C also used reverse Polish notation. In 1988, Hewlett-Packard introduced a business calculator, the HP-19B, without reverse Polish notation, but its 1990 successor, the HP-19BII, gave users the option of using algebraic notation or reverse Polish notation.
Around 1987, HP introduced RPL, an object-oriented successor to reverse Polish notation. It deviates from classical reverse Polish notation by utilizing a stack only limited by the amount of available memory (instead of three or four fixed levels) and which can hold all kinds of data objects (including symbols, strings, lists, matrices, graphics, programs, etc.) instead of just numbers. 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 the ↵ Enter key so that it no longer duplicates values into Y under certain conditions, both part of the specific ruleset of the so-called automatic memory stack or operational (memory) stack in classical reverse Polish notation in order to ease some calculations and to save keystrokes, but which had shown to also sometimes cause confusion among users not familiar with these properties. From 1990 to 2003 HP manufactured the HP-48 series of graphing RPL calculators and in 2006 introduced the HP 50g.
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. While calculators emulating classical models continue to support classical reverse Polish notation, new reverse Polish notation models feature a variant of reverse Polish notation, where the ↵ Enter key behaves as in RPL. This latter variant is sometimes known as entry RPN. In 2013, the HP Prime introduced a 128-level form of entry RPN called advanced RPN. By late 2017, only the 12C, 12C Platinum, 17bii+, 35s and Prime remain active HP models supporting reverse Polish notation.
WP 31S and WP 34SEdit
The community-developed calculators WP 31S and WP 34S, which are based on the HP 20b/HP 30b hardware platform, support Hewlett-Packard-style classical reverse Polish notation with either a four- or an eight-level stack. A seven-level stack had been implemented in the MITS 7400C scientific desktop calculator in 1972 and an eight-level stack was already suggested by John A. Ball in 1978.
In 1974 Commodore produced the Minuteman *6 (MM6) without ↵ Enter key and the Minuteman *6X (MM6X) with ↵ Enter key, both implementing a form of two-level RPN. The SR4921 RPN came with a variant of four-level RPN with stack levels named X, Y, Z, and W (rather than T). In contrast to Hewlett-Packard's reverse Polish notation implementation, W filled with 0 instead of its contents being duplicated on stack drops.
Prinz and Prinztronic were own-brand trade names of the British Dixons photographic and electronic goods stores retail chain, which was later rebranded as Currys Digital stores, 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 Programmable Scientific Calculator which featured reverse Polish notation.
Soviet programmable calculators (MK-52, MK-61, B3-34 and earlier B3-21 models) used reverse Polish notation for both automatic mode and programming. Modern Russian calculators MK-161 and MK-152, designed and manufactured in Novosibirsk since 2007 and offered by Semico, are backward compatible with them. Their extended architecture is also based on reverse Polish notation.
Existing implementations using reverse Polish notation include:
- Any stack-oriented programming language, such as:
- PostScript page description language
- Lotus 1-2-3 and Lotus Symphony formulas
- RPL (aka Reverse Polish Language), a programming language for the Commodore PET around 1979/1981
- RPL (aka Reverse Polish Lisp), a programming language for Hewlett-Packard calculators between 1984 and 2015
- RPNL (Reverse Polish Notation Language)
- Hardware calculators:
- Software calculators:
- Mac OS X Calculator
- Several Apple iPhone applications e.g. "reverse polish notation calculator"
- Several Android applications e.g. "RealCalc"
- Several Windows 10 Mobile applications e.g. "RPN9"
- Unix system calculator program dc
- Emacs lisp library package calc
- Xorg calculator (xcalc)
- grpn scientific/engineering calculator using the GIMP Toolkit (GTK+)
- F-Correlatives in MultiValue dictionary items
- RRDtool, a widely used tabulating and graphing software
- grdmath, a program for algebraic operations on NetCDF grids, part of Generic Mapping Tools (GMT) suite
- galculator, a GTK desktop calculator
- Mouseless Stack-Calculator scientific/engineering calculator including complex numbers.
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[…] In their advertisements and also in a letter to me, Hewlett-Packard Company (HP), the best known manufacturer of RPN calculators, says that RPN is based on a suggestion by Jan 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 old Friden EC-130 desktop 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, the Monroe Epic calculator 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 or Jan Łukasiewicz. […]
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[…] Hamblin soon 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 was Jan 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. […]
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- HP35 User's Manual. Hewlett-Packard. 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. […]
- HP Calculators
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- Rosettacode.org providing many implementations in several programming languages.
- http://rpn.codeplex.com/ Implementation of RPN with custom functions support and flexible list of operators.
- https://xrjunque.nom.es/ConvertAlg2RPN_RPL.aspx Free online Algebraic expression to RPN Converter