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Designed byKonrad Zuse
First appeared1948; 76 years ago (1948) – concept first published
Major implementations
Plankalkül-Compiler by the FU Berlin in 2000
Influenced by
Superplan, ALGOL 58[2]

Plankalkül (German pronunciation: [ˈplaːnkalkyːl]) is a programming language designed for engineering purposes by Konrad Zuse between 1942 and 1945. It was the first high-level programming language to be designed for a computer.

Kalkül (from Latin calculus) is the German term for a formal system—as in Hilbert-Kalkül, the original name for the Hilbert-style deduction system—so Plankalkül refers to a formal system for planning.[3]

History of programming[edit]

In the domain of creating computing machines, Zuse was self-taught, and developed them without knowledge about other mechanical computing machines that existed already – although later on (building the Z3) being inspired by Hilbert's and Ackermann's book on elementary mathematical logic (see Principles of Mathematical Logic).[4]: 113, 152, 216  To describe logical circuits, Zuse invented his own diagram and notation system, which he called "combinatorics of conditionals" (German: Bedingungskombinatorik). After finishing the Z1 in 1938, Zuse discovered that the calculus he had independently devised already existed and was known as propositional calculus.[5]: 3  What Zuse had in mind, however, needed to be much more powerful (propositional calculus is not Turing-complete and is not able to describe even simple arithmetic calculations[6]). In May 1939, he described his plans for the development of what would become Plankalkül.[4]: 113, 152, 216  He wrote the following in his notebook:

Historical marker on house in Hinterstein [de] where Zuse worked on Plankalkül

While working on his doctoral dissertation, Zuse developed the first known formal system of algorithm notation[7]: 9  capable of handling branches and loops.[8]: 18 [4]: 56  In 1942 he began writing a chess program in Plankalkül.[4]: 216–217  In 1944, Zuse met with the German logician and philosopher Heinrich Scholz, who expressed appreciation for Zuse's utilization of logical calculus.[9] In 1945, Zuse described Plankalkül in an unpublished book.[10] The collapse of Nazi Germany, however, prevented him from submitting his manuscript.[8]: 18 

At that time the only two working computers in the world were ENIAC and Harvard Mark I, neither of which used a compiler, and ENIAC needed to be reprogrammed for each task by changing how the wires were connected.[5]: 3 

Although most of his computers were destroyed by Allied bombs, Zuse was able to rescue one machine, the Z4, and move it to the Alpine village of Hinterstein[7]: 8  (part of Bad Hindelang).

The very first attempt to devise an algorithmic language was undertaken in 1948 by K. Zuse. His notation was quite general, but the proposal never attained the consideration it deserved.

— Heinz Rutishauser, creator of ALGOL

Unable to continue building computers – which was also forbidden by the Allied Powers[11] – Zuse devoted his time to the development of a higher-level programming model and language.[8]: 18  In 1948, he published a paper in the Archiv der Mathematik and presented at the Annual Meeting of the GAMM.[4]: 89  His work failed to attract much attention.[citation needed] In a 1957 lecture, Zuse expressed his hope that Plankalkül, "after some time as a Sleeping Beauty, will yet come to life."[5]: 3  He expressed disappointment that the designers of ALGOL 58 never acknowledged the influence of Plankalkül on their own work.[8]: 18 [7]: 15 

Plankalkül was republished with commentary in 1972.[12] The first compiler for Plankalkül was implemented by Joachim Hohmann in his 1975 dissertation.[13] Other independent implementations followed in 1998[14] and 2000 at the Free University of Berlin.[5]: 2 


Plankalkül has drawn comparisons to the language APL, and to relational algebra. It includes assignment statements, subroutines, conditional statements, iteration, floating-point arithmetic, arrays, hierarchical record structures, assertions, exception handling, and other advanced features such as goal-directed execution. The Plankalkül provides a data structure called generalized graph (verallgemeinerter Graph), which can be used to represent geometrical structures.[15]

Many features of the Plankalkül reappear in later programming languages; an exception is its idiosyncratic two-dimensional notation using multiple lines.

Some features of the Plankalkül:[4]: 217 

  • only local variables
  • functions do not support recursion
  • only supports call by value
  • composite types are arrays and tuples
  • contains conditional expressions
  • contains a for loop and a while loop
  • no goto

Data types[edit]

The only primitive data type in the Plankalkül is a single bit or Boolean (German: Ja-Nein-Werte – yes-no value in Zuse's terminology). It is denoted by the identifier . All the further data types are composite, and build up from primitive by means of "arrays" and "records".[16]: 679 

So, a sequence of eight bits (which in modern computing could be regarded as byte) is denoted by , and Boolean matrix of size by   is described by . There also exists a shortened notation, so one could write instead of .[16]: 679 

Type could have two possible values and . So 4-bit sequence could be written like L00L, but in cases where such a sequence represents a number, the programmer could use the decimal representation 9.[16]: 679 

Record of two components and is written as .[16]: 679 

Type (German: Art) in Plankalkül consists of 3 elements: structured value (German: Struktur), pragmatic meaning (German: Typ) and possible restriction on possible values (German: Beschränkung).[16]: 679  User defined types are identified by letter A with number, like – first user defined type.


Zuse used a lot of examples from chess theory:[16]: 680 

Coordinate of chess board (it has size 8x8 so 3 bits are just enough)
square of the board (for example L00, 00L denotes e2 in algebraic notation)
piece (for example, 00L0 — white king)
piece on a board (for example L00, 00L; 00L0 — white king on e2)
board (pieces positions, describes which piece each of 64 squares contains)
game state ( — board,  — player to move,  — possibility of castling (2 for white and 2 for black),  — information about cell on which en passant move is possible


Identifiers are alphanumeric characters with a number.[16]: 679  There are the following kinds of identifiers for variables:[10]: 10 

  • Input values (German: Eingabewerte, Variablen) — marked with a letter V.
  • Intermediate, temporary values (German: Zwischenwerte) — marked with a letter Z.
  • Constants (German: Constanten) — marked with a letter С.
  • Output values (German: Resultatwerte) — marked with a letter R.

Particular variable of some kind is identified by number, written under the kind.[16]: 679  For example:

, , etc.

Programs and subprograms are marked with a letter P, followed by a program (and optionally a subprogram) number. For example , .[16]: 679 

Output value of program saved there in variable is available for other subprograms under the identifier , and reading value of that variable also means executing related subprogram.[16]: 680 

Accessing elements by index[edit]

Plankalkül allows access for separate elements of variable by using "component index" (German: Komponenten-Index). When, for example, program receives input in variable of type (game state), then  — gives board state,  — piece on square number i, and bit number j of that piece.[16]: 680 

In modern programming languages, that would be described by notation similar to V0[0], V0[0][i], V0[0][i][j] (although to access a single bit in modern programming languages a bitmask is typically used).

Two-dimensional syntax[edit]

Because indexes of variables are written vertically, each Plankalkül instruction requires multiple rows to write down.[citation needed]

First row contains variable kind, then variable number marked with letter V (German: Variablen-Index), then indexes of variable subcomponents marked with K (German: Komponenten-Index), and then (German: Struktur-Index) marked with S, which describes variable type. Type is not required, but Zuse notes that this helps with reading and understanding the program.[16]: 681 

In the line types and could be shortened to and .[16]: 681 


variable V3 — list of pairs of values of type
Row K could be skipped when it is empty. Therefore, this expression means the same as above.
Value of eights bit (index 7), of first (index 0) pair, of і-th element of variable V3, has Boolean type ().

Indexes could be not only constants. Variables could be used as indexes for other variables, and that is marked with a line, which shows in which component index would value of variable be used:

Using variable as index for other variable, in 2d Plankalül notation Z5-th element of variable V3. Equivalent to expression V3[Z5] in many modern programming languages.[16]: 681 

Assignment operation[edit]

Zuse introduced in his calculus an assignment operator, unknown in mathematics before him. He marked it with «», and called it yields-sign (German: Ergibt-Zeichen). Use of concept of assignment is one of the key differences between mathematics and computer science.[7]: 14 

Zuse wrote that the expression:

is analogous to the more traditional mathematical equation:

There are claims that Konrad Zuse initially used the glyph as a sign for assignment, and started to use under the influence of Heinz Rutishauser.[16]: 681  Knuth and Pardo believe that Zuse always wrote , and that was introduced by publishers of «Über den allgemeinen Plankalkül als Mittel zur Formulierung schematisch-kombinativer Aufgaben» in 1948.[7]: 14  In the ALGOL 58 conference in Zürich, European participants proposed to use the assignment character introduced by Zuse, but the American delegation insisted on :=.[16]: 681 

The variable that stores the result of an assignment (l-value) is written to the right side of assignment operator.[7]: 14  The first assignment to the variable is considered to be a declaration.[16]: 681 

The left side of assignment operator is used for an expression (German: Ausdruck), that defines which value will be assigned to the variable. Expressions could use arithmetic operators, Boolean operators, and comparison operators ( etc.).[16]: 682 

The exponentiation operation is written similarly to the indexing operation - using lines in the two-dimensional notation:[10]: 45 

Exponentiation notation in Plankalkül

Control flow[edit]

Boolean values were represented as integers with FALSE=0 and TRUE=1. Conditional control flow took the form of a guarded statement A -> B, which executed the block B if A was true. There was also an iteration operator, of the form W { A -> X; B -> Y} which repeats until all guards are false.[17]


Zuse called a single program a Rechenplan ("computation plan"). He envisioned what he called a Planfertigungsgerät ("plan assembly device"), which would automatically translate the mathematical formulation of a program into machine-readable punched film stock, something today would be called a translator or compiler.[4]: 45, 104, 105 


The original notation was two-dimensional.[clarification needed] For a later implementation in the 1990s, a linear notation was developed.

The following example defines a function max3 (in a linear transcription) that calculates the maximum of three variables:

P1 max3 (V0[:8.0],V1[:8.0],V2[:8.0]) → R0[:8.0]
max(V0[:8.0],V1[:8.0]) → Z1[:8.0]
max(Z1[:8.0],V2[:8.0]) → R0[:8.0]
P2 max (V0[:8.0],V1[:8.0]) → R0[:8.0]
V0[:8.0] → Z1[:8.0]
(Z1[:8.0] < V1[:8.0]) → V1[:8.0] → Z1[:8.0]
Z1[:8.0] → R0[:8.0]

See also[edit]


  1. ^ "Early Programming Languages / CS208e: Great Ideas in Computer Science" (PDF). Stanford University. Archived (PDF) from the original on 2023-11-02. Retrieved 2023-11-02. (46 pages)
  2. ^ Rojas, Raúl; Hashagen, Ulf [in German] (2002). The First Computers: History and Architectures. MIT Press. p. 292. ISBN 978-0-26268137-7. Retrieved 2013-10-25.
  3. ^ Zenil, Héctor [at Wikidata], ed. (2012). A Computable Universe: Understanding and Exploring Nature As Computation - with a Foreword by Sir Roger Penrose. Singapore: World Scientific Publishing Company. p. 791.
  4. ^ a b c d e f g Hellige, Hans Dieter, ed. (January 2004) [November 2002]. Written at Bremen, Germany. Geschichten der Informatik. Visionen, Paradigmen, Leitmotive (in German) (1 ed.). Berlin / Heidelberg, Germany: Springer-Verlag. pp. 45, 56, 89, 104–105, 113, 152, 216–217. doi:10.1007/978-3-642-18631-8. ISBN 978-3-540-00217-8. ISBN 3-540-00217-0. (xii+514 pages)
  5. ^ a b c d e Rojas, Raúl; Göktekin, Cüneyt; Friedland, Gerald; Krüger, Mike; Scharf, Ludmila; Kuniß, Denis; Langmack, Olaf (January 2004) [November 2002]. "Konrad Zuses Plankalkül — Seine Genese und eine moderne Implementierung" (PDF). In Hellige, Hans Dieter (ed.). Geschichten der Informatik. Visionen, Paradigmen, Leitmotive. Teil 3: Leitideen und Paradigmen in der Entwicklung der Programmiersprachen und der Programmierung (in German) (1 ed.). Berlin / Heidelberg, Germany: Springer-Verlag. pp. 215–235 [2–4]. doi:10.1007/978-3-642-18631-8_9. ISBN 978-3-642-62208-3. Archived from the original (PDF) on 2006-05-01. (21 [24] pages)
  6. ^ "Why is propositional logic not Turing complete?". Mathematics. StackExchange. 2013-04-01. Archived from the original on 2023-11-02. Retrieved 2023-11-02.
  7. ^ a b c d e f Knuth, Donald Ervin; Pardo, Luis Isidoro Trabb [in Portuguese] (August 1976). "The Early Development of Programming Languages" (PDF). Stanford University, Computer Science Department. pp. 8, 9, 14, 15. Archived from the original (PDF) on 2017-09-12. Retrieved 2017-12-28.
  8. ^ a b c d Giloi, Wolfgang K. [in German] (April–June 1997). "Konrad Zuse's Plankalkül: The First High-Level "non von Neumann" Programming Language". IEEE Annals of the History of Computing. 19 (2). IEEE: 17–24. doi:10.1109/85.586068. (8 pages)
  9. ^ Petzold, Hartmut [in German] (1992). Moderne Rechenkünstler. Die Industrialisierung der Rechentechnik in Deutschland (in German). München, Germany: C.H. Beck Verlag.
  10. ^ a b c Zuse, Konrad (1946) [1945]. Rojas, Raúl; Wagner, G.; Scharf, Ludmila; Schöttker-Söhl [Schötke-Suhl], Susanne (eds.). Der Plankalkül (In der Fassung von 1945) (Manuscript) (in German). Konrad Zuse Internet Archive. pp. 10, 45. ZIA ID: 0233. Archived from the original on 2015-04-16. Retrieved 2023-11-01. (1+1+180 pages)
  11. ^ Coy, Wolfgang [in German] (January 2004) [November 2002]. "Was ist Informatik? Zur Entstehung des Faches an den deutschen Universitäten". In Hellige, Hans Dieter (ed.). Geschichten der Informatik. Visionen, Paradigmen, Leitmotive. Teil 5: Wandel der Leitkonzepte in der Wissenschaftsdisziplin Informatik (in German) (1 ed.). Berlin / Heidelberg, Germany: Springer-Verlag. pp. 473–498 [474]. doi:10.1007/978-3-642-18631-8_17. ISBN 978-3-540-00217-8. ISBN 3-540-00217-0. [1]
  12. ^ Zuse, Konrad (1972). Der Plankalkül. Kommentierter Nachdruck der Fassung von 1945 (in German). Vol. 63. Sankt Augustin, Germany: Gesellschaft für Mathematik und Datenverarbeitung (GMD) / Bundesministerium für Bildung und Wissenschaft (BMBW). BMBW-GMD-63.
  13. ^ Hohmann, Joachim (1979). Der PLANKALKÜL im Vergleich mit algorithmischen Sprachen. Reihe Informatik und Operations Research (in German). Vol. 7 (1 ed.). Darmstadt, Germany: S. Toeche-Mittler-Verlag (stmv). ISBN 3-87820-028-5. (136 pages) Contents
  14. ^ Description of Plankalkül-Compiler by Wolfgang Mauerer; Mauerer, Wolfgang (2016-06-03). "Der Plankalkül von Konrad Zuse" (in German). Implementation in German. Archived from the original on 2016-06-03. Retrieved 2017-10-03.
  15. ^ Giloi, Wolfgang K. [in German] (November 1990), Konrad Zuses Plankalkül als Vorläufer moderner Programmiermodelle (in German)
  16. ^ a b c d e f g h i j k l m n o p q r Bauer, Friedrich L.; Wössner, Hans (1972). "The "Plankalkül" of Konrad Zuse: A Forerunner of Today's Programming Languages" (PDF). Communications of the ACM. 15 (7): 678–685. doi:10.1145/361454.361515. S2CID 17681101. Archived from the original (PDF) on 2009-02-20. (HTML)
  17. ^ Rojas, Raúl (2001). "Plankalkül" (PDF). Encyclopedia of computers and computer history. Chicago / London: Fitzroy Dearborn Publishers. p. 634. ISBN 1-57958235-4. Retrieved 2023-05-26.

Further reading[edit]

  • Zuse, Konrad (1943). Ansätze einer Theorie des allgemeinen Rechnens unter besonderer Berücksichtigung des Aussagenkalküls und dessen Anwendung auf Relaisschaltungen [Inception of a universal theory of computation with special consideration of the propositional calculus and its application to relay circuits] (unpublished manuscript) (in German). Zuse Papers 045/018.
  • Zuse, Konrad (1948-12-06) [November 1948]. Written at Hopferau bei Füssen, Germany. "Über den allgemeinen Plankalkül als Mittel zur Formulierung schematisch-kombinativer Aufgaben". Archiv der Mathematik (in German). 1 (6). Karlsruhe / Stuttgart / Basel, Germany: Birkhäuser Verlag: 441–449. doi:10.1007/BF02038459. eISSN 1420-8938. ISSN 0003-889X. (9 pages)
  • Rojas, Raúl; Göktekin, Cüneyt; Friedland, Gerald; Krüger, Mike; Kuniß, Denis; Langmack, Olaf (February 2000). Plankalkül: The First High-Level Programming Language and its Implementation (PDF). Berlin, Germany: Institut für Informatik, Freie Universität Berlin & Technical Report B-3/2000. Archived from the original on 2006-05-01.[2] (22 pages)
  • Bruines, Bram (2010-01-08). "Plankalkül" (PDF) (Thesis). Archived (PDF) from the original on 2023-11-02. Retrieved 2023-11-02. (24 pages)

External links[edit]

  • "Plankalkül-Programme". Konrad Zuse Internet Archive (in German and English). 2014-08-21. Archived from the original on 2014-08-21. Retrieved 2017-10-04. (NB. Plankalkül Java applets and documents.)