Data types

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 ${\displaystyle S0}$. 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 ${\displaystyle 8\times S0}$, and Boolean matrix of size ${\displaystyle m}$ by ${\displaystyle n}$  is described by ${\displaystyle m\times n\times S0}$. There also exists a shortened notation, so one could write ${\displaystyle S1\cdot n}$ instead of ${\displaystyle n\times S0}$.^[16]: 679

Type ${\displaystyle S0}$ could have two possible values ${\displaystyle 0}$ and ${\displaystyle L}$. 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 ${\displaystyle \sigma }$ and ${\displaystyle \tau }$ is written as ${\displaystyle (\sigma ,\tau )}$.^[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 ${\displaystyle A1}$ – first user defined type.

Examples

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

${\displaystyle A1}$	${\displaystyle S1\cdot 3}$	Coordinate of chess board (it has size 8x8 so 3 bits are just enough)
${\displaystyle A2}$	${\displaystyle 2\times A1}$	square of the board (for example L00, 00L denotes e2 in algebraic notation)
${\displaystyle A3}$	${\displaystyle S1\cdot 4}$	piece (for example, 00L0 — white king)
${\displaystyle A4}$	${\displaystyle (A2,A3)}$	piece on a board (for example L00, 00L; 00L0 — white king on e2)
${\displaystyle A5}$	${\displaystyle 64\times A3}$	board (pieces positions, describes which piece each of 64 squares contains)
${\displaystyle A10}$	${\displaystyle (A5,S0,S1\cdot 4,A2)}$	game state (${\displaystyle A5}$ — board, ${\displaystyle S0}$ — player to move, ${\displaystyle S1\cdot 4}$ — possibility of castling (2 for white and 2 for black), ${\displaystyle A2}$ — information about cell on which en passant move is possible

Identifiers

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:

${\displaystyle {\begin{matrix}V\\0\end{matrix}}}$, ${\displaystyle {\begin{matrix}Z\\2\end{matrix}}}$, ${\displaystyle {\begin{matrix}C\\31\end{matrix}}}$ etc.

Programs and subprograms are marked with a letter P, followed by a program (and optionally a subprogram) number. For example ${\displaystyle P13}$, ${\displaystyle P5\cdot 7}$.^[16]: 679

Output value of program ${\displaystyle P13}$ saved there in variable ${\displaystyle {\begin{matrix}R\\0\end{matrix}}}$ is available for other subprograms under the identifier ${\displaystyle {\begin{matrix}R17\\0\end{matrix}}}$, and reading value of that variable also means executing related subprogram.^[16]: 680

Accessing elements by index

Plankalkül allows access for separate elements of variable by using "component index" (German: Komponenten-Index). When, for example, program receives input in variable ${\displaystyle {\begin{matrix}V\\0\end{matrix}}}$ of type ${\displaystyle A10}$ (game state), then ${\displaystyle {\begin{matrix}V\\0\\0\end{matrix}}}$ — gives board state, ${\displaystyle {\begin{matrix}V\\0\\0\cdot i\end{matrix}}}$ — piece on square number i, and ${\displaystyle {\begin{matrix}V\\0\\0\cdot i\cdot j\end{matrix}}}$ 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

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 ${\displaystyle S}$ types ${\displaystyle S0}$ and ${\displaystyle S1}$ could be shortened to ${\displaystyle 0}$ and ${\displaystyle 1}$.^[16]: 681

Examples:

${\displaystyle {\begin{array}{r|l}&V\\V&3\\K&\\S&m\times 2\times 1\cdot n\end{array}}}$	variable V3 — list of ${\displaystyle m}$ pairs of values of type ${\displaystyle S1\cdot n}$
${\displaystyle {\begin{array}{r|l}&V\\V&3\\S&m\times 2\times 1\cdot n\end{array}}}$	Row K could be skipped when it is empty. Therefore, this expression means the same as above.
${\displaystyle {\begin{array}{r|l}&V\\V&3\\K&i\cdot 0\cdot 7\\S&0\end{array}}}$	Value of eights bit (index 7), of first (index 0) pair, of і-th element of variable V3, has Boolean type (${\displaystyle S0}$).

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:

Assignment operation

Zuse introduced in his calculus an assignment operator, unknown in mathematics before him. He marked it with «${\displaystyle \Rightarrow }$», 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:

${\displaystyle {\begin{array}{r|lll}&Z+1&\Rightarrow &Z\\V&1&&1\\\end{array}}}$

is analogous to the more traditional mathematical equation:

${\displaystyle {\begin{array}{r|lll}&Z+1&=&Z\\V&1&&1\\K&i&&i+1\\\end{array}}}$

There are claims that Konrad Zuse initially used the glyph as a sign for assignment, and started to use ${\displaystyle \Rightarrow }$ under the influence of Heinz Rutishauser.^[16]: 681 Knuth and Pardo believe that Zuse always wrote ${\displaystyle \Rightarrow }$, 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 (${\displaystyle =,\neq ,\leq }$ etc.).^[16]: 682

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

Control flow

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]

Terminology

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}

Example

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]
END
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]
END