Plankalkül

The only primitive data type in the Plankalkül is a single bit or boolean (German: Ja-Nein-Werte - yes-no value in Zuses 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".[23]

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 ${\displaystyle n\times S0}$.[23]

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.[23]

Record of two components ${\displaystyle \sigma }$ and ${\displaystyle \tau }$ is written as ${\displaystyle (\sigma ,\tau )}$.[23]

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).[23] User defined types are identified by letter A with number, like ${\displaystyle A1}$ - first user defined type.

Zuse used a lot of examples from chess theory:[24]

${\displaystyle A1}$	${\displaystyle S1\cdot 3}$	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, ${\displaystyle S0}$ — who moves, ${\displaystyle S1\cdot 4}$ — possibility of castling (2 for white and 2 for black), A2 — information about cell on which En passant move is possible

Identifiers are alphanumeric characters with a number.[23] There are following kinds of identifiers for variables:[25]

• Input values (German: Eingabewerte, Variablen) — marked with a letter V.
• Indermediate, 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.[23] 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 ${\displaystyle P13}$, ${\displaystyle P5\cdot 7}$.[23]

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 readin value of that variable also means executing related subprogram.[24]

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.[24]

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).

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

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.[26]

In the line ${\displaystyle S}$ types ${\displaystyle S0}$ and ${\displaystyle S1}$ could be shortened to ${\displaystyle 0}$ and ${\displaystyle 1}$. [26]

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:

Z5-th element of variable V3. Equivalent to expression V3[Z5] in many modern programming languages.[26]

Zuse introduced in his calculus operator, unknown for mathematics before him - assignment. 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 math and computer science.[27]

Zuse wrote that expression:

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

is analogous to more traditional mathematical equation:

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

Exists opinion, that Konrad Zuse initially used for assignment sign , and started to use ${\displaystyle \Rightarrow }$ under the influence of Heinz Rutishauser.[26] Knuth and Pardo believe that Zuse always wrote ${\displaystyle \Rightarrow }$, and was introduced by publishers of «Über den allgemeinen Plankalkül als Mittel zur Formulierung schematisch-kombinativer Aufgaben».[27] On the ALGOL 58 conference in Zurich, European participants proposed to use for assignment character introduced by Zuse, but the American delegation insisted on :=.[26]

The variable that stores the result of an assignment (l-value) is written to the right side of assignment operator.[27] First assignment to the variable is considered to be a declaration.[26]

The left side of assignment operator is used for expression (German: Ausdruck), that defines which value will be assigned to variable. Expressions could use arithmetic operators, boolean operators, and comparison operators (${\displaystyle =,\neq ,\leq }$ etc.).[28]

Exponentiation operation is written similarly to the indexing operation - using lines in 2d notation:[29]

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.[30]

The original notation was two dimensional. 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