Switchboard
Switchboard is a Turing tarpit with zero instructions and one operation. It is comprised of data in a "board" and a data pointer which moves and operates according the board. It was created in September 2025 by User:CreeperBomb.
Model
The board is an array of cells with some dimension, each cell storing a some valid position on the board. A cell's "neighbor" is the cell to its right - that is, if a cell is located at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P = \left(p_1, p_2, \ldots \right)}
, its neighbor is the cell at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P' = \left(p_1 + 1, p_2, \ldots \right)}
. Unspecified cells are assumed to exist with a value pointing to the origin, and there can only be finitely many nonzero cells. If a coordinate is 0 across all cells, its dimension is effectively nonexistent; as such, all boards are considered to have infinitely many dimensions that are unused. Each step, the data pointer piecewise-adds the current cell's nieghbor's value to the current cell's value, and then goes to the cell whose position matches the current cell's (updated) value; in other words, the language repeatedly executes *P += *(P + (1,0,0,…)); P = *P;. The data pointer begins at the origin.
Note that there is no specification in this description for what each coordinate can be. In fact, a coordinate's value can come from any computable near-semiring with a multiplicative identity (computability here applying to both the operations and the set elements). The origin is the cell whose coordinates are all the additive identity, and a cell's neighbor is found by right-adding the multiplicative identity to the lowest dimensional coordinate. "Infinite dimensions" also does not place a restriction on the dimensions being limited to positive integers; in fact, there can be any cardinality of dimensions as long as a minimal dimension exists. Thus, a board with a dimension for every real number and each coordinate of a dimension being a 2x2 complex matrix is just as valid a board as one with two dimensions and integer coordinates. Another example is a 1-dimensional board with boolean coordinates who operate under Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \oplus} as addition and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \wedge} as multiplication. In fact, within a board, a dimension may have a different near-semiring than another dimension.
The board mathematical specifications are denoted by the notation for the near-semiring and multiplicative identity of each dimension; that is, by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (M, +, \cdot, 0, 1)} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M} is the set, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle +} is the symbol for the additive operator, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdot} is the symbol for the multiplicative operator, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0} is the additive identity, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1} is the multiplicative identity. The notations for the dimensions are then concatenated, with superscripts being used for repeats. For example, the board with a dimension for every real number mentioned above would be notated as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathcal{M}_2(\mathbb{C}),+,\cdot,0_2,I_2)^{\beth_1}} . The superscript should be a cardinal number greater than 1 to be canonical (equal to 1 means it doesn't need to be written and equal to 0 means the dimension doesn't exist in the board).
Turing completeness
Switchboard is Turing complete, as it can iterate on arbitrary functions:
- Assume a 1-dimensional board operating in the near-semiring satisfying:
- Reverse function composition Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \circ} as the additive operation whose identity is the indentity function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{id}_\circ}
- An operation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \ast} as the multiplicative operation whose identity is some function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{id}_\ast} and is defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f \ast g = \text{id}_\circ} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{id}_\ast \ast f = f \ast \text{id}_\ast = f} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f, g \neq \text{id}_\ast}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{id}_\ast} has no additive inverse within the near-semiring
- The origin's neighbor has the value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{id}_\ast} , the origin has some constant function as its value, and all other cells have the identity function as their value (the identity function being the near-semiring's 0)
The choice of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{id}_\ast} determines the function being iterated. The dimensionality of the board is unimportant to the Turing completeness. The set of functions is also arbitrary, but must only contain computable functions and form a near-semiring with a multiplicative identity with the specified operators.
Integer boards
It is unknown if boards of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathbb{Z},+,\cdot,0,1)^n} are Turing complete. However, they do bear resemblance to Three Star Programmer and an I/D machine.