1.19  GCalc – Visual λ-Calculus

To play GCalc, run the PLT Games program. (Under Unix, it’s called plt-games).

GCalc is a system for visually demonstrating the λ-Calculus (not really a game).

See the following for the principles:

1.19.1 The Window Layout

The window is divided into three working areas, each made of cells. Cells hold cube objects, which can be dragged between cells (with a few exceptions that are listed below). The working areas are as follows:

• The right side is the storage area. This is used for saving objects – drag any cube to/from here. Note that cubes can be named for convenience.

• The left side is a panel of basic color cubes. These cells always contain a set of basic cubes that are used as the primitive building blocks all other values are made of. They cannot be overwritten. (Note that this includes a transparent cell.)

• The center part is the working panel. This is the main panel where new cubes are constructed. The center cell is similar to a storage cell, and the surrounding eight cells all perform some operation on this cell.

1.19.2 User Interaction

Right-click any cell except for the basic colors on the left panel, or hit escape or F10 for a menu of operations. The menu also includes the keyboard shortcuts for these operations.

1.19.3 Cube operations

There are six simple operations that are considered part of the simple graphic cube world. The operations correspond to six of the operation cells: a left-right composition is built using the left and the right cells, a top-bottom using the top and the bottom, and a front-back using the top-left and bottom-right. Dragging a cube to one of these cells will use the corresponding operator to combine it with the main cell’s cube. Using a right mouse click on one of these cells can be used to cancel dragging an object to that cell, this is not really an undo feature: a right-click on the right cell always splits the main cube to two halves and throws the right side.

The colored cubes and the six basic operators make this simple domain, which is extended to form a λ-Calculus-like language by adding abstractions and applications. Right-clicking on a basic cube on the left panel creates an abstraction which is actually a lambda expression except that colors are used instead of syntactic variables. For example, if the main cell contains R|G (red-green on the left and right), then right-clicking the green cube on the left panel leaves us with λ G . R|G, which is visualized as R|G with a green circle. The last two operator cells are used for application of these abstractions: drag a function to the top-right to have it applied on the main cube, or to the bottom-left to have the main cube applied to it. As in the λ-Calculus, all abstractions have exactly one variable, use currying for multiple variables.

So far the result is a domain of colored cubes that can be used in the same way as the simple λ-Calculus. There is one last extension that goes one step further: function cubes can themselves be combined with other functions using the simple operations. This results in a form of "spatial functions" that behave differently in different parts of the cube according to the construction. For example, a left-right construction of two functions f|g operates on a given cube by applying f on its left part and g on its right part. You can use the preferences dialog to change a few aspects of the computation.

Use the Open Example menu entry to open a sample file that contains lots of useful objects: Church numerals, booleans, lists, Y-combinator, etc.