Program to perform and display calculations with arithmetical combinators
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A program to perform and display calculations with arithmetical combinators, based on the arithmetic (addition, multiplication, exponentiation and nought) of Church numerals.

You might also use the program to do your shopping, though it lacks some amenities, (:-) eg. convenient numerical display and input. It is more useful for investigating algebraic laws.

These "AMEN" combinators are combinatorially complete, ie. define the same numerical functions as the \lambda-calculus. They satisfy interesting algebraic laws, slightly weaker than the arithmetic of the same operations on transfinite ordinals (except 1^a = 1 and 0*a = 0.) One can take them as a basis for Turing-complete computation. A (bracket-)abstraction operation is defined, based on "logarithmic" ideas due to Boehm.

The program evaluates arithmetical expressions built out of the 4 constants (+),(*),(^),0 by four corresponding binary operators, by rewriting according to computationally oriented algebraic equations. Crucially, we can examine the reduction sequences, and assess expressions for \zeta-equality.

The program is best run from ghci, as in

$ ghci amen-calulator/arithmetic-code.lhs

The most useful function is called test. It takes an expression as argument, and prints a representation of its first reduction sequence in a certain order. The expression to be tested can be assembled in the syntax:

  • binary infix operators :+:, :*:, :^:, :<>:,

  • unary constants V"+", V"*", V"^", V"0", and

  • variables va, vb, ..., given by alphabetical strings.

In ghci one can build up expressions making use of let-expressions. A rudimentary parser from strings is also available. Something like

let (e,[]):_ = prun expression (tokens "(eggs + bacon) ^ egg") in test $ vc :^: vb :^: va :^: blog "egg" e

should either

  • complain of a parse error, or

  • construct the indicated expression (using Boehm's logarithm inbuilt as blog) and display a particular reduction sequence.

Various combinators are defined, for example the standard combinator-sets IBCKW and SKI (also combinatorially complete). But there are also combinators for pairing, currying, and projection, combinators related to currying, permutation of arguments, fixed points, the continuation monad, and lots more. You have to browse the the code to find these.