# Cancellation Based Pattern Matching

In Comma is a Product: the Algebra of Concatenative Programming we described the algebra of concatenative programming. In this paper, we introduce a way to define the concatenative pattern matching without using variables, by investigating the cancellation properties of the algebra.

## Introduction

An algebraic data type is a data type defined by a list of data constructors. For example:

data Stack[T] where
Nil : -> Stack[T]
Cons : Stack[T], T -> Stack[T]

Any value of the ADT is can be constructed by using data constructors and primitive values. Like so:

Cons[Cons[Cons[Nil, 1], 2], 3]

Pattern matching is a way to deconstruct an ADT value by matching it against a pattern:

case stack:
Cons[$xs,$x] ->
...
Nil ->
...

Here, the value of stack is matched against two patterns: Cons[$xs,$x] and Nil. In the Cons[$xs,$x] branch, the corresponding values are binded to variables $xs and $x. For example, if stack is Cons[Cons[Cons[Nil, 1], 2], 3], then $xs is Cons[Cons[Nil, 1], 2] and $x is 3.

In functional languages, algebraic data types and pattern matching rely on application and variable binding. Our approach relies on the cancellation properties of the concatenative algebra.

## Pattern matching as cancellation

Expression like (((Nil,1 Cons),2 Cons),3 Cons) are built using only data constructors and values. Let's call such expressions as simple constructors.

Proposition: every simple constructor can be rewritten as vs cs, where vs is a concatenation of values and cs is a function built from data constructors and id.

Proof: The value elimination theorem and the substructural properties of the concatenative algebra.

For example,(((Nil,1 Cons),2 Cons),3 Cons) can be written as 1,2,3 (((Nil,id Cons),id Cons),id Cons).

In an algebra with operation $$*$$, the element $$c$$ is right-cancellative if for any $$a$$ and $$b$$ the following is true:

$a * c = b * c \quad ⇒ \quad a = b$

In other words, $$c$$ is cancellative if it can be undone in an equation. An algebra is right-cancellative if all elements in the algebra are right-cancellative.

We will use a different, more restricted definition. We say that $$c$$ is right-cancellative if there exists such function $$\tilde\, c$$ so the following is true for any $$a$$:

$\tilde\, c (a * c) = a$

Note that $$\tilde\, c$$ is only valid for values obtained by applying $$(\cdot * c)$$.

Proposition: Let $$C$$ be a concatenative algebra of values, data constructors and id. Then $$C$$ is right-cancellative over the composition.

Proof:

1. The cancellation property of data constructors and id is trivial
2. The cancellation of a,b is ~a,~b, where ~a and ~b are cancellations of a and b
3. The cancellation of a b is ~b ~a

For example, the cancellation of (id,id Cons),3 Cons is ~Cons (~Cons id,id),drop.

This is the essence of concatenative pattern matching. We can define the case statement, where patterns are expressions, and each branch cancels the corresponding expression. Like so:

case stack:
Cons ->
f    -- evaluates as ~Cons f
Nil ->
g    -- evaluates as ~Nil g

This is a variable-free equivalent of the case statements in applicative languages.