This module provides unification and matching in an Abelian group.

In this module, an Abelian group is a free algebra over a signature with three function symbols:

- the binary symbol +, the group operator,
- a constant 0, the identity element, and
- the unary symbol -, the inverse operator.

The algebra is generated by a set of variables. Syntactically, a
variable is an identifer such as x and y (see `isVar`

).

The axioms associated with the algebra are:

- Communtativity
- x + y = y + x
- Associativity
- (x + y) + z = x + (y + z)
- Identity Element
- x + 0 = x
- Cancellation
- x + -x = 0

A substitution maps variables to terms. A substitution s is applied to a term as follows.

- s(0) = 0
- s(-t) = -s(t)
- s(t + t') = s(t) + s(t')

The unification problem is given the problem statement t =? t', find a most general substitution s such that s(t) = s(t') modulo the axioms of the algebra. The matching problem is to find a most general substitution s such that s(t) = t' modulo the axioms. Substitition s is more general than s' if there is a substitition s" such that s' = s" o s.

- data Term
- ide :: Term
- isVar :: String -> Bool
- var :: String -> Term
- mul :: Int -> Term -> Term
- add :: Term -> Term -> Term
- assocs :: Term -> [(String, Int)]
- newtype Equation = Equation (Term, Term)
- data Substitution
- subst :: [(String, Term)] -> Substitution
- maplets :: Substitution -> [(String, Term)]
- apply :: Substitution -> Term -> Term
- unify :: Equation -> Substitution
- match :: Monad m => Equation -> m Substitution

# Terms

A term in an Abelian group is represented by the identity element, or as the sum of factors. A factor is the product of a non-zero integer coefficient and a variable. No variable occurs twice in a term. For the show and read methods, zero is the identity element, the plus sign is the group operation, and the minus sign is the group inverse.

A variable is an alphabetic Unicode character followed by a
sequence of alphabetic or numeric digit Unicode characters. The
show method for a term works correctly when variables satisfy
the `isVar`

predicate.

assocs :: Term -> [(String, Int)]Source

Return all variable-coefficient pairs in the term in ascending variable order.

# Equations and Substitutions

An equation is a pair of terms. For the show and read methods, the two terms are separated by an equal sign.

data Substitution Source

A substitution maps variables into terms. For the show and read methods, the substitution is a list of maplets, and the variable and the term in each element of the list are separated by a colon.

subst :: [(String, Term)] -> SubstitutionSource

Construct a substitution from a list of variable-term pairs.

maplets :: Substitution -> [(String, Term)]Source

Return all variable-term pairs in ascending variable order.

apply :: Substitution -> Term -> TermSource

Return the result of applying a substitution to a term.

# Unification and Matching

unify :: Equation -> SubstitutionSource

Given `Equation`

(t0, t1), return a most general substitution s
such that s(t0) = s(t1) modulo the equational axioms of an Abelian
group. Unification always succeeds.