Algebra.AbelianGroup.UnificationMatching

Contents

Description

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)
Group Identity: 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.

Synopsis

Terms

data Term Source

A term in an Abelian group is represented by the group 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 group identity, the plus sign is the group operation, and the minus sign is the group inverse.

Instances

Eq Term
Read Term
Show Term

ide :: Term Source

ide represents the identity element (zero).

isVar :: String -> Bool Source

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.

var :: String -> Term Source

Return a term that consists of a single variable.

mul :: Int -> Term -> Term Source

Multiply every coefficient in a term by an integer.

add :: Term -> Term -> Term Source

Add two terms.

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

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

Equations and Substitutions

newtype Equation Source

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

Constructors

Equation (Term, Term)

Instances

Eq Equation
Read Equation
Show Equation

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.

Instances

Eq Substitution
Read Substitution
Show Substitution

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

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 -> Term Source

Return the result of applying a substitution to a term.

Unification and Matching

unify :: Monad m => Equation -> m Substitution Source

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

match :: Monad m => Equation -> m Substitution Source

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