agum-2.1: Unification and Matching in an Abelian Group

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

ide :: TermSource

ide represents the identity element (zero).

isVar :: String -> BoolSource

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 -> TermSource

Return a term that consists of a single variable.

mul :: Int -> Term -> TermSource

Multiply every coefficient in a term by an integer.

add :: Term -> Term -> TermSource

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) 

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 :: Monad m => Equation -> m 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.

match :: Monad m => Equation -> m SubstitutionSource

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