Delia KESNER IRIF, CNRS et Universit ´e Paris kesner@irif.fr www.irif.fr/˜kesner
Unification
Delia KESNER IRIF, CNRS et Universit ´e Paris kesner@irif.fr www.irif.fr/˜kesner
Unification
Two termssandtareunifiableiff there exists a substitution (calledunifier) s.t.θ(s)=θ(t).
Example
f(x,g(x,a))and f(f(a),y)are unifiable withθ={x/f(a),y/g(f(a),a)}
f(x,g(x,a)) f(f(a),y) f(f(a),g(f(a),a)) = f(f(a),g(f(a),a)) f(x,g(x,a))and f(f(a),f(b,a))are not unifiable.
Delia KESNER IRIF, CNRS et Universit ´e Paris kesner@irif.fr www.irif.fr/˜kesner
Principal substitutions
Letθandτbe two substitutions andSbe a set of substitutions.
Thecompositionofθandτis(θ◦τ)(x)=bθ(τ(x))for every variablex∈ X.
θis aninstanceofτ(orτismore generalthanθ) iff there exists a substitutionρs.t.
for every variablex∈ X,(ρ◦τ)(x)=θ(x).
τ∈ Sisprincipal(ormost general) iff every substitutionθ∈ Sis an instance ofτ.
Example
Letσ1={y/b,z/h(c)}andσ2={x/f(y),y/z}. σ1◦σ2={x/f(b),y/h(c),z/h(c)}.
σ2is more general thanσ1◦σ2.
Delia KESNER IRIF, CNRS et Universit ´e Paris kesner@irif.fr www.irif.fr/˜kesner
Principal unifier
Theorem
LetSbe a non-empty set of unifiers ofsandt. Then, there exists aprincipal unifierθ∈ S s.t. for everyτ∈ S,θis more general thanτ. Moreover, this principal unifier isunique modulo renaming.
Delia KESNER IRIF, CNRS et Universit ´e Paris kesner@irif.fr www.irif.fr/˜kesner
Idempotent unifiers
A substitutionθisidempotentiffθ◦θ=θ. Example
{y/b,z/h(c)}is idempotent.
{x/f(y),y/z}is not idempotent.
Theorem
Ifsandtare unifiable, then there exists aprincipal unifier ofsandtwhich isidempotent.
How we can construct this unifier?
Delia KESNER IRIF, CNRS et Universit ´e Paris kesner@irif.fr www.irif.fr/˜kesner
Equational systems
An(Σ)equational systemis a set of (Σ)equations of the formst.
An equational systemEisunifiableiff there exists a unifier (calledsolution) for all the equations ofE.
Finiteequational systems are denoted{s1t1, . . . ,sntn}.
Delia KESNER IRIF, CNRS et Universit ´e Paris kesner@irif.fr www.irif.fr/˜kesner
Solved forms
The equational systemE={s1t1, . . . ,sntn}is insolved formiff All thesiaredistinct variables.
Nosiappears intj.
Example
E0={xy,zf(a)}is in solved form butE1={xy,xf(a)}, E2={xy,yf(a)},
E3={xz,yf(y)}do not.
Notation :For the solved systemE={x1t1, . . . ,xntn}we noteE~the substitution {x1/t1, . . . ,xn/tn}.
Delia KESNER IRIF, CNRS et Universit ´e Paris kesner@irif.fr www.irif.fr/˜kesner
The transformation rules
E∪{ss}
E (erase)
E∪{tx} t<X
E∪{xt} (orient)
E∪{f(s1, . . . ,sn)f(t1, . . . ,tn)}
E∪{s1t1, . . . ,sntn} (decompose)
E∪{xs} x∈Var(E) x<Var(s)
E{x\s}∪ {xs} (substitute)
Delia KESNER IRIF, CNRS et Universit ´e Paris kesner@irif.fr www.irif.fr/˜kesner
The unification algorithm
1 Take an equational systemE
2 Compute a new systemPby applying thetransformation rulesas far as possible.
3 If the systemPis insolved form then send the answerP~ else fail
Delia KESNER IRIF, CNRS et Universit ´e Paris kesner@irif.fr www.irif.fr/˜kesner
Example
Unification of the system{p(a,x,f(g(y)))p(z,f(z),f(u))}
p(a,x,f(g(y)))p(z,f(z),f(u)) d az,x f(z),f(g(y))f(u)
o za,xf(z),f(g(y))f(u)
s za,xf(a),f(g(y))f(u)
d za,x f(a),g(y)u
o za,xf(a),ug(y)(solved f orm) yields the (idempotent) substitution{z/a,x/f(a),u/g(y)}.
Delia KESNER IRIF, CNRS et Universit ´e Paris kesner@irif.fr www.irif.fr/˜kesner
Soundness and completeness of the algorithm
Theorem
The algorithm terminates.
Theorem
(Soundness)If the algorithm finds a substitution~Sfor the problemP, thenPis unifiable andS~is a m.g.u. ofP.
That is,
IfPis not unifiable, then the algorithm fails.
Theorem
(Completeness)If the systemPis unifiable, then the algorithm computes the m.g.u. of P.
That is,
If the algorithm fails, then the systemPis not unifiable.
