Negation elimination in equational formulae

Abstract

An equational formula is a first order formula over an alphabet F of function symbols and the equality predicate symbol. Such formulae are interpreted in the algebra T(F) of finite trees. An equational formula is w.e.d. (without equations in disjunctions) if its solved forms do not contain any subformula s = t V u ≠ v. A unification problem is any equational problem which does not contain any negation (in particular, it should not contain any disequation). We give a terminating set of transformation rules such that a w.e.d. formula φ is (semantically) equivalent to a unification problem iff its irreducible forn is a unification problem. This result can be formulated in another way: our set of transformation rules computes a finite complete set of “most general unifiers” for a w.e.d. formula each time such a finite set exists. Such results extend Lassez and Marriott results on “explicit representation of terms defined by counter-examples” [10]. The above results are extended to quotients of the free algebra by a congruence =e which can be generated by a set of shallow permulative equations E.