Unification in the combination of disjoint theories

Abstract

We consider unifaction modulo some equational theory E: Given are terms s, t ε Τ (E) built from the signature ε(E) of E and from variables x in V. A substitution unifies s,t if σ(s) ≡E σ(t), i.e. σ(s), σ(t) are equivalent modulo theory E.

In particular we give a unification algorithm for theories E = E ₁ ∪ ⋯ ∪ E _n which are combinations of theories with disjoint signatures, ε(E _i ) ∩ ε(E _j ) = Φ for i ≠ j. Our method works if for each theory E _i there exists a restricted unification algorithm: Given a set of equations P = {s ₁

t ₁, ..., s _m

t _m }, a linear ordering < of the variables in P, a set L of locked variables, the algorithm returns solutions σ with the following properties: • σ(s _j ) ≡ E_i(t _j ) • x does not occur in σ(y) if y < x • σ(x) = x if x ε L. No other restrictions are needed for the theories E _i .