Unication Problems Modulo a Theory of Until

We refer to this equational theory as W . We came across these axioms while studying identities or equivalences in Linear Temporal Logic (LTL) — a logic for reasoning about time with the “until” operator U , but without the “next-time” operator ©. It is not hard to see that the until operator U satisfies these identities. However, note that there are other models for these identities as well: the logical and (∧) and or (∨) operations in boolean algebra are examples. Another is the “max” function over a domain such as Z. In fact, any operator that has the properties of associativity, commutativity and idempotence (an ACI-operator) will suffice as a model. A more trivial (and hence less interesting) model is where f is interpreted as a projection function into the second argument. In this paper we consider three computational problems for W . The first is the (usual) unifiability problem with constants, i.e., the terms to be unified will have free (uninterpreted) constants besides f . We show that the W -unifiability problem can be done in polynomial time if the input equations contain at most two constants. This was somewhat surprising, since the unifiability problem modulo idempotency (the first axiom) alone is NP-complete in the two constant case (see the reduction in [11]). The W -unifiability problem is NP-complete if the terms have 3 or more constants. The disunification problem modulo W is NP-complete even in the two-constant case. The asymmetric unifiability problem (a concept that was introduced only recently [7, 8]) modulo W is also NP-complete in the two-constant case. We would like to note here that semantic unification problems modulo LTL have been investigated by Vladimir Rybakov (see [4, 14]). However, Rybakov’s results cover a much more general theory—the entire LTL, in fact. This is a preliminary report of our ongoing research. Some of the proofs are omitted due to lack of space. For more details and proofs please see our tech report [5].