Towards Deciding Second-order Unification Problems Using Regular Tree Automata

The second-order unification problem is undecidable. While unification procedures, like Huet’s pre-unification, terminate with success on unifiable problems, they might not terminate on non-unifiable ones. There are several decidability results for infinitary unification, such as for monadic second-order problems. These results are based on the regular structure of the solutions of these problems and by computing minimal unifiers. Beyond the importance of the knowledge that searching for unifiers of decidable problems always terminates, one can also use this information in order to optimize unification algorithms, such as in the case for pattern unification [6]. Nevertheless, being able to prove that the unification problem of a certain class of unification constraints is decidable is far from easy. Some results were obtained for certain syntactic restrictions on the problems (see Levy [4] for some results and references) or on the unifiers (see Schmidt-Schauß [7] and Schmidt-Schauß and Schulz [8, 9] for some results). Infinitary unification problems, like the ones we are considering, might suggest that known tools for dealing with the infinite might be useful. One such tool is the regular tree automaton. The drawback of using regular automata for unification is, of course, their inability to deal with variables. In this talk we try to overcome this obstacle and describe an on-going work about using regular tree automata [1] in order to decide more general second-order unification problems. The second-order unification problems we will consider are of the form λzn.x0t . = λzn.C(x0s) where C is a context [2] and x0 does not occur in t or s. We will call such problems cyclic problems. A sufficient condition for the decidability of second-order unification problems was given by Levy [4]. This condition states that if we can never encounter, when applying Huet’s pre-unification procedure [3] to a problem, a cyclic equation, then the unification problem is decidable. It follows from this result that deciding second-order unification problems depends on the ability to decide cyclic problems. The rules of Huet’s procedure (PUA) are given in Fig. 1. Imitation partial bindings and projection partial bindings are defined in [10] and are denoted, respectively, by PB(f, α) and PB(i, α) where α is a type, Σ a signature f ∈ Σ and 0 < i. The following technical definitions, taken from our previous work on extending PUA to deal with some non-termination [5], describe the change in the unification constraints set when we start with a cyclic problem and execute certain rules of PUA. Let e be a cyclic equation as above where C = C1 . . . Cm such that for all 0 < i ≤ m, Ci = fi(r 1 i , . . . , [.], . . . , r ni i ) where ni = arity(fi) − 1. Define also, for all m < i, Ci = fk(y 1 i−ms, . . . , [.], . . . , y nk i−ms) where k = ((i − 1) mod m) + 1 and y j i−m for 0 < j ≤ nk are new variables. We define the progressive context D i for all 0 ≤ i as D i = Ci+1 . . . Ci+m. In the rest of this talk, e will refer to equations of this form and t, s, C,m, k, ni, r j i and y j i will refer to the corresponding values in e. In order to clarify the definitions, we will use the following (non-unifiable) cyclic equation as an example: x0f(a, a) . = f(x0a, f(f(a, a), b)). Note that PUA does not terminate on this problem.