On the Complexity of H-Subsumption

Abstract

The importance of subsumption as a redundancy elimination method in automated theorem proving is generally acknowledged. For a given Herbrand universe H, it can be further strengthened to the so-called H-subsumption, i.e.: A clause D is H-subsumed by a clause set \({\cal C}\), iff for every H-ground instance Dθ of D there is a clause \(C \in {\cal C}\), s.t. C subsumes Dθ. In recent time, H-subsumption has gained increasing importance especially in the field of automated model building (cf. e.g. [5], [4], [6]). Furthermore, it can be easily shown that H-subsumption may be incorporated as a redundancy deletion rule into many familiar (resolution- and paramodulation-based) inference systems without destroying the refutational completeness.

However, no satisfactory algorithm for checking H-subsumption has been presented so far. We therefore have to investigate the inherent complexity of H-subsumption in order to explain this lack of efficient algorithms: The main result of this work is a Π\(_{\rm 2}^{p}\)-completeness proof for H-subsumption even if it is subjected to some strong restrictions. Hence, unless the polynomial hierarchy collapses to the first level, H-subsumption is non-polynomially more complex than ordinary subsumption.

Finally we present a new algorithm for H-subsumption whose complexity is compared with previously known algorithms (i.e.: from [5] and [4] on the one hand and from [6] on the other hand). The main advantage of our approach is that the total size of an H-subsumption problem (i.e.: in particular, the term depth of the expressions involved) only has polynomial influence on the overall (time and space) complexity. This is in great contrast to the other two approaches.