Undecidability of the unification and admissibility problems for modal and description logics

Frank Wolter, Michael Zakharyaschev
2008 ACM Transactions on Computational Logic  
We show that the unification problem 'is there a substitution instance of a given formula that is provable in a given logic?' is undecidable for basic modal logics K and K4 extended with the universal modality. It follows that the admissibility problem for inference rules is undecidable for these logics as well. These are the first examples of standard decidable modal logics for which the unification and admissibility problems are undecidable. We also prove undecidability of the unification and
more » ... admissibility problems for K and K4 with at least two modal operators and nominals (instead of the universal modality), thereby showing that these problems are undecidable for basic hybrid logics. Recently, unification has been introduced as an important reasoning service for description logics. The undecidability proof for K with nominals can be used to show the undecidability of unification for Boolean description logics with nominals (such as ALCO and SHIQO). The undecidability proof for K with the universal modality can be used to show that the unification problem relative to role boxes is undecidable for Boolean description logics with transitive roles, inverse roles and role hierarchies (such as SHI and SHIQ). set of equations axiomatising the variety of Boolean algebras with operators and additional equations corresponding the axioms of L. A closely related algorithmic problem for L is the admissibility problem for inference rules: given an inference rule ϕ 1 , . . . , ϕ n /ϕ, decide whether it is admissible in L, that is, for every substitution s, we have L s(ϕ) whenever L s(ϕ i ), for 1 ≤ i ≤ n. It should be clear that if the admissibility problem for L is decidable, then the unification problem for L is decidable as well. Indeed, the rule ϕ/⊥ is not admissible in L iff there is a substitution s for which L s(ϕ). As was observed in [Ghilardi 1999] , in some cases the admissibility problem can be reduced to the unification problem. More precisely, suppose that for a unifiable formula ϕ in L one can compute a finite complete set S of unifiers in the sense that each unifier s for ϕ in L is less general than some s ∈ S (i.e., there exists a substitution s such that L s(p) ↔ s (s (p)), for all variables p in ϕ). Then to decide whether the rule ϕ/ψ is admissible in L it is enough to check whether L s (ψ) for all s ∈ S. It follows from the results of V. Rybakov (see [Rybakov 1997 ] and references therein) that the unification and admissibility problems are decidable for propositional intuitionistic logic and such standard ('transitive') modal logics as K4, GL, S4, S4.3. The computational complexity of the admissibility problem for these logics has been investigated in [Jerabek 2007] . For example, for intuitionistic logic, S4, and GL, the problem was shown to be NExpTime-complete. For further studies on unification and admissibility of rules in intuitionistic and modal logics, in particular, the problem of finding a finite basis for admissible rules and the existence of finite complete sets of unifiers, we refer the reader to []. Unfortunately, nearly nothing has been known about the decidability status of the unification and admissibility problems for other important modal logics such as the ('non-transitive') basic logic K, various multi-modal, hybrid and description logics. In fact, only one-rather artificial-example of a decidable uni modal logic for which the admissibility problem is undecidable has been found [Chagrov 1992] (see also [Chagrov and Zakharyaschev 1997] ). The first main result of this paper shows that for the standard modal logics K and K4 (and, in fact, all logics between them) extended with the universal modality the unification problem and, therefore, the admissibility problem are undecidable. The universal modality, first investigated in [Goranko and Passy 1992] , is regarded nowadays as a standard constructor in modal logic; see, e.g., [Blackburn et al. 2007 ]. Basically, the universal box is an S5-box whose accessibility relation contains the accessibility relations for all the other modal operators of the logic. The undecidability result formulated above also applies to those logics where the universal modality is definable, notably to propositional dynamic logic with the converse; see, e.g., [Harel et al. 2000 ]. The unification and admissibility problems for K itself still remain open. Observe that K4 is an example of a logic for which the unification and admissibility problems are decidable, but the addition of the universal modality makes them undecidable. This might be regarded as a surprising result: recall that the satisfiability problem for K4 with the universal modality can be easily reduced (in polynomial time) to the satisfiable problem for K4 itself. Given the fact that K4 is decidable in PSpace, this shows that K4 with the univer-
doi:10.1145/1380572.1380574 fatcat:ts3c3wnptnboravpvrl3nzspli