The Price of Selfishness: Conjunctive Query Entailment for ALCSelf is 2ExpTime-hard

TL;DR

This paper demonstrates that adding the Self operator to the description logic ALC makes conjunctive query entailment 2ExpTime-hard, significantly increasing computational complexity, which was previously not the case.

Contribution

It proves that the Self operator in ALC leads to 2ExpTime-hardness for conjunctive query entailment, a surprising complexity jump not observed with other extensions.

Findings

Adding Self operator increases complexity to 2ExpTime

Reduction from exponential space alternating Turing machines

Novel encoding techniques for restricted logic setting

Abstract

In logic-based knowledge representation, query answering has essentially replaced mere satisfiability checking as the inferencing problem of primary interest. For knowledge bases in the basic description logic ALC, the computational complexity of conjunctive query (CQ) answering is well known to be ExpTime-complete and hence not harder than satisfiability. This does not change when the logic is extended by certain features (such as counting or role hierarchies), whereas adding others (inverses, nominals or transitivity together with role-hierarchies) turns CQ answering exponentially harder. We contribute to this line of results by showing the surprising fact that even extending ALC by just the Self operator - which proved innocuous in many other contexts - increases the complexity of CQ entailment to 2ExpTime. As common for this type of problem, our proof establishes a reduction from alternating Turing machines running in exponential space, but several novel ideas and encoding tricks are required to make the approach work in that specific, restricted setting.