On the Complexity of Answer Set Programming with Aggregates

Abstract

The addition of aggregates has been one of the most relevant enhancements to the language of answer set programming (ASP). They strengthen the modelling power of ASP in terms of natural and concise problem representations. In this paper, we carry out an in-depth study of the computational complexity of the language. The analysis pays particular attention to the impact of syntactical restrictions on programs in the form of limited use of aggregates, disjunction, and negation. While the addition of aggregates does not affect the complexity of the full language with negation and disjunction, it turns out that their presence does increase the complexity of non-disjunctive ASP programs up to the second level of the polynomial hierarchy. Interestingly, under cautious reasoning nonmonotone aggregates are even harder than disjunction (Π2p -complete vs co-NP-complete on positive programs). However, we show that there are large classes of aggregates the addition of which does not cause any complexity gap even for normal programs, including the fragment allowing for arbitrary monotone, arbitrary antimonotone, and stratified (i.e., non-recursive) nonmonotone aggregates. Moreover, we also prove that for positive programs with arbitrary monotone, stratified antimonotone, and stratified nonmonotone aggregates the complexity remains polynomial. This analysis provides some useful indications on the possibility to implement aggregates in existing reasoning engines.