Answer Set Programming (ASP) has seen several extensions by generalizing the notion of atom used in these programs, for example dl-atoms, aggregate atoms, HEX atoms, generalized quantifiers, and abstract constraints, referred to collectively as generalized atoms in this paper. The idea common to all of these constructs is that their satisfaction depends on the truth values of a set of (non-generalized) atoms, rather than the truth value of a single (non-generalized) atom. In a previous work, it was argued that for some of the more intricate generalized atoms, the previously suggested semantics provide unintuitive results, and an alternative semantics called supportedly stable was suggested. Unfortunately, this semantics had a few issues on its own and also did not have a particularly natural definition. In this paper, we present a family of semantics called Chain Answer Sets, which has a simple, but somewhat unusual definition. We show several properties of the new semantics, including the computational complexity of the associated reasoning tasks.
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