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Partial equilibrium logic (PEL) is a new nonmonotonic reasoning formalism closely aligned with logic programming under well-founded and partial stable model semantics. In particular it provides a logical foundation for these semantics as well as an extension of the basic syntax of logic programs. In this paper we describe PEL, study some of its logical properties and examine its behaviour on disjunctive and nested logic programs. In addition we consider computational features of PEL and study different approaches to its computation.