In the current paper we consider theories with vocabulary containing a number of binary and unary relation symbols. Binary relation symbols represent labeled edges of a graph and unary relations represent unique annotations of the graph's nodes. Such theories, which we call annotation theories, can be used in many applications, including the formalization of argumentation, approximate reasoning, semantics of logic programs, graph coloring, etc. We address a number of problems related to annotation theories over finite models, including satisfiability, querying problem, specification of preferred models and model checking problem. We show that most of considered problems are NPTime- or co-NPTime-complete. In order to reduce the complexity for particular theories, we use second-order quantifier elimination. To our best knowledge none of existing methods works in the case of annotation theories. We then provide a new second-order quantifier elimination method for stratified theories, which is successful in the considered cases. The new result subsumes many other results, including those of [2, 28, 21].
Bibliographical noteFunding Information:
This work has been supported in part by the grant N N206 399134 of Polish Ministry of Science and Higher Education and by grants from the Swedish Foundation for Strategic Research (SSF) Strategic Research Center MOVIII and the Swedish Research Council (VR) Linnaeus Center CADICS.
- Argumentation theory
- Labeled graphs
- Second-order quantifier elimination
- Semantics of logic programs