The answer set semantics may assign a logic program to model, due to logical contradiction or unstable negation, which is caused by cyclic dependency of an atom on its negation. While logical contradictions can be handled with traditional techniques from paraconsistent reasoning, instability requires other methods. We consider resorting to a paracoherent semantics, in which 3-valued interpretations are used where a third truth value besides true and false expresses that an atom is believed true. This is at the basis of the semi-stable model semantics, which was defined using a program transformation. In this paper, we give a model-theoretic characterization of semi-stable models, which makes the semantics more accessible. Motivated by some anomalies of semi-stable model semantics with respect to basic epistemic properties, we propose an amendment that satisfies these properties. The latter has both a transformational and a model-theoretic characterization that reveals it as a relaxation of equilibrium logic, the logical reconstruction of answer set semantics, and is thus called the semi-equilibrium model semantics. We consider refinements of this semantics to respect modularity in the rules, based on splitting sets, the major tool for modularity in modeling and evaluating answer set programs. In that, we single out classes of canonical models that are amenable for customary bottom-up evaluation of answer set programs, with an option to switch to a paracoherent mode when lack of an answer set is detected. A complexity analysis of major reasoning tasks shows that semi-equilibrium models are harder than answer sets (i.e., equilibrium models), due to a global minimization step for keeping the gap between true and believed true atoms as small as possible. Our results contribute to the logical foundations of paracoherent answer set programming, which gains increasing importance in inconsistency management, and at the same time provide a basis for algorithm development and integration into answer set solvers.
Answer Set Programming (ASP) is a well-established formalism for nonmonotonic reasoning. An ASP program can have no answer set due to cyclic default negation. In this case, it is not possible to draw any conclusion, even if this is not intended. Recently, several paracoherent semantics have been proposed that address this issue, and several potential applications for these semantics have been identified. However, paracoherent semantics have essentially been inapplicable in practice, due to the lack of efficient algorithms and implementations. In this paper, this lack is addressed, and several different algorithms to compute semi-stable and semi-equilibrium models are proposed and implemented into an answer set solving framework. An empirical performance comparison among the new algorithms on benchmarks from ASP competitions is given as well.
We propose a model of random quantified boolean formulas and their natural random disjunctive logic program counterparts. The model extends the standard models for random SAT and 2QBF. We provide theoretical bounds for the phase transition region in the new model, and show experimentally the presence of the easy-hard-easy pattern. Importantly, we show that the model is well suited for assessing solvers tuned to real-world instances. Moreover, to the best of our knowledge, our model and results on random disjunctive logic programs are the first of their kind.
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