In this paper, we analyze the decision version of the NK landscape model from the perspective of threshold phenomena and phase transitions under two random distributions, the uniform probability model and the fixed ratio model. For the uniform probability model, we prove that the phase transition is easy in the sense that there is a polynomial algorithm that can solve a random instance of the problem with the probability asymptotic to 1 as the problem size tends to infinity. For the fixed ratio model, we establish several upper bounds for the solubility threshold, and prove that random instances with parameters above these upper bounds can be solved polynomially. This, together with our empirical study for random instances generated below and in the phase transition region, suggests that the phase transition of the fixed ratio model is also easy.

Fusions are a simple way of combining logics. For normal modal logics, fusions have been investigated in detail. In particular, it is known that, under certain conditions, decidability transfers from the component logics to their fusion. Though description logics are closely related to modal logics, they are not necessarily normal. In addition, ABox reasoning in description logics is not covered by the results from modal logics. In this paper, we extend the decidability transfer results from normal modal logics to a large class of description logics. To cover different description logics in a uniform way, we introduce abstract description systems, which can be seen as a common generalization of description and modal logics, and show the transfer results in this general setting.

Simple conceptual graphs are considered as the kernel of most knowledge representation formalisms built upon Sowa's model. Reasoning in this model can be expressed by a graph homomorphism called projection, whose semantics is usually given in terms of positive, conjunctive, existential FOL. We present here a family of extensions of this model, based on rules and constraints, keeping graph homomorphism as the basic operation. We focus on the formal definitions of the different models obtained, including their operational semantics and relationships with FOL, and we analyze the decidability and complexity of the associated problems (consistency and deduction). As soon as rules are involved in reasonings, these problems are not decidable, but we exhibit a condition under which they fall in the polynomial hierarchy. These results extend and complete the ones already published by the authors. Moreover we systematically study the complexity of some particular cases obtained by restricting the form of constraints and/or rules.

Cox's well-known theorem justifying the use of probability is shown not to hold in finite domains. The counterexample also suggests that Cox's assumptions are insufficient to prove the result even in infinite domains. The same counterexample is used to disprove a result of Fine on comparative conditional probability.

A general notion of algebraic conditional plausibility measures is defined. Probability measures, ranking functions, possibility measures, and (under the appropriate definitions) sets of probability measures can all be viewed as defining algebraic conditional plausibility measures. It is shown that algebraic conditional plausibility measures can be represented using Bayesian networks.

We investigate the problem of reasoning in the propositional fragment of MBNF, the logic of minimal belief and negation as failure introduced by Lifschitz, which can be considered as a unifying framework for several nonmonotonic formalisms, including default logic, autoepistemic logic, circumscription, epistemic queries, and logic programming. We characterize the complexity and provide algorithms for reasoning in propositional MBNF. In particular, we show that entailment in propositional MBNF lies at the third level of the polynomial hierarchy, hence it is harder than reasoning in all the above mentioned propositional formalisms for nonmonotonic reasoning. We also prove the exact correspondence between negation as failure in MBNF and negative introspection in Moore's autoepistemic logic.

Starting with a likelihood or preference order on worlds, we extend it to a likelihood ordering on sets of worlds in a natural way, and examine the resulting logic. Lewis earlier considered such a notion of relative likelihood in the context of studying counterfactuals, but he assumed a total preference order on worlds. Complications arise when examining partial orders that are not present for total orders. There are subtleties involving the exact approach to lifting the order on worlds to an order on sets of worlds. In addition, the axiomatization of the logic of relative likelihood in the case of partial orders gives insight into the connection between relative likelihood and default reasoning.