|M.Sc Student||Grimberg Benjamin|
|Subject||Minimal Model semantics for First-Order Ground Non-Monotonic|
|Department||Department of Computer Science||Supervisor||Professor Michael Kaminski|
|Full Thesis text|
Non-monotonic logics are intended to simulate the process of human reasoning by providing a formalism for deriving consistent conclusions from an incomplete description of the world. Roughly speaking, the main idea lying behind non-monotonic modal logics is using an inference rule of the form ``if ~p is not derivable conclude that p is possible''.
Ground non-monotonic modal logics are based on the idea of characterizing the knowledge of the agent by allowing him/her to make non-monotonic assumptions only with respect to his/her knowledge about the world.
As of today, the main effort has been invested into the investigation of propositional non-monotonic logics. Only recently, Zbar came with the first sound and well-based definition of first-order ground non-monotonic modal logic. Her definition characterizes the knowledge of the agent by a fixpoint equation.
In our work we present an alternative semantical characterization of first-order ground non-monotonic modal logic, by defining a preference relation on possible-world models. The preference relation is based on the minimization of knowledge expressed by non-modal formulas. We prove the equivalence of our definition to Zbar's definition. We also show that other well-known semantical characterizations of propositional ground non-monotonic modal logic, which are also based on the minimization of knowledge expressed by non-modal formulas, are not suitable for the first-order case.
In addition, we present two semantical definitions of minimal sets for first-order default theories. The definitions are similar to the definitions of extensions for first-order default theories. We prove that our two definitions are equivalent and that the relationship between minimal sets and extensions in the first-order case is similar to that of the propositional case.
Finally, we prove that the relationship between minimal sets for first-order default theories and first-order ground non-monotonic modal logic based on S4 and S5 is similar to the corresponding one in the propositional case.