M.Sc Student | Grimberg Benjamin |
---|---|

Subject | Minimal Model semantics for First-Order Ground Non-Monotonic Modal Logic |

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.