362,765 research outputs found
Axiomatizing modal inclusion logic
Modal inclusion logic is modal logic extended with inclusion atoms. It is the modal variant of first-order inclusion logic, which was introduced by Galliani (2012). Inclusion logic is a main variant of dependence logic (Väänänen 2007). Dependence logic and its variants adopt team semantics, introduced by Hodges (1997). Under team semantics, a modal (inclusion) logic formula is evaluated in a set of states, called a team. The inclusion atom is a type of dependency atom, which describes that the possible values a sequence of formulas can obtain are values of another sequence of formulas. In this thesis, we introduce a sound and complete natural deduction system for modal inclusion logic, which is currently missing in the literature.
The thesis consists of an introductory part, in which we recall the definitions and basic properties of modal logic and modal inclusion logic, followed by two main parts. The first part concerns the expressive power of modal inclusion logic. We review the result of Hella and Stumpf (2015) that modal inclusion logic is expressively complete: A class of Kripke models with teams is closed under unions, closed under k-bisimulation for some natural number k, and has the empty team property if and only if the class can be defined with a modal inclusion logic formula. Through the expressive completeness proof, we obtain characteristic formulas for classes with these three properties. This also provides a normal form for formulas in MIL. The proof of this result is due to Hella and Stumpf, and we suggest a simplification to the normal form by making it similar to the normal form introduced by Kontinen et al. (2014).
In the second part, we introduce a sound and complete natural deduction proof system for modal inclusion logic. Our proof system builds on the proof systems defined for modal dependence logic and propositional inclusion logic by Yang (2017, 2022). We show the completeness theorem using the normal form of modal inclusion logic
Euler characteristic and Akashi series for Selmer groups over global function fields
Let be an abelian variety defined over a global function field of
positive characteristic and let be a -adic Lie extension with
Galois group . We provide a formula for the Euler characteristic
of the -part of the Selmer group of over . In
the special case and a constant ordinary variety, using
Akashi series, we show how the Euler characteristic of the dual of
is related to special values of a -adic -function
The patchy Method for the Infinite Horizon Hamilton-Jacobi-Bellman Equation and its Accuracy
We introduce a modification to the patchy method of Navasca and Krener for
solving the stationary Hamilton Jacobi Bellman equation. The numerical solution
that we generate is a set of polynomials that approximate the optimal cost and
optimal control on a partition of the state space. We derive an error bound for
our numerical method under the assumption that the optimal cost is a smooth
strict Lyupanov function. The error bound is valid when the number of subsets
in the partition is not too large.Comment: 50 pages, 5 figure
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