Types of Frames
In full generality, general frames are hardly more than a fancy name for Kripke models; in particular, the correspondence of modal axioms to properties on the accessibility relation is lost. This can be remedied by imposing additional conditions on the set of admissible valuations.
A frame is called
- differentiated, if implies ,
- tight, if implies ,
- compact, if every subset of V with the finite intersection property has a non-empty intersection,
- atomic, if V contains all singletons,
- refined, if it is differentiated and tight,
- descriptive, if it is refined and compact.
Kripke frames are refined and atomic. However, infinite Kripke frames are never compact. Every finite differentiated or atomic frame is a Kripke frame.
Descriptive frames are the most important class of frames because of the duality theory (see below). Refined frames are useful as a common generalization of descriptive and Kripke frames.
Read more about this topic: General Frame
Famous quotes containing the words types of, types and/or frames:
“Our children evaluate themselves based on the opinions we have of them. When we use harsh words, biting comments, and a sarcastic tone of voice, we plant the seeds of self-doubt in their developing minds.... Children who receive a steady diet of these types of messages end up feeling powerless, inadequate, and unimportant. They start to believe that they are bad, and that they can never do enough.”
—Stephanie Martson (20th century)
“The American man is a very simple and cheap mechanism. The American woman I find a complicated and expensive one. Contrasts of feminine types are possible. I am not absolutely sure that there is more than one American man.”
—Henry Brooks Adams (1838–1918)
“The bird would cease and be as other birds
But that he knows in singing not to sing.
The question that he frames in all but words
Is what to make of a diminished thing.”
—Robert Frost (1874–1963)