Invariant Definability (Extended Abstract)

) J.A. Makowsky ? Department of Computer Science Technion---Israel Institute of Technology IL-32000 Haifa, Israel [email protected] Abstract. We define formally the notion of invariant definability in a logic L and study it systematically. We relate it to other notions of definability (implicit definability, \Delta-definability and definability with built-in relations) and establish connections between them. In descriptive complexity theory, invariant definability is mostly used with a linear order (or a successor relation) as the auxiliary relation. We formulate a conjecture which spells out the special role linear order plays in capturing complexity classes with logics and prove two special cases. 1 Introduction This paper initiates an analysis of the notion of invariant definability special cases of which have been used in various contexts of finite model theory and descriptive complexity theory. We assume the reader is familiar with the basics of complexity theory as given ..