We investigate Diophantine definability and decidability over some subrings of algebraic numbers contained in quadratic extensions of totally real algebraic extensions of Q. Among other results we prove the following. The big subring definability and undecidability results previously shown by the author to hold over totally complex extensions of degree 2 of totally real number fields, are shown...

Probability logic has contributed to significant developments in belief types for game-theoretical economics. We present a new probability logic for Harsanyi Type spaces, show its completeness, and prove both a denesting property and a unique extension theorem. We then prove that multi-agent interactive epistemology has greater complexity than its single-agent counterpart by showing that if the...

Could we define " I " ? Throughout this article we give a negative answer to this question. More exactly, we show that there is no definition for " I " in a certain way. But this negative answer depends on our definition of definability. Here, we try to consider sufficient generalized definition of definability. In the middle of paper a paradox will arise which makes us to modify the way we use...

We give a general technique on how to produce counterexamples to Beth’s definability (and weak definability) theorem. The method is then applied for various infinitary, cardinality quantifier logics and ∆-closure of such logics. I would like to thank Alice Leonhardt for the beautiful typing. AMS-Classification: 03C95, 03C20, 03E55, 03C55, 03C30. The author would like to thank the United States ...

D We investigate the notion of “semicomputability,” intended to generalize the notion of recursive enumerability of relations to abstract structures. Two characterizations are considered and shown to be equivalent: one in terms of “ partial computable functions’ ’ (for a suitable notion of computability over abstract structures) and one in terms of definability by means of Horn programs over su...

We consider some classical maps from the theory of abelian varieties and their moduli spaces, and prove their definability on restricted domains in the o-minimal structure Ran,exp. In particular, we prove that the projective embedding of the moduli space of the principally polarized abelian variety Sp(2g,Z)\Hg is definable in Ran,exp when restricted to Siegel’s fundamental set Fg . We also prov...

In standard propositional logic, logical definability is the ability to derive the truth value of some propositional symbols given a propositional formula and the truth values of some propositional symbols. Although appearing more or less informally in various AI settings, a computation-oriented investigation of the notion is still lacking, and this paper aims at filling the gap. After recallin...

We introduce some notions of invariant elementary definability which extend the notions of first-order order-invariant definability, and, more generally, definability invariant with respect to arbitrary numerical relations. In particular, we study invariance with respect to expansions which depend not only on (an ordering of) the universe of a structure, but also on the particular relations whi...