In this paper we present some of our school’s results in the area of building up relativity theory (RT) as a hierarchy of theories in the sense of logic. We use plain first-order logic (FOL) as in the foundation of mathematics (FOM) and we build on experience gained in FOM. The main aims of our school are the following: We want to base the theory on simple, unambiguous axioms with clear meaning...

We investigate the possibility of developing a decidable logic which allows expressing a large variety of real world specifications. The idea is to define a decidable subset of many-sorted (typed) firstorder logic. The motivation is that types simplify the complexity of mixed quantifiers when they quantify over different types. We noticed that many real world verification problems can be formal...

In this paper we show that Ω(n) variables are needed for first-order logic with counting to identify graphs on n vertices. The k-variable language with counting is equivalent to the (k − 1)-dimensional Weisfeiler-Lehman method. We thus settle a long-standing open problem. Previously it was an open question whether or not 4 variables suffice. Our lower bound remains true over a set of graphs of ...

I describe metarules which order predicates contained in first order logic rules. These metarules are applied to rules created by a learning Go program. After the ordering of the predicates, the rules are matched orders of magnitude faster than the original learned rules.

We develop several aspects of local and global stability in continuous first order logic. In particular, we study type-definable groups and genericity.

We introduce an atomic formula y ⊥ x z intuitively saying that the variables y are independent from the variables z if the variables x are kept constant. We contrast this with dependence logic D based on the atomic formula =( x, y), actually equivalent to y ⊥ x y, saying that the variables y are totally determined by the variables x. We show that y ⊥ x z gives rise to a natural logic capable of...

We prove that the first-order theory of a finite extension of the field of p-adic numbers is model-complete in the language of rings, for any prime p.

We consider order-generic queries, i.e., queries which commute with every order-preserving automorphism of a structure’s universe. It is well-known that first-order logic has the natural order-generic collapse over the rational and the real ordered group for the class of dense order constraint databases (also known as finitely representable databases). I.e., on this class of databases over 〈Q, ...

In the beginning of the 20th century, many prominent logicians and mathematicians, such as Frege, Russell, Hilbert, and many others, felt that mathematics needed a very rigorous foundation in logic. The standard approach in the early part of the 20th century was to use a syntactic or proof-theoretic definition of logical consequence. This says that “for one sentence to be a logical consequence ...