In this article, we shall show that the Plotkin-Statman conjecture [?, ?] is false. The conjecture was that, in a model of the simply typed λ-calculus with only finitely many elements at each type, definability (by a closed term of the calculus) is decidable. This conjecture had been shown to imply many things, for example, Statman [?] (see also Wolfram’s book [?]) has shown it implies the deci...

Definability and approximations are two important notions of the theory of rough sets. In many studies, one is used to define the other. There is a lack of an explicit interpretation of the physical meaning of definability. In this paper, the definability is used as a more primitive notion, interpreted in terms of formulas of a logic language. A set is definable if there is a formula that defin...

We survey recent work on the use of Hankel matrices H(f,2) for real-valued graph parameters f and a binary sum-like operation 2 on labeled graphs such as the disjoint union and various gluing operations of pairs of laeled graphs. Special cases deal with real-valued word functions. We start with graph parameters definable in Monadic Second Order Logic MSOL and show how MSOL-definability can be r...

This paper investigates logical aspects of combining linear orders as semantics for modal and temporal logics, with modalities for possible paths, resulting in a variety of branching time logics over classes of trees. Here we adopt a unified approach to the Priorean, Peircean and Ockhamist semantics for branching time logics, by considering them all as fragments of the latter, obtained as combi...

Let ML( u) denote the fragment of modal logic extended with the universal modality in which the universal modality occurs only positively. We characterize the definability of ML( u) in the spirit of the well-known Goldblatt-Thomason-Theorem. We show that an elementary class F of Kripke frames is definable in ML( u) if and only if F is closed under taking generated subframes and bounded morphic ...

Game semantics has renewed denotational semantics. It offers among other things an attractive classification of programming features, and has brought a bunch of new definability results. In parallel, in the denotational semantics of proof theory, several full completeness results have been shown since the early nineties. In this note, we review the relation between definability and full abstrac...

In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called “uniform definability of types over finite sets” (UDTFS). We explore UDTFS and show how it relates to well-known properties in model theory. We recall that stable theories and weakly o-minimal theories have UDTFS and UDTFS implies dependence. We then show th...

We present some results concerning definability of classes of topological spaces in hybrid languages. We use language Lt described in [9] to establish notion of “elementarity” for classes of topological spaces. We use it to prove the analogue of Goldblatt-Thomason theorem in topological spaces for hybrid languages H(E) and H(@). We also prove a theorem that allows to reformulate definability re...

This paper gives two definability results in the local theory of the ω-enumeration degrees. First we prove that the local structure of the enumeration degrees is first order definable as a substructure of the ω-enumeration degrees. Our second result is the definability of the the classes Hn and Ln of the highn and lown ω-enumeration degrees. This allows us to deduce that the first order theory ...