Definability in structures of finite valency

Arithmetical definability over finite structures

Arithmetical definability has been extensively studied over the natural numbers. In this paper, we take up the study of arithmetical definability over finite structures, motivated by the correspondence between uniform and . We prove finite analogs of three classic results in arithmetical definability, namely that and TIMES can first-order define PLUS, that and DIVIDES can first-order define TIM...

Inductive Definability with Counting on Finite Structures

Infinitary Logic and Inductive Definability over Finite Structures

The extensions of first-order logic with a least fixed point operators (FO + LFP) and with a partial fixed point operator (FO + PFP) are known to capture the complexity classes P and PSPACE respectively in the presence of an ordering relation over finite structures. Recently, Abiteboul and Vianu [AV91b] investigated the relation of these two logics in the absence of an ordering, using a mchine ...

Natural Definability in Degree Structures

A major focus of research in computability theory in recent years has involved definability issues in degree structures. There has been much success in getting general results by coding methods that translate first or second order arithmetic into the structures. In this paper we concentrate on the issues of getting definitions of interesting, apparently external, relations on degrees that are o...

Definability within structures related to

Let Sq denote the set of squares, and let SQn be the squaring function restricted to powers of n; let ⊥ denote the coprimeness relation. Let Bn(x, y) = (x+y x ) MOD n. For every integer n ≥ 2 addition and multiplication are definable in the structures 〈N;Bn,⊥〉 and 〈N;Bn,Sq〉; thus their elementary theories are undecidable. On the other hand, for every prime p the elementary theory of 〈N;Bp,SQp〉 ...