First-order definability on finite structures

Definability by Programs in First-Order Structures

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ti 1. Preliminarv detinitilms and results . . . . . . . . . . . . . . . . . . . . . . . . . Algebraic characterizations of the unwind property .................. A structure with the unwind property for every iterative (but not every recursive 1 program . The truth-table property ............................. Struc...

Inductive Definability with Counting on Finite Structures

Using Non-Standard Techniques to Analyze first-Order Definability over Finite Structures

We present a positive solution to McColm’s conjecture for the special case of decidable theories. This gives a uniform explanation of the separation of fixed-point from first-order definability over a wide range of commonly studied classes of finite structures (including ordered ones). We demonstrate why this result generalizes the standard constructive proofs, and show why it is unlikely that ...

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...

Successor-invariant first-order logic on finite structures

We consider successor-invariant first-order logic (FO+ succ)inv, consisting of sentences Φ involving an “auxiliary” binary relation S such that (A, S1) |= Φ ⇐⇒ (A, S2) |= Φ for all finite structures A and successor relations S1, S2 on A. A successorinvariant sentence Φ has a well-defined semantics on finite structures A with no given successor relation: one simply evaluates Φ on (A, S) for an a...