On the Size of Logical Descriptions

There are structures A which can be described up to isomorphism by a single sentence φ (in some logic), that is, B |= φ ⇐⇒ B ∼= A for all B. If the structure A is finite, then φ can be taken as a first-order sentence. We want to have φ as succinct as possible. (Therefore, a sentence ψ in some non-first-order logic could be a better choice.) 1 Atomic Complexity of structures Let A = 〈A,S〉 be a structure, or a model, in the usual model-theoretic sense. The signature S consists of relations, functions on A, and individuals from A, i.e. S = {R, . . . f, . . . c, . . . }, where {R, . . . f, . . . , c, . . . } is a vocabulary of relation, function, and individual symbols. Some of these categories might be empty. (When we consider questions of atomic complexity, we usually replace functions by relations.) Let L be a model-theoretic logic more or less in the sense of [2]. Suppose that the complete Ltheory of A can be axiomatized by one single sentence as is, e.g., always the case when A is finite (i.e. both A and S are finite) and L is first-order logic. But e.g. also every א0-categorical linear order has a finitely axiomatized first-order theory. (There are א0 many א0-categorical linear orders.) Let φ be a sentence of the logic (language) L. As usual, a sentence is a closed formula, i.e. a formula without free variables. The atomic complexity of φ is the number of occurrences of atomic formulae in φ. Thus, if one and the same atomic formula occurs at several places all these occurrences are counted. One may distinguish between nonlogical and logical atomic formulae. E.g. R(x, y) and R(y, x) are two different nonlogical atomic formulae (provided that R is different from the equality symbol = ). Atomic formulae containing only variables and the equality sign = like x = x, x = y are counted as logical. If c and d are individual constants then c = y, x = d, c = d are counted as nonlogical atoms. We denote the atomic complexity of φ by atc(φ), and define it as the number of occurrences of atomic fromulae. If A is a structure with a finitely axiomatizable (complete) L-theory, then the L-atomic complexity of the structure A is also denoted by L − atc(A) and is defined by L− atc(A) := min{atc(φ) : theL − sentenceφ is equivalent to the L − theory of A} For finite structures A this definition comes to this: L − atc(A) := min{atc(φ) : φ is an L − sentence and for all finite B : B |= φ ⇐⇒ B ∼= A} Here ∼= means isomorphic.