Model Theory. These are non-empty families I of partial isomorphisms between models M and N , closed under taking restrictions to smaller domains, and satisfying the usual Back-and-Forth properties for extension with objects on either side -- restricted to apply only to partial isomorphisms of size at most k . 'Invariance for k--partial isomorphism' means having the same truth value at tuples of objects in any two models that are connected by a partial isomorphism in such a set. The precise sense of this is spelt out in the following proof. 21 Proof (Outline.) k-variable formulas are preserved under partial isomorphism, by a simple induction. More precisely, one proves, for any assignment A and any partial isomorphism IÎI which is defined on the A-values for all variables x 1 , ..., x k , that M, A |= f iff N , IoA |= f . The crucial step in the induction is the quantifier case. Quantified variables are irrelevant to the assignment, so that the relevant partial isomorphism can be res...

We show that the existential preservation theorem fails for FO², two-variable first-order logic. It is known that for all k 3, FO^k does not have an existential preservation theorem, so this settles the last open case, answering a question of Andr'eka, van Benthem, and N'emeti. In contrast, we prove that the homomorphism preservation theorem holds for FO².

. Logic preservation theorems often have the form of a syntax /semantics correspondence. For example, the / Los-Tarski theorem asserts that a first-order sentence is preserved by extensions if and only if it is equivalent to an existential sentence. Many of these correspondences break when one restricts attention to finite models. In such a case, one may attempt to find a new semantical characterization of the old syntactical property or a new syntactical characterization of the old semantical property. The goal of this paper is to provoke such a study. 1 Introduction It is well known that famous theorems about first-order logic fail in the case when only finite structures are allowed (see, for example, [?]). A more careful examination shows that it is wrong to lump all these failing theorems together. On one side we have theorems like completeness or compactness where the failure is really and truly hopeless. On the other side there are theorems like the / LosTarski theorem, which we...

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The homomorphism preservation theorem (h.p.t.), a result in classical model theory, states that a first-order formula is preserved under homomorphisms on all structures (finite and infinite) if and only if it is equivalent to an existential-positive formula. Answering a long-standing question in finite model theory, we prove that the h.p.t. remains valid when restricted to finite structures (unlike many other classical preservation theorems, including the ̷Lo´s-Tarski theorem and Lyndon’s positivity theorem). Applications of this result extend to constraint satisfaction problems and to database theory via a correspondence between existential-positive formulas and unions of conjunctive queries. A further result of this article strengthens the classical h.p.t.: we show that a first-order formula is preserved under homomorphisms on all structures if and only if it is equivalent to an existential-positive formula of equal quantifier-rank.

We prove that the homomorphism preservation theorem (h.p.t.), a classical result of mathematical logic, holds when restricted to finite structures. That is, a first-order formula is preserved under homomorphisms on finite structures if, and only if, it is equivalent in the finite to an existential-positive formula. This result, which contrasts with the known failure of other classical preservation theorems on finite structures, answers a longstanding question in finite model theory. The relevance of this result, however, extends beyond logic to areas of computer science, including constraint satisfaction problems and database theory; the database connection arises from a correspondence between existential-positive formulas and unions of conjunctive queries (also known as select-project-join-union queries). A second result of this article strengthens the classical h.p.t. by showing that a firstorder formula is preserved under homomorphisms on all structures if, and only if, it is equivalent to an existential-positive formula of equal quantifier-rank. Unlike traditional proofs of the classical h.p.t., the proof of this stronger “equirank ” theorem is compactnessfree and constructive. While these results are logical in nature, the technical development of the article takes place almost entirely within a combinatorial framework. The concept of tree-depth, a graph parameter related to tree-width, plays an important role in our analysis (as a combinatorial counterpart to quantifier-rank). We introduce new notions of n-homomorphism and n-core, which approximate the familiar concepts of homomorphism and core “up to tree-depth n”. The key technical lemmas take a pair of n-homomorphically equivalent [finite] relational structures and construct corresponding [finite] co-retracts which satisfy a certain back-andforth property.

Abstract. This paper investigates the class of k-universal finite graphs, a local analog of the class of universal graphs, which arises naturally in the study of finite variable logics. The main results of the paper, which are due to Shelah, establish that the class of k-universal graphs is not definable by an infinite disjunction of first-order existential sentences with a finite number of variables and that there exist k-universal graphs with no k-extendible induced subgraphs. 1.

We consider L | first order logic restricted to k variables, and interpreted in finite structures. The study of classes of finite structures axiomatisable with finitely many variables has assumed importance through connections with computational complexity. In particular, we investigate the relationship between the size of a finite structure and the number of distinct types it realizes, with respect to L . Some open questions, formulated as finitary Löwenheim-Skolem properties, are presented regarding this relationship. This is also investigated through finitary versions of an Ehrenfeucht-Mostowski property.