Homomorphisms and First-Order Logic

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. ∗Supported by an MIT Akamai Presidential Fellowship and a National Defense Science and Engineering Graduate Fellowship. This article was partially written during an internship at IBM’s Almaden Research Center.