Lindström theorems for fragments of first-order logic

Theorem 2.4 An abstract logic extending F O 3 is containedin F O iff it satisfies Compactness and invariance forpotential isomorphisms.Theorem 2.5 A “concrete” abstract logic extending F O 3is contained in F O iff it satisfies the Löwenheim-Skolemproperty and is recursively enumerable for validity.Here, by “concrete” we mean that formulas can be codedas finite strings over some alphabet, in such a way that negation,conjunction, and relativisation are computable operations,and there is a computable translation from F O 3 -formulas to formulas of the logic. The proof of Theorem 2.5uses the fact that satisfiability of F O 3 formulas on finitemodels is undecidable [3].2.2 First application: Tarski’s relation algebraTarski’s relation algebra RA [11] is an algebraic languagein which the terms denote binary relations. It hasatomic terms R, S, . . . ranging over binary relations (oversome domain), constants δ and ⊤ denoting the identity relationand the total relation, and operators ∩, −, •, (·) ⌣ fortaking the intersection, complement, composition and converseof relations. Thus, the syntax of RA is as follows:α ::= R | δ | ⊤ | α ∩ β |− α | α • β | α ⌣with R an element from some countably infinite set of variablesstanding for binary relations. An interpretation forthis language is a set X together with an assignment of binaryrelations over X to the atomic terms. In other words, itis a first-order structure for the vocabulary that contains theatomic terms as binary relation symbols. We write α ≡ βif, in each interpretation, α and β denote the same binaryrelation, and we write α ⊆ β if, in each interpretation, αdenotes a subrelation of the relation denoted by β.In this section, we provide a Lindström-theorem for extensionsof relation algebra. By an extended relation algebrawe will mean any language obtained by extending thesyntax of relation algebra with zero or more additional logicaloperations, where a logical operation is any operationthat takes as input a fixed finite number of binary relationsR 1 , . . . , R n (over some set X), and that produces a newbinary relation S over the same set. We also require logicaloperations to respect isomorphisms, and to be domainindependent in the following sense (familiar from databasetheory): the output relation S does not depend on elementsof the domain X that do not participate in any pair belongingto any input relation R i . This excludes for instancecomplementation as a logical operation, but, in terms of expressivity,it is not an essential restriction: one can alwaysrelativise such operations, by introducing an additional argument.For instance, absolute complementation as in −Rmay be replaced by relative complementation as in ⊤ − R.One example of an extended relation algebra, RA F O , isthe extension of relation algebra with all first-order definablelogical operations (see e.g. [15]). Another example isRAT , the extension of relation algebra with the transitiveclosure operation [10].The compactness and Löwenheim-Skolem propertiescan be defined for extended relation algebras as usual.For instance, we say that an extended relation algebra Lhas the Löwenheim-Skolem property if every set of L-expressions Φ, if there is an interpretation under which ⋂ Φis a non-empty relation, then there is such an interpretationover a countable domain. As is not hard to see, RAand RA F O satisfy both Compactness and the Löwenheim-Skolem property, whereas RAT satisfies the Löwenheim-Skolem property but lacks Compactness.The following Lindström-style theorem shows that allextended relation algebras containing non-first-order definableoperations lack either Compactness or the Löwenheim-Skolem property.Theorem 2.6 Let L be any extended relation algebra withthe Compactness and Löwenheim-Skolem properties. Thenevery logical operation of L is first-order definable.Proof: Lemma 2.2 can be adapted to the relation algebrasetting, allowing us to show that every extended relationalgebra containing a non-elementary logical operation andhaving the Löwenheim-Skolem property can projectivelydefine finiteness, and hence lacks Compactness. We willnot spell out the details, but merely mention the followingkey points of the proof:◮ For every first-order sentence φ containing only threevariables, in a signature consisting only of binary relations,there is a relation algebra expression α such that for everymodel M, M |= φ iff α ≡ ⊤ holds in M [11].◮ Unary relations can be mimicked by binary relations bysystematically intersecting them with the identity relation.◮ In this way, every extended relation algebra gives rise toan abstract logic extending F O 3 . Closure under relativisationof the logic is guaranteed by the domain independenceof the logical operations of L.✷In other words, the extension of relation algebra with allelementary operations is the greatest extension that satisfiesLöwenheim-Skolem and Compactness. The same holds ifwe replace the Löwenheim-Skolem property by invariancefor potential isomorphisms, or if we replace Compactnessby recursive enumerability.Theorem 2.6 nicely complements a known result: everyextended relation algebra with Craig interpolation candefine all first-order definable operations [12]. Together,these results show that RA F O is the unique (up to expressiveequivalence) extension of RA satisfying Compactness,Löwenheim-Skolem, and Craig Interpolation.4