Spectrum Hierarchies and Subdiagonal Functions Aaron Hunter Department of Computing Science Simon Fraser University Burnaby, Canada [email protected] Abstract been done on the properties of the classes in Fagin’s pro- posed spectrum hierarchy. The spectrum of a first-order sentence is the set of cardi- Similarly, very little is known about the classes of spec- nalities of its finite models. Relatively little is known about tra obtained by varying the number of binary relation sym- the subclasses of spectra that are obtained by looking only bols, or the number of unary relation symbols allowed in at sentences with a specific signature. In this paper, we sentences. In this paper, we will use model-theoretic tech- study natural subclasses of spectra and their closure prop- niques to explore the properties of these restricted classes erties under simple subdiagonal functions. We show that of spectra. many natural closure properties turn out to be equivalent to the collapse of potential spectrum hierarchies. We prove all 1.2. Closure Properties of our results using explicit transformations on first-order structures. Given a set of natural numbers S and a function f over the natural numbers, we write f(S) as a shorthand notation for {f(n)|n ∈ S}. We say that a class of spectra S is closed 1. Introduction under f if f(S) ∈Swhenever S ∈S. Using explicit transformations on graphs, More proves 1.1. Historical Motivation that F2 is closed under every polynomial with rational co- efficients that is assymptotically greater than the identity The spectrum of a first-order sentence φ is the set of car- function [12]. In this paper, we study closure properties un- dinalities of its finite models. Let Sp(φ) denote the spec- der functions that are assymptotically less than the identity trum of φ and let SPEC denote the set of all spectra. function. It turns out that establishing these closure proper- Let BIN1 denote the class of spectra of sentences involv- ties is more difficult. ing a single binary relation symbol. In [5], Fagin asks if every spectrum is in BIN1. Actually, Fagin restricts the sin- 1.3. Summary of Main Results gle relation to be irreflexive and symmetric, but we will al- low an arbitrary relation in this paper. Fagin also introduces In order to summarize our results, we introduce a more a potential spectrum hierarchy. Let Fk denote the class of general notation for subclasses of spectra. spectra of sentences involving relation symbols of arity at m ,...,m S ∈F k 1 S = (φ) most k. Clearly Fk ⊆Fk+1 for all k. Fagin proposes that Definition k if Sp for some sentence φ this hierarchy is strict, yet no one has been able to prove that which satisfies the following conditions: it does not collapse to the second level. 1. for each i ≤ k, φ involves at most mi relation symbols F Relatively little is known about the classes k.Partof of arity i the reason that they are not understood is because research on spectra has tended to rely heavily on the techniques 2. φ involves no other non-logical symbols. of descriptive complexity theory. Many interesting results p p,0 It will be convenient to write F2 as a shorthand for F2 . have been proved by characterizing subclasses of spectra in 1 1 We remark that BIN = F2 . terms of complexity, and then applying the results of com- We study three potential spectrum hierarchies: plexity theory [7, 8, 11]. However, no natural characteriza- 1,0 1,1 1,2 tion of Fk has been given. As a result, little research has (1) F2 ⊆F2 ⊆F2 ⊆ ... Proceedings of the 18th Annual IEEE Symposium on Logic in Computer Science (LICS’03) 1043-6871/03 $17.00 © 2003 IEEE 1 2 3 (2) F2 ⊆F2 ⊆F2 ⊆ ... spectra. For example, it has been shown that, if there is a spectrum whose complement is not a spectrum, then there F ⊆F ⊆F ⊆ ... (3) 2 3 4 is one in BIN1 [4]. Establishing general arithmetic closure 1 The first hierarchy is based on the number of unary relation conditions for BIN and related classes may provide a first symbols in the sentence and the second hierarchy is based step towards an improved understanding of these important on the number of binary relation symbols. The third hier- subclasses of spectra. archy is Fagin’s hierarchy, and it is based on the arity of The results that we present give arithmetic conditions for the relation symbols. The inclusions given above are trivial, the collapse of some potential spectrum hierarchies. We do but it is not clear if any of them are strict. We remark that not know of any other attempts to systematically study the (1) is the simplest spectrum hierarchy that is worth study- arithmetic closure properties of SPEC under natural subdi- ing, because it is well-known that F1 is the set of finite and agonal functions. We prove our results using simple, de- co-finite sets. finable transformations on first-order structures. This kind In this paper, we prove that the collapse of each hierar- of proof was employed in Fagin’s early work[5] and more chy is equivalent to an arithmetic closure condition on the recently this approach was explicitly studied by More[12]. lowest level. In particular, we establish the following re- sults. 2 Preliminaries • (1) collapses ⇐⇒ BIN1 is closed under n → n − 1 1 n 2.1. Definitions and Notation • (2) collapses ⇐⇒ BIN is closed under n →2 √ • (3) collapses ⇐⇒ F 2 is closed under n → n . A finite set of non-logical symbols together with an arity for each symbol will be called a vocabulary (or a signature). In the last section of the paper, we turn our attention to We will assume that all vocabularies contain only relation 1 the class BINbo introduced in [3]. This is the class of spectra symbols. Given a vocabulary L = {R1,...,Rm},anL- of sentences involving a single binary relation symbol with structure M = M,P1,...,Pm is composed of a finite 1 bounded outdegree. More precisely, S ∈ BINbo if S = set M called the universe and, for each k-ary relation sym- k Sp(φ) for some φ involving a single binary relation symbol bol Ri, a relation Pi ⊆ M called the interpretation of Ri. R and there is some k such that every model of φ assigns R Note that, by convention, M is used to denote the universe a relation with outdegree bounded by k. Although this class of the structure M. Normally, RM will denote the interpre- 1 of spectra is clearly a subset of BIN , it is not clear if it is a tation of R in the structure M. Following the conventions 1 1 strict subset. We prove that, if BIN = BINbo,thenFagin’s of set theory, we identify n with the set {0,...,n−1} when hierarchy collapses. we are specifying the universe of a structure. Throughout the paper, we will generally assume that all The notation a¯ will be used as an abbreviation for a tuple signatures are relational. We remark that some related work (a1,...,an) when the value of n is understood or unimpor- has been done on a proposed spectrum hierarchy based on tant. If φ(¯x) is a formula with free variables x¯,thenwe signatures involving only unary function symbols[3, 6]. write M|= φ[¯a] to indicate that a¯ satisfies φ(¯x) in M.The satisfaction relation is defined rigorously in [2]. 1.4. Significance of the Results For some familiar functions f, we use a natural short- hand notation for f(S). For example, if f is n → n +1, The most famous problem in the theory of spectra was then S +1denotes f(S). posed by Asser, when he asked if SPEC is closed under complement [1]. This problem has proven to be very dif- 2.2. Methodology ficult, due to a well-known characterization of SPEC in terms of complexity theory. Given a set of natural numbers Typically, we will start with a vocabulary L1 and a func- S Sˆ , there is a corresponding language over the alphabet tion f. Given an L1-spectrum S, we will be interested in { , } 0 1 that contains the binary representation of every ele- proving that f(S) is the spectrum of a sentence in some tar- S S ∈ ment of . It has been shown that SPEC if and only get vocabulary L2. For example, we might be interested in Sˆ if is accepted in exponential time by a non-deterministic proving that, if S is an {R}-spectrum, then S +1is also an Turing machine [4, 9]. Hence, if Asser’s problem has a neg- {R}-spectrum. The prototypical question we are interested ative solution, it follows by a straightforward argument that in can be stated as follows: P = NP. This fact has motivated a great deal of research on spectra. (*) Given an L1-spectrum S and a function f Understanding BIN1 and the hierarchies we have out- whose domain includes S, does it follow lined in this section is important for our understanding of that f(S) is an L2-spectrum? Proceedings of the 18th Annual IEEE Symposium on Logic in Computer Science (LICS’03) 1043-6871/03 $17.00 © 2003 IEEE If L1 = L2, then we are simply asking if the class of L1- 3 Extra Unary relations spectra is closed under the function f. Our proofs will be constructive. Starting with an arbitrary L1-sentence φ, We now turn to our proposed spectrum hierarchy based ∗ ∗ we will contruct an L2-sentence φ such that Sp(φ )= on a single binary relation and an arbitrary number of unary f φ 1,p 1,p+1 (Sp( )).