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A Logical Approach to Asymptotic Combinatorics I. First Order Properties

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A Logical Approach to Asymptotic Combinatorics I. First Order Properties View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ADVANCES IN MATI-EMATlCS 65, 65-96 (1987) A Logical Approach to Asymptotic Combinatorics I. First Order Properties KEVIN J. COMPTON ’ Wesleyan University, Middletown, Connecticut 06457 INTRODUCTION We shall present a general framework for dealing with an extensive set of problems from asymptotic combinatorics; this framework provides methods for determining the probability that a large, finite structure, ran- domly chosen from a given class, will have a given property. Our main concern is the asymptotic probability: the limiting value as the size of the structure increases. For example, a common problem in elementary probability texts is to show that the asymptotic probabity that a per- mutation will have no fixed point is l/e. We shall say nothing about the closely related problem of determining rates of convergence, although the methods presented here may extend to such problems. To develop a general approach we must fix a language for specifying properties of structures. Thus, our approach is logical; logic is the branch of mathematics that deals with problems of language. In this paper we con- sider properties expressible in the language of first order logic and speak of probabilities of first order sentences rather than properties. In the sequel to this paper we consider properties expressible in the more general language of monadic second order logic. Clearly, we must restrict the classes of structures we consider in order for questions about asymptotic probabilities to be meaningful and significant. Therefore, we choose to consider only classes closed under disjoint unions and components (see Sect. 1 for definitions). This condition is general enough to include the classes of graphs, permutations, unary functions, and many other important examples. The main theorems of the paper are about classes satisfying this condition: Theorems 5.7 and 5.8 state that the existence of asymptotic probabilities for a special set of sentences called component-bounded sentences (defined in Sect. 5) is equivalent to a con- dition on the growth of such a class; Theorem 5.9 gives necessary and suf- ficient conditions for every first order sentence to have an asymptotic probability of 0 or 1 in nonfast growing classes. In Section 8 we present a long list of examples from the literature. ’ Present address: University of Michigan, Ann Arbor, Michigan 48109. 65 OOOl-8708/87 $7.50 Copyright Cl 1987 by Academic Press, Inc. All rights of reproduction in any form reserved. 66 KEVINJ.COMPTON Parts of this paper appeared in the author’s Ph.D. thesis [9]. The author would like to thank H. J. Keisler, Richard Askey, Ward Henson, Doug Hoover, and Jim Lynch for advice and helpful discussions. 1. PRELIMINARIES We will assume a familiarity, but not an expertise, with basic concepts of model theory; the first chapter of Chang and Keisler [S] introduces the essential ideas. Throughout, L will denote a finite language without con- stant symbols, L-structures (models) will be denoted by upper case fraktur letters (‘U, %?I*, ‘23, A, etc.) and their universes by the corresponding upper case italic letters (A, A*, B, K, etc.), and the set of first order sentences from L will be denoted L,,],. There are two natural ways to choose a structure randomly when assign- ing a probability to a first order sentence. The first is to choose from a set of structures with a fixed universe; the second is to choose from a set of isomorphism types. We make those ideas precise in the following definitions. DEFINITION. Let % be a class of L-structures closed under isomorphism, G$ the set of structures in % with universe n = {0, l,..., n - 1 }. For any sen- tence cp define p,,(q) to be the fraction of structures in &n in which rp is true. Combinatorialists often refer to this kind of enumeration as labeled enumeration since every element in the universe of a structure in Z$ is labeled with an ordinal number. Accordingly, we refer to the members of the <G&‘)sas the labeled structures of V and define to be the labeled asymptotic probability of cp whenever this limit exists. a,,, is a representative set from the isomorphism classes in 4,; members of the 2l,,‘s are the unlabeled structures of %?and the unlabeled asymptotic probability v is defined in the same way as p. Asymptotic problems are the focus of much research in combinatorics. Erdos and Spencer [ 151 contains a wealth of examples, most having to do with classes of graphs and matrices. Many of the results in this area are due to Erdiis and his collaborators (especcially Renyi, cf. Chap. 14 or ErdGs C141). Other classes have been studied. Metropolis and Ulam [30] asked about the asymptotic probability of connectivity in random functions. Katz [25] answered the question (he derived an asymptotic expression for the number of connected unary functions, a computation requiring more sophisticated LOGICAL ASYMPTOTIC COMBINATORICS I 67 analysis than a simple calculation of the asymptotic probability). Goncharov [20] showed that the asymptotic distribution of cycles in ran- dom permutations is Poisson. Shepp and Lloyd [37] extended his results; they calculated asymptotic probabilities of many statements about random permutations. Fagin [16] proved that when %? is the class of all L-structures, the asymptotic probability of any L,, sentence exists and, in fact, is either equal to 0 or 1. We will be especially interested in this kind of phenomenon. DEFINITION. We say that % has an L,, labeled tX1 law if I exists and is equal to either 0 or 1 for all cp in L,,. We define L,, unlabeled O-l law analogously. Mycielski [32] and Lynch [28,29] have looked at the question of the existence of asymptotic probabilities and the presence of O-l laws for a language with a relation symbol interpreted by total order and orientation relations. Grandjean [21] has determined the computational complexity required to decide, in the case of Fagin’s Cl law, which sentences have probability 0 and which have probability 1. Our results relate Cl laws to growth conditions on the classes &‘,, and B,,. The following types of generating series will figure prominently in the endeavor. DEFINITION. Let a,, = Iz$l, 6, = lBnI. In the labeled case the appropriate series to use is the exponential generating series for V: a(x)= f $xn. n=O . In the unlabeled case we use the ordinary generating series for %?: b(x)= f b,x”. fl=O Asymptotic probabilities do not always exist for first order sentences. One reason is that there may be infinitely many positive integers that are not cardinalities of structures in %, the class we have fixed. This is a minor technicality and in Section 6 we propose a slight change in the definition of asymptotic probability to overcome this difficulty. A more profound reason is that for some sentences cp, p,, (cp) or v, (cp) may fail to converge. We need only consider a few familiar examples-the class of groups, for exam- ple-to see that this is not an uncommon occurrence. Hoover [24] has shown that % may be finitely axiomatizable by first order universal senten- ces and still have first order cp such that p,, (cp) does not converge (this answers a question posed by Fagin [16]). 68 KEVIN J.COMPTON We want to restrict our attention to classes for which asymptotic probabilities are likely to exist, but not so narrowly that our examples are unnatural or uncommon. Our restrictions should not exclude structures, such as graphs, permutations, and unary functions, often considered in asymptotic combinatorics, These structures share the property that they may be uniquely represented as disjoint unions of connected structures (we will make this notion precise presently) from their respective classes: any graph is a disjoint union of connected graphs; beginning abstract algebra students learn that finite permutations can be decomposed into disjoint cycles; unary functions behave similarly. The classes we consider will have this property. We formalize this idea with the following: DEFINITION. Let 2l be an L-structure. Define a binary relation - as follows. Let a, b E A. a-b iffor some relation symbol R in L and sequences X, y, z of element variables a b (3~ Y, 2) W, a, Y, b, z). Let - * be the least equivalence relation extending -. The - * equivalence classes are called components of YI. If % is a graph, this corresponds to the graph theoretic notion of component. Sometimes we will say that a structure si is a component of ?I. By this, we mean that R is a substructure of ‘?I and K is a component of ‘?I. We will also say that 21 is connected if it has just one component. Let 9I and 23 be L-structures. 23 is a closed substructure of 2I (equivalen- tly, ‘S is a closed extension of %3) if 23 is a substructure of ‘% and is union of components of 2I. The classes we will consider are those closed under disjoint unions and components; i.e., classes 55’ which satisfy (i) If rU, 23 E V then ‘% II !Z3, the disjoint union of (L[ and 23, is in %; and (ii) If 2I E %?and A is a component of 9I then R E V. Combinatorialists have studied these classes extensively. In the next sec- tion, we will see that their generating series behave quite nicely and provide a useful means for enumeration. 2. PROPERTIES OF THE CLASSES Cayley [7, Vol. 3, pp. 242-2461 was first to use the properties of classes closed under disjoint unions and components for enumeration. In counting LOGICAL ASYMPTOTIC COMBINATORICS I 69 the number of unlabeled oriented trees, he found it useful to consider unlabeled oriented forests (disjoint unions of trees).
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