T L  C S C Z-o L: T  P C

The 0-1 law for first-order properties of finite structures and its proof via extension axioms were first obtained in the context of arbitrary finite structures for a fixed finite vocabulary. But it was soon observed that the result and the proof continue to work for structures subject to certain restrictions. Examples include undirected graphs, tournaments, and pure simplicial complexes. We discuss two ways of formalizing these extensions, Oberschelp’s (1982) parametric conditions and our (2003) thesauri. We show that, if we restrict thesauri by requiring their probability distributions to be uniform, then they and parametric conditions are equivalent. Nevertheless, some situations admit more natural descriptions in terms of thesauri, and the thesaurus point of view suggests some possible extensions of the theory. ∗Mathematics Department, University of Michigan, Ann Arbor, MI 48109–1043, U.S.A., [email protected] Quisani: I’ve been thinking about zero-one laws for first-order logic. I know it’s a rather old topic, but I noticed something in the literature that I’d like to understand better. The first proof [5] established that, for any first-order sentence σ in a finite relational vocabulary Υ, the proportion of models of σ among all Υ-structures with base set {1, 2, . . . , n} approaches 0 or 1 as n tends to infinity. Fagin [4] rediscovered the result (with a simpler proof) and added, near the end of his paper, some remarks about what happens if, instead of considering all Υ-structures, we consider only those satisfying some specified sentence τ. He pointed out that for some but not all choices of τ, there is still a 0-1 law: The proportion of models of σ among models of τ with base set {1, 2, . . . , n} approaches 0 or 1 as n tends to infinity. He gave two examples of such τ, both in the language with just a single binary relation symbol E, the language of digraphs. One example was the sentence saying that τ is symmetric and irreflexive, so the models are undirected loopless graphs. The other example defined the class of tournaments. The case of undirected graphs was rediscovered in [2], where another example was added, pure d-dimensional simplicial complexes, formulated using a completely symmetric and completely irreflexive (d + 1)-ary relation. I’d think there should be some general result explaining these variants of the 0-1 law. Authors: There is such a result in Oberschelp’s paper [6], but it seems he never published the proof. His result, expressed in terms of what he calls parametric conditions, covers the variants that you mentioned as well as others, for example involving graphs with several colors of edges. It is based on the same approach, via extension axioms, as the work in [4] and many later works. So it doesn’t cover the 0-1 laws obtained by other methods, for example by Compton (see [3] and the references given there) for slowly-growing classes of structures. Later, not knowing of Oberschelp’s work, we introduced in [1] the notion of a thesaurus as a suitable context for Shelah’s proof of the 0-1 law for choiceless polynomial time. It also provides a suitable context for the 0-1 law in the more restrictive context of first-order logic, once that logic is appropriately defined for thesauri. Q: Does the thesaurus approach also depend on the extension axioms? And should it be combined with parametric conditions to produce a common generalization? A: Both of your questions are answered — the first affirmatively and the second negatively — by the fact that the two approaches, parametric conditions and thesauri, are essentially equivalent, at least when applied to the case of uniform probability distributions on structures of any given size. Q: That leaves me with a lot of questions: What are parametric conditions? What are thesauri? What exactly does “essentially equivalent” mean in this context? And what happens when the probability distributions aren’t uniform? A: Let’s start with Oberschelp’s parametric conditions. These are conjunctions of first-order universal formulas, for a relational vocabulary, having the special form ∀x1 . . .∀xk (D(~x)→ C(~x)), where ~x stands for the n-tuple of variables x1, . . . , xk, where D(~x) is the formula ∧ 1≤i< j≤k xi , x j saying that the values of these variables are distinct, and where C(~x) is a propositional combination of atomic formulas such that, in each atomic subformula of C(~x), all k of the variables occur. Q: I assume that k is allowed to vary from one conjunct to another in a parametric condition, and that when k ≤ 1 the empty conjunction D is interpreted as true. So, for example, irreflexivity of a binary relation is expressed by the parametric condition ∀x(true→ ¬R(x, x)). A: That’s right, and it’s easy to express the other conditions you mentioned earlier — symmetry for undirected graphs, asymmetry for tournaments, and complete irreflexivity and symmetry for simplicial complexes — as parametric conditions. Q: I see that, but I don’t yet see the significance of the requirement that all atomic subformulas of C use all the variables. A: The simplest explanation is that if you drop this requirement then the extension axioms need not have asymptotic probability 1. For example, for almost all finite partially ordered sets, the longest chain has length three (see the proof of [3, Theorem 5.4]). So there are configurations, like a four-element chain, that can arise in partial orders but are absent with asymptotic probability 1. Traditional extension axioms, in contrast, imply that any configuration permitted by the underlying assumption τ must occur. The trouble comes from the transitivity clause in the definition of partial orders; it involves three variables but each atomic subformula uses only two of them. Q: The example shows that some requirement is needed to eliminate the case of partial orders, but how does the “use all the variables” requirement connect with extension axioms? I guess what I’m really asking for is a sketch of Oberschelp’s proof. A: The crucial contribution of parametricity is that it permits a reformulation of the uniform probability measure on structures of a fixed size n in terms of independent choices of the truth values of instances of the relations. Recall that, when we consider the class of all structures (of a given relational vocabulary) with universe {1, 2, . . . , n}, the uniform probability measure on these structures can be described by saying that each instance R(a1, . . . , ar) (where R ranges over the relations of the structure and ~a over tuples of appropriate length from {1, 2, . . . , n}) is independently assigned truth value true or false, with equal probability. When we deal with, say, loopless undirected graphs, this description must be modified, since R(a, a) must be false and since R(a1, a2) must have the same truth value as R(a2, a1). Nevertheless, the uniform distribution can still be described in terms of independent flips of a fair coin: flip a coin for each 2element subset {a1, a2} of {1, 2, . . . , n} to determine both R(a1, a2) and R(a2, a1). Similarly in the case of tournaments, a single flip of a fair coin decides which one of R(a1, a2) and R(a2, a1) shall hold. And similarly in the other examples. Something similar happens for arbitrary parametric conditions τ. To describe it, we need the notion of a k-type relative to τ. Temporarily fix a positive integer k, less than or equal to the maximum arity of the relation symbols in the vocabulary of τ. Consider all the atomic formulas that use exactly the variables x1, . . . , xk, possibly more than once. A k-type is an assignment of truth values to these atomic formulas that makes C(~x) true whenever ∀~x (D(~x) → C(~x)) is a conjunct of τ (up to renaming bound variables, so that ~x is x1, . . . , xk). In other words, a k-type is an assignment of truth values that can be realized by a k-tuple of distinct elements in a model of τ. Now the uniform distribution on models of τ with base set {1, 2, . . . , n} admits the following equivalent description: For each k and each k-element subset {a1 < a2 < · · · < ak} ⊆ {1, 2, . . . , n}, choose, uniformly at random, a k-type to be realized by the k-tuple (a1, . . . , ak). This works because each of these types is realized by (a1, . . . , ak) in equally many models of τ with base set {1, 2, . . . , n} and because different increasing tuples (a1, . . . , ak) behave independently. Furthermore, once the k-types of increasing tuples ~a are chosen, they determine all the relations of the structure. Q: What about instances of the relations where the arguments are not in increasing order? A: They’re included, because the atomic formulas to which a k-type assigns truth values include those in which the variables ~x occur out of order. Once one has this alternative description of the uniform distribution on models, one can easily imitate the traditional proof of 0-1 laws. There is an extension axiom for each k-type with k > 0; it says that, for any distinct x1, . . . , xk−1, there is an xk, distinct from all of them, such that the tuple (x1, . . . , xk−1, xk) realizes the given k-type. It is easy to check that each extension axiom has asymptotic probability 1 and that the theory axiomatized by the extension axioms is complete. (To prove completeness, one can proceed as in [4] because the theory is א0-categorical, or one can eliminate quantifiers as in [2].) The role of parametricity in this argument is to ensure that all the information about any k-tuple of distinct elements can be isolated in its k-type and the k′types of its subtuples, a finite amount of information, whose size is independent of the size n of the base set. That allows us to formulate extension axioms and verify their asymptotic validity. Contrast this with the situation for, say, partial orders. Here the requirement of transitivity imposes correlations between a truth value R(a, b) and many other truth values R(a, c) and R(b, c), for all c in the base set. The number of relation instances correlated with a single R(a, b) thus grows with the structure and the proof described above breaks down. Oberschelp [6] summarizes this (in the case of a vocabulary with only one relation symbol) by saying “A parametric property defines a class of relations which can be determined by the independent choice of values (parameters) in fixed regions of the adjacency array.” Q: OK. I see that parametricity seems to be just what’s needed to carry out the traditional proof of the 0-1 law via extension axioms. Now what are thesauri? A: A thesaurus is a finite set of signa, so of course we have to say what a signum is, but let’s first deal with a simplified notion of signum, which turns out to have the same generality as parametric conditions. A signum in this simplified sense consists of • a symbol R (assumed to be different for the different signa in a thesaurus), • a natural number j called the arity, • a finite set V called the value set1, • a group G of permutations of {1, 2, . . . , j}, and • a homomorphism from G to the group of permutations of V . For notational convenience, one often writes simply the symbol R when one really means the whole signum. Q: That sounds pretty complicated; what’s really going on here? A: The symbol R and the arity j are analogous to what you have in a relational vocabulary of ordinary first-order logic — a symbol and the number of its argument places. Our R’s, however, will not necessarily be 2-valued as in first-order logic, but v-valued, where v is the cardinality of the value set V . So we could, for 1In [1], the value set was always of the form {1, 2, . . . , v} for some positive integer v. Allowing arbitrary finite sets of values makes no essential difference but is technically convenient. example, treat a graph with colored edges by having a single binary signum where V is the set of colors plus one additional value to indicate the absence of an edge. The other two constituents of the signum, the group G and the homomorphism h, describe the symmetry properties that we intend R to satisfy. The idea is that permuting the j arguments of R by a permutation π in G results in a change of the value given by h(π). More precisely, a structure A for a thesaurus consists of a base set A together with, for each signum 〈R, j, v,G, h〉 (often abbreviated as just R), an interpretation R assigning to each j-tuple of distinct elements a1, . . . , a j in A a value R(~a), subject to the symmetry requirement R(a1, . . . , a j) = h(π)(R (aπ(1), . . . , aπ( j))). Q: The following variation seems more natural to me: R(aπ(1), . . . , aπ( j)) = h(π)(R (a1, . . . , a j)) It explains how to obtain R(aπ(1), . . . , aπ( j)) from R(a1, . . . , a j). A: This doesn’t work unless you either put π−1 on one side of the equation or make h an anti-homomorphism. Here’s the calculation, using your proposed variation. Let π and σ be two permutations in the group, let ~a be a j-tuple of elements of A, and let ~b be the j-tuple defined by bi = aσi. h(π)h(σ)R(a1, . . . , a j) = h(π)R (aσ1, . . . , aσ j) = h(π)R(b1, . . . , b j) = R(bπ1, . . . , bπ j) = R(aσπ1, . . . , aσπ j) = h(σπ)R(a1, . . . , a j), where we’ve applied your variation three times, once with σ, once with π, and once with σπ. So for this to work, we’d need h(σπ) = h(π)h(σ), i.e., h should be an anti-homomorphism. Q: I suppose using an anti-homomorphism wouldn’t be a disaster, but it would defeat the purpose of my suggestion, increased naturality. Now why does R apply only to j-tuples of distinct elements? A: Distinctness is technically convenient. For example, in tournaments, one wants the truth value of R(a, b) to be negated if a and b are interchanged, except when a = b. So we think of a binary relation R as being given by two signa, one binary signum for distinct arguments and one unary signum for equal arguments. Similarly, a relation of higher arity would be represented by several signa, one for each way of partitioning the argument places into blocks with equal arguments. Q: I see that, just as with parametric conditions, you can represent the uniform probability distributions on structures with base set {1, 2, . . . , n} in terms of independent random choices for some instances R(~a). For each signum R, choose a representative from each G-orbit of j-tuples of distinct elements, and assign R random values at these representatives. Then propagate these assignments through the whole orbits by means of the symmetry requirement. A: That’s right. To be precise about these G-orbits, one should say that G acts naturally on the set of j-tuples of elements from any set by π(a1, . . . , a j) = (aπ−1(1), . . . , aπ−1( j)). Q: With this formulation in terms of independent random choices, it should be possible to prove something analogous to extension axioms for the thesaurus context. I’d expect almost all Υ-structures to have the following property, for each n: Given any n distinct points a1, . . . , an, there is a point b, distinct from all the ai, and giving prescribed values for all signum instances R(c1, . . . , c j) where one of the ci is b and the others are distinct elements of {a1, . . . , an}. A: That’s right, provided the prescribed values obey the symmetry requirement for thesaurus models. We’re pleased that you remembered that the arguments of a signum are supposed to be distinct, so that b should occur only once in R(c1, . . . , c j) and each ai should occur at most once. Q: This result should yield 0-1 laws for thesauri, except that you haven’t yet defined first-order logic in the context of thesauri. A: Indeed, we have not introduced a syntax to go with these semantical notions in [1], but it is not difficult to do so. Take atomic formulas to be R(~x) = c where R is a signum (or the symbol part of it), ~x is a sequence of variables of length equal to the arity of R, and c ∈ V . Also allow equality as usual in first-order logic. Then form compound formulas using propositional connectives and quantifiers, just as in ordinary first-order logic. The semantics is obvious. (If the values of the variables in ~x are not all distinct, then R(~x) = c is naturally taken to be false.) If one is willing to stretch the notion of syntax a bit, then it would be appropriate to identify the atomic formulas R(x1, . . . , x j) = c and R(xπ−1(1), . . . , xπ−1( j)) = h(π)(c) for any π in the group of the signum R, since the symmetry requirement for structures implies that these will always have the same truth value. Once these definitions are in place, it is, as you said, not difficult to show, via extension axioms, that first-order sentences have asymptotic probabilities 0 or 1 over the class of all structures of a thesaurus. Q: This should also be clear for another reason, once you explain how thesauri and parametric conditions are essentially equivalent. Having the 0-1 law for parametric conditions, we should be able to use the essential equivalence to deduce the 0-1 law for thesauri. But what exactly did you mean by essential equivalence? A: Essential equivalence has several components. First, for each thesaurus Υ, there is a parametric condition τ (in some vocabulary) such that the Υ-structures with any particular base set (for example {1, 2, . . . , n}) are in (natural) one-to-one correspondence with models of τ on the same base set. Second, for each first-order sentence of the thesaurus, there is a first-order sentence of the vocabulary of τ such that the models of these sentences match up under the correspondence above. Third, conversely, for each parametric condition τ there is a thesaurus Υ with a one-to-one correspondence as before. And fourth, for every first-order sentence of the vocabulary of τ, there is a first-order sentence of Υ with the corresponding models. Q: That seems to be exactly what’s needed in order to convert 0-1 laws from either of the two contexts to the other. So how do these correspondences work? A: One direction is implicit in the syntax for thesauri described above. Given a thesaurus Υ, form a first-order vocabulary Υ′ with the same atomic formulas. That is, for each j-ary signum R of Υ and each value c, let Rc be a j-ary relation symbol in Υ′. The intended interpretation is that Rc(~a) should mean R(~a) = c. Every Υ-structure gives, in this way, a structure (in the ordinary, first-order sense) for Υ′. The converse is in general false, but the collection of Υ′-structures arising from Υ-structures in this way can be described by a parametric condition τ. The conjuncts in τ express the symmetry requirements of the thesaurus, i.e., ∀~x (D(~x)→ (Rc(~x)→ Rh(π)(c)(xπ−1(1), . . . , xπ−1( j)))) for each signum R and each π in its group. There are also conjuncts saying that every j-tuple of distinct elements satisfies Rc for exactly one c ∈ V and that Rc is false whenever two of its arguments are equal. We trust this makes the first two parts of “essentially equivalent” clear. Q: Yes; “essentially equivalent” is now half clear. But I suspect that this was the easier half. How do you handle the reverse direction? A: Here we have to convert a parametric condition τ into a thesaurus Υ. Let Υ consist of one j-ary signum T j for each j up to the maximum arity of the relation symbols in the vocabulary of τ. Q: Just one signum per arity, no matter how rich the vocabulary of τ is? A: That’s right. We compensate by using a rich set of values. Take the values of the j-ary signum T j to be the j-types relative to τ. (This is one place where it’s convenient to allow a signum to have any finite set of values, rather than only an initial segment of the positive integers as in [1].) The group associated to the j-ary signum T j is the symmetric group of all permutations of {1, 2, . . . , j}. To describe its action on the set of values, i.e., on the set of j-types, just let it act on the j variables occurring in the types. That is, the truth value assigned to an atomic formula θ by the type h(π)(c) is the same as the truth value assigned by c to the formula obtained from θ by substituting xπ−1(i) for xi for all i. A structure for this thesaurus provides, for each j-tuple ~a of elements of the base set, a j-type to be realized by this j-tuple. This specifies which atomic formulas are to be true of this ~a and of all permutations of ~a. The symmetry requirement on thesaurus structures is exactly what is needed to ensure that these specified truth values for the various permutations of ~a are consistent and thus describe a structure for the vocabulary of τ. Furthermore, since we use only j-types relative to τ, the resulting structures will be models of τ. And it is easy to check that every model of τ arises from exactly one Υ-structure. Q: That takes care of the third part of essential equivalence. For the fourth part, you have to translate formulas in the vocabulary of τ into Υ-formulas. Since the syntax and semantics of thesauri treats connectives and quantifiers the same way as first-order logic does, it suffices to consider atomic formulas. A: Right, and handling these is mainly just bookkeeping. Given an atomic formula θ in the vocabulary of τ, let {x1, . . . , xk} be the set of distinct free variables in it. For each equivalence relation E on this set of variables, we can write a quantifier-free formula θE, in the syntax associated to the thesaurus Υ, with variables x1, . . . , xk, saying that • the equality pattern of the xi is given by E, i.e., ∧ (i. j)∈E (xi = x j) ∧ ∧ (i, j)<E ¬(xi = x j),