Negation and Inductive Norms

In 1982, N. Immerman proved that (positive) least fixed point logic was closed under negation. He used a construction similar to that of Moschovakis [34]: if a logic admits an “inductive norm” that partitions a relation into blocks labelled by integers, then an appropriate “stage comparison relation” might be used to construct a negation of that relation within that logic. In this paper, we generalize this construction to many fragments of positive least fixed point logic, and in particular we will find that if such a fragment is closed under ∨, ∧, ∃, ∀ (on a class M of finite structures), and admits “stage comparison relations” (on M), then it is closed under negation (on M). We will have several examples, but we will spend a lot of time on the “positive stratified existential fixed point logic” introduced in [28]. In a separate paper ([32]) we prove that while connectivity can be expressed in this logic, non-connectivity cannot; so by this generalized Moschovakis-Immerman construction, the stage comparison relations for reachability cannot be expressed in this logic. But the hypotheses for Immerman’s Theorem are sharp: in this paper, we find that over a class of unions of Hamming Graphs — over which non-connectivity is not expressible in this logic — a relation very close to the stage comparison relation is expressible in this logic. An inductive norm is a device for dissecting relations developed by recursion. If a relation R is generated by repeated iterations of some process, then the tuples ~x satisfying R(~x) can be labelled with the number of iterations required to generate R(~x); this number |~x|R is 1 the R-norm of ~x. (If ¬R(~x), then we might write |~x|R = ∞. ) The basic idea is probably ancient, and the use of inductive norms in logic goes back to [25], and was developed further in [36] and generalized in [34]. In [33] and [34], Moschovakis developed his Stage (or norm) Comparison relations and showed how they could be used to take a relation R, expressible in a nice logic L, to show that ¬R is also L-expressible. The idea was to have a norm partition R, and another norm petition another relation S, and then, if the logic admitted “comparison relations” that allowed one to compare the R-norm of ~x with the S-norm of ~y, and if S admitted tuples of greater finite norm than R did, we would have ~y such that S(~y) (so that |~y|S <∞) and yet, for each ~x such that R(~x), |~x|R < |~y|S. This gives us ¬R(~x) ≡ ∃~y [∀~z(|~z|R < |~y|S ∨ |~y|S < |~z|R) ∧ (|~y|S < |~x|R)] : by the Boundedness Theorem ([34, Theorem 3C.1]), ¬R is L-expressible. Moschovakis was primarily interested in the norm |~x|φ = “the iteration ~x appears in the ‘least fixed point’ induction. ” In [21], Immerman proved that for every Least Fixed Point R, there existed a Least Fixed Point S such that over the class of all finite structures (of the same schema), the Least Fixed Point induction of S took one more iteration than that of R. Immerman concluded that over the class of finite structures, Least Fixed Point logic is closed under negation. This result was somewhat surprising: the leading exposition of the 1970s, Moschovakis’s [34], notes that although Least Fixed Point logic is closed under ∨, ∧, ∃ and ∀, there are infinite structures on which Least Fixed Point logic is not closed under negation. Indeed, before Immerman’s theorem, situations where Least Fixed Point logic was closed under negation were regarded as a bit esoteric: on most popular structures (e.g., the natural numbers with the usual arithmetic operations), Least Fixed Point logic is not closed under negation. But after Immerman’s theorem, research in infinite structures became less fashionable, while 2 (inspired by developments in computer science) research in Least Fixed Point logic on classes of finite structures became more popular. Several popular logics are fragments of Least Fixed Point logic; some of these popular logics are known to be closed under negation while others are known to be not closed (and for a few, the question is not settled). We would like to organize this diversity with a more general application of the Boundedness Theorem and a more general statement of Immerman’s Theorem. We find that a re-examination of Immerman’s proof, in game-theoretic terms, will give us the following generalization of his Theorem: Given a fragment of Least Fixed Point logic that is closed under ∀, ∃, ∨, ∧, if this logic admits stage comparison relations (over a class of finite structures) for its relations, then it is closed under negation (over that same class of finite structures). We will look at a few examples: in particular, we will look at one example that will show us that the hypotheses of Immerman’s Theorem are sharp. One popular fragment of Least Fixed Point logic is Existential Fixed Point logic, of [4] and [3], [27], [11], and elsewhere. This logic could be regarded as a sort of upgraded Datalog (a popular language in itself: see the expository paper [24], the text [1], or the tome [38]); recall that Datalog is a recursive call language permitting only existential quantification and no negation. In [28], we introduced a variant that we will call “positive stratified existential fixed point logic,” which essentially consists of those queries definable by least fixed points permitting only a bounded number of universal quantifications (we will define this logic more precisely later). In [28], this logic, which we will denote FO + pos EFP, was proven strictly weaker than the least fixed point logic of bounded alternation of [27]. In [28], we proved that FO + pos EFP is not closed under negation. In [32], we proved that 3 although connectivity is expressible in FO + pos EFP, non-connectivity is not: we proved this over a class of unions of “Hamming graphs” (to be described later in this article). It follows from Immerman’s Theorem that the Stage Comparison relation R(u, v, x, y) ≡ dist(u, v) < dist(x, y) (where dist(r, s) = the minimum distance along paths) is not FO + pos EFP expressible over this class of unions of Hamming graphs. However, we will find that over that class of Hamming graphs, the “bounded stage comparison relation” R(u, v, x, y) ≡ dist(u, v) < dist(x, y) <∞ is FO + pos EFP expressible. Thus we need the full stage comparison relations, not just the bounded ones, to get Immerman’s Theorem. Here is an outline of this article. We will first look at the notion of an inductive norm, a generalization of the stages-ofan-induction measure, and at applications of Moschovakis’s Boundedness Theorem. We then turn to fragments of Least Fixed Point logic. We will eschew the usual approach (which is to define these logics in terms of “least fixed points” of “operative systems” of “positive elementary” formulas), and instead define our logics using Datalog-like programs in a system similar to the Game Theoretic Semantics of [17]; this system was introduced in [28] and explored in [29], but this paper is self-contained and will contain all the necessary machinery. Thus in Subsection 2.3, we will present Least Fixed Point logic as a “game theoretic logic,” and FO + pos EFP as a sublogic characterized by imposing a “topological restriction” on the games. In Section 3, we will devise a game-theoretic “naive” stage comparison relation, and in Subsection 3.2, we will prove a general version of Immerman’s Theorem using