The Hierarchy Theorem for Generalized Quantifiers

The Hierarchy Theorem for Generalized Quanti ers  Lauri Hella

y

Kerkko Luosto

z

Jouko V a an anen

x

July 16, 1999

Abstract

The concept of a generalized quanti er of a given similarity type was de ned in [Lin66]. Our main result says that on nite structures dierent similarity types give rise to dierent classes of generalized quanti ers. More exactly, for every similarity type t there is a generalized quanti er of type t which is not de nable in the extension of rst order logic by all generalized quanti ers of type smaller than t. This was proved for unary similarity types by Per Lindstrom [Wes] with a counting argument. We extend his method to arbitrary similarity types.

1 Introduction According to Lindstrom [Lin66], generalized quanti ers are simply classes of structures of a xed similarity type such that the class is closed under isomorphisms. We identify similarity types with nite sequences of positive integers. A structure A of (similarity) type t = (t ; : : :; tu) consists of a nite 1



Key words: generalized quanti er, nite model theory, abstract model theory,

y Partially supported by a grant from the University of Helsinki. This research was completed while the rst author was a Junior Researcher at the Academy of Finland. z Partially supported by a grant from the Emil Aaltonen Foundation. x Partially supported by grant 1011040 from the Academy of Finland.

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universe A together with relations Ri  Ati for i = 1; : : : ; u. We shall use the notation (A; R ; : : :; Ru ) for such a structure A. A generalized quanti er - or a model class - of (similarity) type t is any class Q of structures of type t so that Q is closed under isomorphisms. Every (generalized) quanti er Q gives rise to a natural logical operation ~Q. We refer to [Lin66] and [BF85] for the de nition of this operation. This leads to the concept of de nablity of quanti ers: A quanti er Q is de nable in terms of quanti ers Q ; : : : ; Qn if Q is the class of all models of a sentence of the extension of rst order logic by the logical operations Q~ ; : : :; Q~ n. The purpose of this paper is to prove as generally as possible that quanti ers of higher type are not de nable in terms of quanti ers of lower type. We call this result the Hierarchy Theorem for generalized quanti ers. In order to de ne the kind of ordering of types that we need, it is helpful to recall another well-known ordering: Let S be the set of nite sequences p = (p ; : : : ; pa) of natural numbers so that pa 6= 0. We then denote a by l(p). If p and q are elements of S , we let p < q hold if either l(p) < l(q) or else l(p) = l(q) and pi < qi for the largest i so that pi 6= qi. It is easy to see that < is a well-ordering of S of order-type !! . The following is an alternative de nition for 0, the quanti er Q r is not de nable in the logic L!! (Q ; : : : ; Q r? ). ( )

(1)

(

1)

There is no diculty in modifying the above proof so that the binary quanti er Q comes out LOGSPACE. Our counting methods do not seem to yield results like Theorem 13 for the branching and Ramsey operations. These questions are considered with dierent means in [HVW]. A question of a dierent kind is, whether the quanti er Q of Theorem 1 can be chosen so that it is not de nable even in terms of quanti ers obtained by one of the above operations from quanti ers of smaller type. We can solve this question as far as branching and Ramsey are concerned, but extending the result to resumption faces serious problems.

Theorem 16 If t is a similarity type  (2), then there is a generalized

quanti er Q of type t so that Q is not de nable in the extension of rst order logic (or xpoint logic) by any nite number of quanti ers of the following kind: 1. Generalized quanti ers of type < t. 2. Arbitrary branchings B (Q1; : : :; Qm), where m 2 !. 3. Arbitrary Ramsey quanti ers Rm(Q0), where m 2 !.

Proof. (Sketch) There is trivial but less informative proof: B (Q ; : : :; Qm) 1

and Rm (Q) are de nable in the extension of second order logic by Q ; : : :; Qm; Q. Hence the claim follows from the Note after Theorem 5. However, we indicate how our general method works in this case since it is needed if we want Q to have other properties, like being LOGSPACE. Let Q be the collection of all quanti ers as in 1-3 above. Consider a potential de nition 1

j (Q ; Q ; Q ) of Q in L!! (Q), where Q is a sequence of quanti ers of type < t, Q is a sequence of branchings of quanti ers and Q is a sequence of Ramsey quanti ers. We have to nd n so that in a universe of size n the quanti er Q is not de ned by this sentence, whatever the quanti ers in Q , whatever the bases of branching in Q , and whatever the bases of the Ramsey quanti ers 1

2

3

1

2

3

1

2

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of Q . For each of Q ; Q and Q we have to count the number of quanti ers of some xed types < t and the number of quanti ers of type t. Thus this proof reduces to a minor modi cation of the proof of Theorem 1. Q.E.D. 3

1

2

3

Corollary 17 There is a quanti er of type (2) which is not de nable in

the extension of rst order logic by any branchings B (Q1; : : : ; Qm) and any Ramsey quanti ers Rm(Q).

In view of our earlier remarks and results, we can replace \ rst order logic" above by e.g. \second order logic" or we can make the quanti er LOGSPACE and monotone. Can it be shown that some quanti ers are not obtained by any resumption from quanti ers of smaller type? Maybe, but a general result in this direction would imply P 6= NP . This follows from a result of A. Dawar as follows: Suppose we could assign with every type t a polynomial time generalized quanti er Qt of type t in such a way that Qt is not de nable in the extension of xpoint logic by any nite number of resumptions of generalized quanti ers of type s < t. Suppose on the other hand P = NP . Then P has a \syntax" and hence by a result of A. Dawar [Daw93] there is a quanti er Q0 of some type s so that every polynomial time quanti er is de nable in the extension of rst order logic by a suitable Q0 r . In particular, if we choose t > s, then the polynomial time quanti er Qt is de nable from some Q0 r , a contradiction. ( )

( )

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