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First-Order Logic vs. Fixed-Point Logic in Finite Set Theory Albert Atserias Computer Science Department University of California Santa Cruz, CA 95064 [email protected] Abstract The ordered conjecture states that least fixed-point logic LFP is strictly more expressive than first-order logic FO on every infinite class of ordered finite structures. It has been established that either way of settling this conjecture would resolve open problems in complexity theory. In fact, this holds true even for the particular instance of the ordered conjecture on the class of BIT-structures, that is, ordered finite structures with a built-in BIT predicate. Using a well known isomorphism from the natural numbers to the hereditarily finite sets that maps BIT to the membership relation between sets, the ordered conjecture on BIT-structures can be translated to the problem of comparing the expressive power of FO and LFP in the context of finite set theory. The advantage of this approach is that we can use set-theoretic concepts and methods to identify certain fragments of LFP for which the restriction of the ordered conjecture is already hard to settle, as well as other restricted fragments of LFP that actually collapse to FO. These results advance the state of knowledge about the ordered conjecture on BITstructures and contribute to the delineation of the boundary where this conjecture becomes hard to settle. 1. Introduction and summary of results The main goal of descriptive complexity theory is to investigate the connections between computational complexity and logic on classes of finite structures. As regards firstorder logic, it is well known that every first-order definable query is computable in LOGSPACE. Moreover, research in descriptive complexity theory has established that essentially all major computational complexity classes can be characterized in terms of definability in natural extensions of first-order logic on classes of finite structures. In particular, Immerman [Imm86] and Vardi [Var82] showed that Research partially supported by NSF Grant CCR-9610257. Phokion G. Kolaitis Computer Science Department University of California Santa Cruz, CA 95064 [email protected] on every class of ordered finite structures, that is to say, if is a class of ordered finite structures, then the class of polynomial-time computable queries on coincides with the class of queries definable in least fixedpoint logic on . Least fixed-point logic LFP is the extension of first-order logic FO obtained by augmenting the syntax and semantics with a least fixed-point operator for positive first-order formulas. As a general rule, least fixed-point logic is strictly more expressive than first-order logic. In particular, this holds true on the class of all ordered finite structures over a fixed vocabulary, as well as on the class of all (unordered) finite structures over a fixed vocabulary. There are, however, classes of unordered finite structures on which LFP collapses to FO, and both are properly contained in PTIME. McColm [McC90] was the first to focus attention on this phenomenon and to formulate a certain conjecture concerning necessary and sufficient conditions for the collapse of LFP to FO on an arbitrary class of finite structures. Although in its full generality McColm’s conjecture was refuted by Gurevich, Immerman and Shelah [GIS94], it sparked a sequence of related investigations in finite model theory [KV92, DH95, KV96, DLW96]. Moreover, the following special case of McColm’s conjecture still remains open: Conjecture 1 If is an infinite class of ordered finite struc tures, then first-orderlogic is properly contained in least fixed-point logic on . This conjecture, which is often called the Ordered Conjecture, was made by Kolaitis and Vardi [KV92]. In view of the aforementioned result of Immerman [Imm86] and Vardi [Var82], the conjecture is equivalent to the ordered assertion that on every infinite class of ordered finite structures. Thus, it enunciates an inherent limitation in the expressive power of first-order logic by stating that first-order logic can never capture polynomial time. All empirical evidence gathered to date supports it. At the same time, researchers have established that either way of settling the ordered conjecture will have sig- nificant complexity-theoretic consequences. Specifically, Dawar and Hella [DH95] if the showed that ordered conjecture is false, then . Furthermore, Dawar, Lindell and Weinstein [DLW96] pointed out that if the conjecture holds, then , where ordered (the Linear-Time Hierarchy) is the class of languages computable by alternating Turing machines in linear time using a constant number of alternations, and is the class of languages computable by deterministic Turing machines from in time. The separation of can be viewed as the linear version of the separation between the Polynomial-Time Hierarchy PH and the full Exponential Time EXPTIME; although it is widely believed that both these separations hold, neither has been established thus far. Intuitively, the difficulty in proving the ordered conjecture arises from the potential presence of powerful builtin arithmetic predicates that may significantly enhance the expressive power of first-order logic on classes of ordered finite structures. One especially powerful such predicate is on$ the natural numbers ! #"% the binary relation , where holds if and only if the -th " bit of the binary expansion of is equal to . The expressive power of first-order logic on finite structures with BIT as a built-in predicate has been investigated in depth by Immerman [Imm89] and by Barrington, Immerman and Straubing [BIS90]. In many respects, the presence of embodies the difficulties encountered with the ordered conjecture; as a matter of fact, a particular instance of the ordered conjecture on finite structures with BIT turns out to be literally equivalent to open ' problems in complexity theory. More precisely, let & 0 class of /all 3 )' (*,+ .be/the ordered finite -structures $ 3 *21 linear order and 54 ,6where 0 is the standard 8 . *71 Question 1 Is on & ? In other words, does the ordered conjecture hold on & ? Gurevich, Immerman and Shelah [GIS94] raised this question and stated that it is a “fascinating question in complexity theory and logic related to uniformity of circuits and logical descriptions.” Indeed, it can be shown that :9 on & if and only if log-time is different uniform 9 than polynomial-time uniform (see [BIS90, Lin92] for the definitions of these circuit-complexity classes). Moreover, it can also be shown that on & if and only if . Our goal in this paper is to advance the state of knowledge about the ordered conjecture on & by seeking to delineate the “boundary” where this conjecture becomes hard to settle. To this effect, we study certain fragments of least fixed-point logic LFP on & and investigate the restriction of the ordered conjecture on these fragments. We first iden- tify a natural proper fragment of LFP for which the ordered conjecture cannot be settled without resolving open problems in complexity at the same time. We then establish that the ordered conjecture actually fails when further re stricted to certain fairly expressive fragments of on & , which means that these fragments collapse to first-order logic on & . To isolate these fragments of , it is more convenient to conduct our investigation in the context of finite set theory and to use set-theoretic concepts and methods. The starting point is a recent paper by Dawar, Doets, Lindell and Weinstein [DDLW98], where it was shown that the standard linear order is first-order BIT. '@?BA definable using More precisely, let &<;>= A the C(D*E+ '@?Bbe class of (unordered) finite BIT-structures, where GH $ ./0 . Dawar et al. [DDLW98] showed *F1 that there is a first-order formula over the vocabulary that defines the standard linear order on the class &;>= . This rather surprising result was established by exploiting the existence of a $ well-known be JI6K LM$ isomorphism I6K O(see N [Bar75]) K I and the tween and , where I QP ’s are the finite ranks, that is to say, finite initial segments of the universe of sets I 9 SR I obtained JI by$ iterating the power-set operation: $ G!I TVK U LM$ . The isomorphism beI K tween and is the function W ( YX defined by the recursion: W /$ ZR "[$ J $ ! #"%$\ W W ( This isomorphism enables us to translate questions about the expressive power of logics on the class &;>'@= ?BA to equiva$ lent G questions ?BA its$ image .$\ HLM(/$*5+ '@on C class /$0 & ^$B] B W , where W . It W W W *_1 is easy to see that the containments ` C ] b a & ` c ] e a d ` ] G !I LM$ hold, where ^] is the class (f*g+ of all finite ranks, and d `] is the class ofLMall $ near fi!h nite ranks, that is, structures of the form such that I h I for some natural number * . Dawar ji i GTVU et al. [DDLW98] showed that there is a first-order definable linear order on d ^] such that its pre-image under the isomorphism W coincides with the standard linear order on . It follows that there is a first-order formula that defines the standard linear order on &<;k= . Consequently, the ordered conjecture on & reduces to the question of whether on &<;> = , which, in turn, is equivalent to the question: is on &^] ? The above set-theoretic framework makes it possible to isolate and study variants of the ordered conjecture for fragments of that are obtained by applying the least fixedpoint operator to collections of set-theoretic formulas with special syntactic properties. In this paper, we focus on the collection of l 9 formulas; these/L are the first-order formulas over a vocabulary containing such that oQ$ every LpoQ$ JmQn Lp rqsn occurrence of a quantifier is of the form or . The collection of l 9 formulas has played a fundamen- tal role in the development of set theory for two reasons: l 9 formulas possess desirable structural properties, known as absoluteness properties, and they can define many frequently encountered set-theoretic predicates. As Barwise [Bar75, page 10] puts it, “The importance of l 9 formulas rests in the metamathematical fact that any predicate defined by a l 9 formula is absolute, and the empirical fact that many predicates occurring in nature can be defined by l 9 formulas.” Both these facts will be of use to us in the sequel. In particular, the proofs of most of our results will rely heavily on the preservation of l 9 formulas under end extensions. $ Let l 9 be the fragment of least fixed-point $ that consists of the least fixed-points of all $ that are positive in the relation symbol l 9 formulas . Consider now the following variant of the ordered con 9 $ jecture: does collapse to on & `] ? Clearly, a l negative answer to this question will imply that the ordered conjecture holds on & and, thus, it is at least as hard to establish as the ordered conjecture on & itself. On the other hand, one may speculate with some reason that the answer to this question is positive, since l 9 formulas constitute a rather small (and well-behaved) fragment of first-order 9 $ logic. Our l first main result establishes that if on i & ^ ] , then , which, in turn, implies that i 9 $ . Thus, the collapse of to l on & ^] can neither be affirmed nor refuted without resolving long-standing open problems in complexity theory at the same time. The above result motivates the further study of the or 9 $ on &^] . To dered conjecture for fragments of l this effect, we isolate a fairly expressive collection of l 9 formulas $ and establish that the corresponding fragment of 9 l collapses indeed to on the class d ^] of near finite ranks and, consequently, on the class & ^] as well. This fragment is inspired from the work of Dawar et al. [DDLW98], who showed that a linear order can be defined in first-order logic on d ^] . Their proof proceeds by first defining a linear order on d ^] in n >a$ natural way as the least fixed-point of a n >$ certain positive l 9 formula , and then showing n >$ that can actually be expressed in firstorder logic on . An inspection of that particular l 9 d ` ] n >$ formula reveals that it has the following special syntactic property: every occurrence of the binary relation symbol involves bound variables only. We turn this propl 9 as erty into a concept and define the class restricted of $ the collection of l 9 formulas such that U every occurrence of the relation symbol involves bound $ variables only. We then show that if is an U arbitrary restricted l 9 formula that is positive in B $ , then its least fixed point is firstU order definable on d `] . This generalizes the results in [DDLW98] and also provides a tool for easily showing that several other basic queries, such as (finite) ordinal addition, are first-order definable on d `] . 9 $ l obtained After this, we consider fragments of by restricting the number of free variables in the l 9 formulas under consideration. $ observe that if is a unary We relation symbol and is a l 9 formula $ is pos that itive in , then the least fixed-point of coincides 9 with the least fixed point of some restricted formula l $ 9 $ . It follows that unary collapses to l on the class d ^] of near finite ranks and, hence, on the class & `] as well. Clearly, this raises the question 9 $ whether l any similar collapses can be proved for formulas of higher arities, while keeping in mind that we cannot 9 $ l hope to show that formulas of arbitrary arities on & `] without simultaneously showing collapse to that 9 .$ Our main result along these lines is that binary l collapses to on & ^] . Finally, we derive tight polylogarithmic bounds for the growth of the closure functions of arbitrary positive l 9 formulas on ^] , where the closure function of a positive formula gives the number of iterations needed to reach the least fixed-point of$ the formula. As a corollary, we show that ev 9 l -definable ery query on `] is a member of the F complexity class of queries computable in polylogarithmic time using a polynomial number of processors. This result appears to be of independent interest and suggests the pursuit of descriptive complexity in the context of finite set theory. 2. Preliminaries Let be a relational vocabulary, a -ary relation sym $ bol not in , and "^ a first-order formula over U the vocabulary ! . For every -structure # with uni-$ h h verse and every -ary h relation $ on , we write % $ for the -ary relation on defined by and $ , that is, $ '& & $ L h +* & B & / % $ $-, U ( #)( U $ The relation $ is a pre fixed-point of on # if % $ i $ ; in a dual manner, $ is a post fixed-point of on # $ if $ # if$ i % % $$ . Finally, $ is a fixed-point ofon $ $ . It is well known that if U is (which means that every occurrence of positive in in is within B an$ even number of negation symbols), $ then has both a$ least fixed-point ."/ # U and a greatest fixed-point 01/ # $ on # . As a matter of fact, the least fixed-point ."/ # is the intersection of all$ pre fixed-points, whereas the greatest fixed-point 02/ # is the union of all post fixed-points. Moreover, they can 6 be computed via transfinite $ % as $#$ ev iterations, !N7follows. 698 For ery ordinal 3 + , let .5/ 4 # . / # and 4 6 $ % 96 8 0/ 4 # 0 / $ 4 and 0 / # 0 / # 4 $ N such 6that ./ # N 698 $ $#$ $ N $ . Then . / # . / # 4 . Furthermore, there is an ordinal 6 6 8 $ $ and hence . / # . / # . / # . The least such ordinal is called$ the closure ordinal of on # and is denoted by$ / # . Note that if # is a finite structure, then h / # is a positive integer less than or equal than ( ( . Least fixed-point logic LFP is the extension of first-order logic FOobtained augmenting $ syntax with a new by B the formula $ every positive U ,for in first-order formula ; naturally, on U every structure # this new formula is interpreted by the $ least fixed-point . / # . It is well known that this syntax gives rise to a robust collection of queries on finite structures. In particular, LFP-definable queries on finite structures are closed under iterated and nested applications of the least fixed-point operator for positive formulas (see [Imm86, GS86]). HL is a firstA l 9 formula over the vocabulary order formula such that all occurrences of quantifiers rq L n $ !m1 L n $ 9 $ are of l the form and . We let denote the fragment of that consists of the least fixed-points 9 $ of positive l 9 formulas; this means that every " l $ formula isof the form , where B $ U 9 is a l formula that is positive in the U B ary relation symbol and$ has as free variables. U 9 Note that every l -formula involves a single application of the least fixed-point operator, that is, it contains no nested or iterated least fixed-points. Furthermore, no additional first-order and second-order parameters are allowed 9 $ in l -formulas. In a series of papers, 9 $ Sazonov has studied definability JI K LM$ in l a variant of on the infinite structure of all hereditarily finite sets (see [Saz97] for a$ survey). Here, 9 l we study uniform definability in on the collection &^] '^ of?Ball finite structures that are images A / $ of the BIT .0 structures under the * 1 I K isomorphism W ( X . In particular, we investigate the 9 $ question: does l collapse to on &^] ? In the process of this investigation, we also study uniform defin 9 $ ability in on the classes d ^] of near finite ranks l and ^] of finite ranks that envelop &^] from above and from below. $# 3. Complexity-theoretic aspects of As explained in the introduction, separating from & ^ ] on is literally equivalent to separating from , an open problem in complexity theory. $ In 9 l this section, we show that even the collapse of to on & ^] would yield important results in complexity theory. Hence, it is difficult to either refute or confirm that 9 $ l collapses to on &^] . i . Proof (sketch): over the 6G Let be a language in . We will show that the language alphabet /8 L D is in * over ! ( $0 ! $ $ ( ( #"$ ; then a de-padding argument puts ! * $0 ! .$#$ $ Theorem 1 If on & ^] , then l 9 i , which in turn implies that in . Here, stands for the * dual word of obtained by interchanging ’s and ’s. By Immerman [Imm86] and Vardi&[Var82], is ),a+ sentence % 3 there 6('* of over the vocabulary , where % is a unary relation symbol, that defines on the class of ordered binary words with . We can assume that $#$ /$ with first-order by the normal3 form theorem for . Moreover, does not occur in , since by [DDLW98] it is first-order definable from . '*),+ Similarly, and need not occur in , because they $ are first-order definable aswell. We turn > $ n n 8 U into a l $ 9 formula , where is a J.U U variable. Replace each occurrence o21$ relation 0/,L o 1 o o0 / -ary % o0/J$ in o0 / Lby , each positive occurrence % o,/J$ n in o0 / Lby , each negative occurrence o0/ 3 o0/ $ U n 8 by in o0/ by o0/ n , and n 8 $ each occurrence . Finally, replace subformulas of the !m oQ$54 Jm U o L n $647 !m o%L n 8 $64 form byoQ$54 , and subformulas q q U o7L n $648 q o[L n 8 $64 of the form . Using the by $:9 U !I K LM$ isomorphism W , it is not difficult to ( prove on the construction L5.byinduction for every of $ * that ( we have that $#$ '^?A 8 $ n n 8 >$ * R <;V $ , $0if and =;M only ( W W ,, if W ;V $ U where stands for the natural number represented in binary by . Since is a l 9 formula, the hypothesis of the theorem implies that the least fixed-point of is definable by a first-order $ <;Mon '@?BA 8 formula $0& ;M^ ] $#. $ It fol-L lows that the query > ( $ ( ( * is first-order definable on &;>= . A result in [BIS90] implies then that a suitable $0 ! $ $ encoding of > is ! computable in #"$ H 8? @ L * . It turns out that ( ( ( * can serve as this encoding. This completes the sketch of the proof 8$ that . Since A * , the i iB space-hierarchy theorem implies that . CD It should be noted that from a result of Dawar and Hella [DH95] it follows that if on & `] , then i . The preceding Theorem 1 shows that the separation of PTIME from PSPACE 9 $ can be derived from the weaker hypothesis that on & `] . l i 4. Restricted E collapses to GF As mentioned earlier, Dawar et al. [DDLW98] showed HL that there is a first-order formula over the vocabulary that defines a linear order on the class d LM`$ ] of near finite Jh I ranks, that such that h I is, structures of the form i k$ this, they considered the following l 9 i GTV U . n For formula !mQn L n $ rq L $ n L 8 L n X n $#$ $ and showed that its least fixed-point is definable by a firstorder formula on d `] . Observe that the occurrence of the relation symbol in involves only the bound variables n and . We now abstract from this observation and introduce the following concept. $ Definition 1 A l 9 formula is restricted U if every occurrence of the relation symbol involves only bound variables of . The main result of this section is that the least fixed-point of every positive restricted l 9 formula is first-order definable on d ^] and, hence, on & `] as well. In fact, we show that it is definable by a first-order formula of low syntactic complexity. The class of formulas is the smallest collection of formulas containing the l 9 formulas and closed!mQunder $ rqsn L $ disjunction, bounded quantifican L conjunction, m n tions , , and existential quantification . The collection of formulas is defined dually by allowq n ing closure under universal quantification . We say that a query > on a class is l -definable if it is definable on by a formula and by a formula. $ B F Theorem 2 Let be a restricted l 9 forU mula that is positive in the -ary relation symbol . The least fixed-point of is first-order definable on d `] . In fact, it is l -definable on d `] . The proof of the above theorem is inspired by the argument of Dawar et [DDLW98] showing that the least fixed n >$ point of the l 9 formula is first-order definable on d ^] . We need, however, to establish certain absoluteness properties of arbitrary l 9 formulas, as well as certain structural properties of arbitrary restricted l 9 formulas that will be used heavily in the sequel. We begin with a basic definition from set theory (see [Bar75, page 34]). /LLet # and be two structures over the vocabulary . We say that is an end extension of # , and write & L h substructure # L &of and L forh every L % & i , if # is a L it is the case that . ( ( Lemma1B(Absoluteness of l 9 formulas [Bar75, page 35]) $ 9 If is a formula and # l & for 'U & & $ L h i * & , then every we have that # ( , U * & B & U if and only if ( ,. U $ If is a l 9 formula, then the set of $ U free indices of , denoted by , is the set of indices of variables that are free in and appear in at least 8 $ one occur . For example, if rence of in the formula rq L $ 8 $ U $ H is , then U $ . Note R that a l 9 formula is restricted if and only if . From L !h LM$ now on, we identify structures over with their unih verse . In the following two lemmas, we establish certain important properties of l 9 and of restricted l 9 formulas on near finite ranks. $ HL F Lemma 2 Let be a l 9 formula over ^ U h , where is a -ary relation symbol, and let be a I h I near finite rank such that . For every " 3 & i GTVU 4e$h i i h , and every tuple & * ,& every $[L Irelation h *& / h TVU * & 4 , we !I have & that U L ( $0.$ $- , ,. ( $ if and only if ( h ( In particular, if is a restricted l 9 formula, then & h & 4[I * $ , if and only if ,. ( * $ Proof : We proceed by induction on the construction of . The only non-trivial case is when is of the form Jm1 / L 1 $ h & .& In L h this case, & ( L & 1 * $ , h if and . if only * $ ,, ( there is some such that and E'& & / & & / B & $ where . Therefore, by U h U TV* & U ( $ , induction& hypothesis, if and L h & L & 1 h only if* there4 is some such that and ( $ L JI $\ $ & 1)L I . Since , we have , & 1 I( & L & 1 TVU that L it is the$0 case I i and,L hence, $0 for every I that ( h +* & I & ( L $\ . Consequently, ( $ , & L h & L & 1 ( if and only if there is some such that and h * . 4 JI & / L $0.$ & / ( L $0.$ , , which means that h ( +* & $ 42!I CD ( $ , , as required. ( The next lemma yields an absoluteness property of l 9 inductions on near finite ranks and also reveals that every positive restricted l 9 formula has a unique fixed-point on near finite ranks. $ HL Lemma 3 Let be a l 9 formula over ^ U that is positive in the -ary relation symbol h I h I , and let be " a near finite rank such that 3 i i Jh $ GTV 4@U I . For ! . # " every and every , we have that / * JI $ Jh $ 4 I + JI $ is . /" 0 /" and 0 / " $ !h $ . If, inJh addition, 9 01/ a restricted l formula, then ."/ . Proof (sketch): The first statement is proved by induction on ! using absoluteness. For the second statement, if is 9 a positive L that Jh $ restricted Jh $ l formula, then it can be shown 05/ . / by induction on the maximum -rank & i B & '& & $ZL Jh $ 05/ of , where , and using U U CD Lemma 2. We now have 9 $ all the necessary tools to show that restricted l collapses to on the class d ^] of near finite ranks. Proof of Theorem 2: The keyidea the proof is that there 4 of $ is a first-order formula Jh $ that approximates the h U 1 0 / greatest fixed-point on near finite ranks ; the for4 Jh $ mula is based on the characterization of 0/ as the union of all post fixed-points of . This approximation can be$ improved to yield an exact first-order definition of 4 !h 05/ by first replacing every occurrence of in by , and then iterating a constant number of times. The result will then follow from Lemma 3, which asserts that the greatest fixed-point of the restricted l 9 formula coincides with its least fixed-point. This type of argument was used by Dawar et al. [DDLW98] n kto$ show that the least fixed-point of the l 9 formula in the beginning of Section 4 is first-order definable on d `] . Here, we have to deploy the machinery of Lemmas 1, 2 and 3 to show that the argument can be extended and applied to every restricted l 9 formula. $ We will construct a formula Jh $ U h L that defines the greatest fixed-point 0 / on every d `] such I h I that for some (for near finite$ h i * 1 i I TVU !h ranks below 8 we can obtain a l 9 definition of 0 / by iterating a sufficient number of times). The construction of is carried out in n $ three steps. For nthe first step, define a l 9 formula " expressing that is the encoding h on . We use the standard encodof a post fixed-point of HB n n ing of a pair by the set , and of B n $ a tuple U " by$?8 o B o . Then is the n rq U U L n $ o o n $ $ formula $ , n U U n where is a l 9 formula expressing that is a set of encodings of -tuples, from and o0/ $ o0/ by o0/ replac o,/ is obtained L n ing atomich formulas by & & . Let $ $ L be such U thath the set $ ( & i & is in ; it is not hard to show that $ h U * fixed-point of ( $ " h $ , if and only if $ is a4 post $ $ on . For the second step, let be the Jm n $ n $ 8 B U L n $ formula expressing " U B $ that belongs to some post fixed-point of . U 4 Jh $ 4[I 8 !h $ Claim 1: 0 / 0 / . TVU i i is l 9 , Lemma 3 implies that Proof of !h $ 4pClaim I 8 1: Since JI 8 $ 0 / 0 / Thus, first Bthe & $p L TVU & . g '& for V T U inclusion it suffices to show that if U !I 8 $ h 4 *& 0 / ( for the 8 a witness $ TV!mQU n ,$ then4 , . We !I need quantifier and in . Put $ 0 / $ . L TVI U I 8 h Since $ . Morei & L TVU , we have that & L i h over, since , we have that ; hence, $ L n $ * & h n $ *( ", . We can now show ( " ", that I 8 h using Lemma 1 and the fact that . For the TVUGJi h $ second inclusion, we use the fact that 0 / is the union h of all post fixed-points of on . 4 Thus, yields fixed!h $ an approximation of the greatest point 01/ of . For the third step, we iterate a number of times (in fact, times) to obtain 4 progressively $ 4 better $ approximations. Define 9 U ( U B 0 4,/ $ L 4,/ U U , for . Note that each is a formula. 4 / !h $ 4 I 8 !h $ / 1 0 / 05/ , for every Claim 2: B i T<UT i . is l 9 , Lemma 3 implies that Proof $ 45Claim / $ Jh of I 8 2: Since JI 8 / 05/ 05/ . We proceed by V T # U T . TVU#T induction on . Claim 1 takes care of the case Assume that& Claim in & for $ L 1 J.h For $ 4Othe & 2holds I first / 8 clusion, if 0 / , h * & JU h $ & L I / 4p h V T U T 8 then ( 0 / , . Since TVUT and is a restricted Lemma 2 implies that h * & / Jh $ 4 l 9 I formula, 8 / ( 0 , . Note that this is the T first place in this proof where we use the assumption that the l 9 $ formula By induction hypothesis, 4 / Jh 4 I 8 / is restricted. h * & 4 / ; therefore, 0 / ( ,, U that h ( 04 / $ * U & , T i It follows since is monotone. h 4 / *& U ( , . The second inclusion can be proved usand ing the induction hypothesis and the monotonicity of ; the details are left to the reader. 4 Q $ Finally, let be the formula 8 $ 2 U J.h Claim 0 / implies that defines the h greatest fixed-point I h I of on every near finite rank such that F$ i i GTVU 1 . Let for some be the dual formula * $ 3 3 of . Note that l 9 for$ Jh $ h is also !ah restricted mula. Moreover,$ 0/ . / (see [Mos74]). 1 !h Therefore, . / is definable $ taking the negation of !h by the $ formula that defines 0/ . The l $ definability Jh Jh / Jh $ of follows immediately, since . / by . / 0 CD Lemma 3. This completes the proof of Theorem 2. 4,/ and //0 $ U ( Example 1: In the full paper we show that, in spite of the stringent syntactic requirements imposed on restricted l 9 formulas, several 9 $ natural queries can be expressed using rel stricted formulas. Thus, Theorem 2 provides a versatile tool for showing that such queries are first-order definable on d `] . Here, we illustrate this technique by considering the query (finite) ordinal addition >, where $ L h Jh $ > if is a near finite rank, then > 3 h - k , , 3 are ordinals such that 3 . The stan( dard recursive specification of ordinal addition does not lead to a restricted l 9 formula. Consider, however, the follow > ing alternate recursive specification: if are ordinals, 3 - then 3 if and only if $ 7 8 8 V $(7 3 L!V 3 L $ $ L $ Jm !m Jm !m" L $ 3 3 ! 8 8 - 3 &8 3 3 " - 8 " - $B 3 This recursive specification can easily be transformed into the least fixed-point of a positive restricted l 9 formula that defines ># ; to this end, we use the fact that an or being - dinal is l 9 definable, and that . 5. Unary E and binary E $ l 9 denote For every positive , let -ary $ integer the fragment of l 9 that allows the formation B of$ least 9 fixed-points of positive l formulas such U that is a relation symbol of arity . The following simple observation reveals that the smallest of these fragments collapses to on d `] . $ Proposition 1 If is a unary l 9 formula that is positive in , then there is a unary restricted l 9 formula $ $ that is positive in and such that . / # $ . # . Consequently, unary / # 9 $ on every structure l collapses to on d ^] . $ Proof: Let be the restricted l 9 formula obtained $ $ by replacing every occurrence of$ from Q o o by , while preserving all occurrences with a bound variable. An easy induction shows that for every $ structure 3 $ we have that .1/ 4 # $ # and every $ ordinal / # . Theorem 2 implies then . . / 4 # ; thus . / # 9 $ CD that unary collapses to on d `] . l $ According toTheorem 1, if l 9 collapses to on & `] , then . The proof of this theorem makes a crucial use of the hypothesis that -ary 9 $ l collapses to for every $ + . In view of Proposition 1, one may investigate whether -ary l 9 collapses to on & `] for particular values of bigger than $ . The main result of this section is that binary 9 also collapses to on & `] . l $ 8 F Theorem 3 Let be a l 9 formula that is posU itive in the binary relation symbol . The least fixed-point of and the greatest fixed-point of are first-order definable on &^] . In fact, the least fixed-point is -definable, whereas the greatest fixed-point is -definable on & ^] . Proof (sketch): For simplicity, we 9 $ only sketch the proof for l the collapse of binary to on the smaller class `] of finite ranks. After presenting the sketch, we outline how the proof can be modified to establish the collapse on the class &^] . As in the proof of Theorem$ 2, the first key idea is that JI I the greatest fixed-point 0 / of on can be approximated by a 4 first-order 8 $ formula. In fact, we can start with the formula featured in that proof, because Claim U 1 uses only the hypothesis that is a l 9 formula (and not the additional hypothesis JI $ 4[ I 4in Theorem JI 2$ that is restricted). Thus 01/ it is not JI i 05/ $ . Moreover, I 4 i 05/ hard to verify that , because is an actual finite rank (instead of a near finite rank). Our goal is to im!I $ prove on this approximation 4,/ JI / $ of 0 4 / by defining / .formu/ las such that 0 , for Since T 8 $ may not be a restricted formula, a difficulty U arises from/# the in /!$of subformulas / oH$ presence o6 1.$ potential L2/of the formo , , and , where and is4 / a bound variable of . Actually, in building the formulas , the is caused by the subfor-o $ serious o difficulty / $ / oHmost is l 9 and mulas and . Note that, since o I , every element of is a bound variable of I witnessing / must be in choice , the set / oQ$ o6 of /!$ I U . Therefore, for every of elements of witnessing (or ) is a sub I I set of U and, hence, a member of . In turn, this makes I it possible to use first-order existential quantifiers over o to quantify the set of all such elements witnessing . Note that this would not be possible, if we had to work with an h arbitrary near finite rank , as the above set may not be in h . 4/ 4,/ We will build the desired formula from 4 4 U , for 9 , where we take to be . Let be the set // . " For each containing , we will 4 / 8$ k 8 $ i define a formula . Then each will be U U the formula / $ 8! / 8 $ 8 " 8 $ U U P / $ Jm o s$ !m o $ / L%o 8 where is the formula each U U o<L[o $ of containing UU TVU and is the set of4/ subsets 8 $ subformulas of . The first two 8 I U / are introduced ! to ensure that and belong to H6 / U n 1 1 T . Let be the be a new variable for each and let set 8 L ! . From now on, we use the abbreviations o o o 8 $ U 8 $ n n n n 8 n 8 $ 9 9 9 9 . Let 8 " n $ $ U " U $ , U !m U n $ " n , $ and be the formula " " and 1 1 where n $B 1 1 / / P / /G oH$ o n $#$B P n $ 1 1 is the formula n 1 1$ 1 1 q 9 L oH$ o n $ $0 4 / o n $ oQ$ 7 o6 n $ $ while is the formula 6 o n U where Jm1 9 L n 1 1 $ o 1 !/ / P o / 8 o 8 U /"$B U P !1 1 8 7 o 8 1 . $0 1 1 n 1 1 with 9 a fresh variable. Note that says that is the set of witnesses for bound variables that relate to 4 4 / U or to 8 . Since 9 is a formula, it is easy to see that each is also a formula. !I !/ $ 4 / B Claim 3: 0 / for . T 4 Proof of Claim , since 9 4 LC/3:/ The # claim holds for is . Fix assume that the claim holds ; we show that itandholds for for . We show first that 4 / 1 JI / $ & '& & 8 $@L I 8 0 / be such that T . Let U i 4 / & & L * L that * & definition of , it follows I 8 / , . I From " the , and with ( , for some . For i T L ! , let 1 1 @L I be a witness every 1 1 8 for the quanti mQn U " fier for in L , and let be theI corresponding n * & & 8 witness & 0$ , we have ( ,. . Since that U & B$ It suffices to show that is a post fixed-point of I on & ; then, the of and Lemma 3 will imply /$ JI monotonicity & 0$ that is in 0 / . We need to verify that & 0$ $ T L I 8 I * & i . Let be such that ,. ( the three disjuncts Then must satisfy one4,of of . As I / * sume first that ( 4 / $3 / Lemmas JI 2 and U , . Using 05/ and the induction hypothesis that T IU , 4 / U it can be verified that is a post fixed-point of on . U 4 / I * (4 / & 0$ , . But Thus and hence, ( i I U * U & 0$ , , as required. Asby monotonicity, ( sume now that satisfies the second disjunct. 8 L ! L 11 & Then, & 1 there & 1 $ exist and 1 9 1 such that . 1 1 U From the choice of and the definition of , we I & 1 & 1 & 0$ I know that ( * , ; therefore, ( B $ & +* , , as required again. The case of the third disjunct is handled in a similar manner. 4 / !I / $ . If Consider next & L JI / $ the inclusion I * & 0 / ! I T /!$ i / , then . Let be 0 / ( 0 , T T the biggest subset of containing and such that for every 8 L it is the case that I ( * & / & / 05/ JI / $ , . & & 8 T .$#8 U !I / $ 4 !I / Let $ be the set 01/ . T 8 L T U U I ( By / Lemma 2, for every we have that 1 1 & & / G U n 1 1 +* $ , . We construct witnesses for in " & B$ such that $ , which will prove the claim. L ! 1 1 Fix 1 1 U 8 & L and $ L if 8U , then I define / & & as1 follows: , then 1 1 & L I T / U[( & 1 & $L $ ; if $ . Using the induc T U^( tion hypothesis ?1 1and the definition of $ , it can be checked that the sets have the aforementioned properties. 4 8$ This concludes the proof that the $ formula JI 8 U $ defines the greatest fixed-point 0/ of on U ` ] . By applying the same argument to the dual formula 8 $ JI $ ./ , we establish that the least fixed-point $ U 8 of is definable on ^] . U Finally, we comment on the modifications needed to extend the proof to the class & `] . The first modification is that we have to add an extra iteration in the construction of the first-order 8 $ formula that defines the greatest fixedpoint of on &^] . Specifically, instead of four 4 (4 4 (4 4 U 8 formulas, we will need five formulas 9 ;4 the U greatest fixed-point will be defined$ by the last formula . 9 l In proving that binary collapses to on ^] , we used twice the assumption that we were working with I structuresL of the form for some *)+ , instead of struc h I h I & `] with 5i tures for some . 4 i $ * + G < T U J I 05/ . This can The first time was to ensure that 4 8 $ be taken care of by considering the formula I ( 4,/ U / / 8 $ 8 4 8 $ / $ . Here, is a formula /U $ U similar to the formula defined in the proof, inL h whose SI / / & ^ tended meaning is that for every / ] $ I h I a such that . / For$ the rest of the proof, [i TVU should be used in place of . h I The second time we used the assumption that for some * + that the existentially quantito n ensure n 9 was 9 8 can fied variables h be witnessed by elements of U these the universe of the structure . As mentioned earlier, h L d ` ] witnesses may noth exist within an arbitrary , or L even an arbitrary & ^ ] . When showing that binary 9 $ l collapses to on & `] , this difficulty is only encountered in4 the last iteration, that is, in the correctness of the formula . This obstacle, however, can be overcome by using the fact that the witnesses can be restricted to be 8 subsets of the transitive closure of the arguments , & U where the transitive closure of a set is defined inductively & $ & N B$ fL & . The reaas follows: ( son is that, since is a l 9 formula, the interpretation of every bound variable of can be restricted to the transitive closure of the sets that interpret the free variables /1 of . Consequently, we may split de / 1 the 08/ 1 witnesses / 1 fined in the proof into three sets U with the fol / 1 / 1 4 & 08/ 1 / 1 4 & 8 lowing U , and / 1 interpretations: / 1 4 # & $ '& 8 $#$ U ,& & 8 $$ . Observe U U L h1 / 1 08/ 1 L h h L & & & `] and U 8 that if , then U , h & '& $ 3 is closed because under taking subsets (if , then /1 L h i B$ W U W U ); observe also that , because its & rank is less than the maximum of the ranks of and . The 08/ 1 /1 /1 U sets , , and capture all relevant information about /1 ; therefore, only these elements need to be existentially quantified. Clearly, several changes in the formulas have to be made; the technical details will be included in the full CD version of the paper. $ 8 The inquisitive reader may wonder whether the argument of Theorem 3 extends to higher arities. The answer is that it does not, for the reason that we cannot encode binary reI I lations on it is possible U by elements of I , whereas I to encode unary relations (sets) on by elements of . U We note, however, that Theorem 3 extends to binary l 9 forL mulas over an expanded vocabulary that, in addition to , contains other relation symbols, as long as they are interpreted by l -definable queries on & ^] . Example 2: The aforementioned extension of Theorem 3 can be used to show that the Even Cardinality query > Jh $ & L h & ( the cardinality of is even is first-order definable L write ` n on$ & ^] . For this, oneHcan a l 9 formula over the vocabulary , n whose least fixed-point defines the binary query “ is an even element of with respect to the linear order ” on &^] . Here, is interpreted by the l -definable linear order on & `] described in the beginning of Section 4. 6. Closure functions of formulas B $ Let be an arbitrary positive first-order U formula. From the preliminaries, recall that if # " is a finite $ structure, then / # is the smallest integer that . # $ N 8 . # $ . In general, the rate ofsuch growth / / $ of / # can be as high as a polynomial in the cardinality of the universe of # . Here, we analyze the rate of growth of the closure on finite of positive l 9 formulas JI LM$ function is a positive ranks ,* + .JI We establish that if $ l 9 formula, then I / I is bounded by a polylogarithm of the cardinality ( ( of . Moreover,$ we show that if !I is a restricted l 9 formula, then 2/ I is bounded by the iterated logarithm of ( ( . $ B Theorem 4 Let be a l 9 formula that is U positive in the -ary relation symbol . Then, for all suffi ciently large * , we have 8 I 3 !I $ 3 I $0 V T U 2 / ( ( U #"$ ( ( * U J I $ 3 l 9 formula, then 2 / Moreover, 3O - if I is a$ restricted #"$ ( ( . * Proof (Sketch): We outline the proof of the first statement only; an easier proof. JI $ 3 the second !I $ has -Y 8 I We show that 2 / 2 / V T U ( U U ( U holds for all 8 by expanding . The result will follow 3 recurrence *e+ I this I $ 8 V T U U and using the fact that ( ( "$ I ( ( * U TVI U ( U ( U , for all sufficiently large * . Put ! " on and let be the closure ordinal of !I $ T " JI U $. . ItSo,is . enough to show that L . / T " TVU / & !I $ i let us assume I * & thatJI $ . / T " TVU , which means I that ( . / T " , . Lemma 2 implies that & JI $ 4 !I ( & B &1 .$ +* . / T " that Jclaim I $ JI $ 4f JI U& U & $ . , .T We . / T " " U . This U U i / will prove I our goal, & because the monotonicity of & im $ L ! I plies that$ . T " JI . ( * . / T " U , and, therefore, / ofJI " $ , it is easy to show Using Lemma the choice JI $ 4e3I and that . / T " . T " U . So, to prove !I $ the 4 U i / those tuples claim, it remains to consider in . T " & & .$ JI !I / $ 4 I that are not in . / JT I " $ 4ZJI U . U U For each + , let $ be the set T U 3 & & .$ JI $ 4 I .. /First . / T note that ( $ " ( 1 U B . $ !I & &1 I U ( 1 U ( . A simple Jcounting U U I the cardinality of the set argument & & shows .$ that I U is U 1 U / / I 3 I ( ( ( ( / / U U U ! U U 3 Therefore ( $ " ( JI $ ! . Now let + be the smallest integer such that $ because $ TVU . Such an exists, . T / JI $ eventually reaches the fixed-point ."/ R 9 . If + ! , then a $ U a a the sequence of proper inclusions a $ " $ " holds. ( $ ( ! Hence , which contradicts the fact that 3 ( $ "9( ! ! . Thus, must hold, from which it JI $ CD follows that $ " $ TVU $ $ " U . / T " U i . In the full paper, we provide examples showing that the 9 above bounds are tight. n >$ As regards restricted l formulas, the formula for the linear order is such an example. We conclude by pointing$ out that the above Theorem 4 9 implies that every l -definable query on `] is in , the parallel complexity class consisting of all queries computable in polylogarithmic time using a polynomial number of processors (see [Pap94, GHR95, BDG90] for a thorough coverage of ). This suggests the systematic pursuit of descriptive complexity in the context of finite set theory. Acknowledgment: We are grateful to Moshe Y. Vardi for his constructive criticism and comments on an earlier draft of this paper. References [Bar75] J. Barwise. Admissible Sets and Structures. Springer-Verlag, 1975. [BDG90] J. L. Balcázar, J. Dı́az, and J. Gabarró. Structural Complexity II. Springer-Verlag, 1990. [BIS90] D. A. M. Barrington, N. Immerman, and H. Straubing. On uniformity within NC U . Journal of Computer and System Sciences, 41:274– 306, 1990. [DDLW98] A. Dawar, K. Doets, S. Lindell, and S. Weinstein. Elementary properties of finite ranks. Mathematical Logic Quarterly, 44:349–353, 1998. [DH95] A. Dawar and L. Hella. The expressive power of finitely many generalized quantifiers. Information and Control, 123:172–184, 1995. [DLW96] A. Dawar, S. Lindell, and S. Weinstein. First order logic, fixed point logic and linear order. In Computer Science Logic ’95, volume 1092 of Lecture Notes in Computer Science, pages 161–177. Springer-Verlag, 1996. [GHR95] R. Greenlaw, H. J. Hoover, and W. L. Ruzzo. Limits to parallel computation : Pcompleteness theory. Oxford University Press, 1995. [GIS94] Y. Gurevich, N. Immerman, and S. Shelah. McColm’s conjecture. In Proc. 7th IEEE Symp. on Logic in Comp. Sci., pages 10–19, 1994. [GS86] Y. Gurevich and S. Shelah. Fixed-point extensions of first-order logic. Annals of Pure and Applied Logic, 32(3):265–280, 1986. [Imm86] N. Immerman. Relational queries computable in polynomial time. Information and Control, 68:86–104, 1986. [Imm89] N. Immerman. Expressibility and parallel complexity. SIAM Journal on Computing, 18:625–638, 1989. [KV92] Ph. G. Kolaitis and M. Y. Vardi. Fixpoint logic vs. infinitary logic in finite-model theory. In Proc. 6th IEEE Symp. on Logic in Comp. Sci., pages 46–57, 1992. [KV96] Ph. G. Kolaitis and M. Y. Vardi. On the expressive power of variable-confined logics. In Proc. 9th IEEE Symp. on Logic in Comp. Sci., pages 348–359, 1996. [Lin92] S. Lindell. A purely logical characterization of circuit uniformity. In Proc. 7th Annual IEEE Structure in Complexity Theory Conference, pages 185–192, 1992. [McC90] G. L. McColm. When is arithmetic possible? Annals of Pure and Applied Logic, 50:29–51, 1990. [Mos74] Y. N. Moschovakis. Elementary Induction on Abstract Structures. North Holland, 1974. [Pap94] C. H. Papadimitriou. Computational Complexity. Addison-Wesley Publishing Company, 1994. [Saz97] V. Yu. Sazonov. On bounded set theory. In M.L. Dalla Chiara et al., editor, Logic and Scientific Methods, Volume One of the Tenth ICLMPS, pages 85–103. Kluwer, 1997. [Var82] M. Y. Vardi. The complexity of relational query languages. In Proc. 14th ACM Symp. on Theory of Computing, pages 137–146, 1982.