Ph. Kolaitis and M. Vardi. Infinitary logics and 0--1 laws. Information and Computation, 98:258--294, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the query language datalog, and practically every aspect of it expressive power, optimization, adding negation, implementation techniques was the subject of numerous papers. For the study of expressive power of query languages, which will interest us most in this paper, a very nice result of [28] showed that the infinitary logic with finitely many variables, L 1 , has a 0 1 law over finite structures. As many fixpoint logics can be embedded into it, this result gives many expressivity bounds for datalog like languages. In the presence of an order relation, it is again a classical ....

Ph. Kolaitis, M. Vardi. Infinitary logic and 0-1 laws. Information and Computation, 98 (1992), 258--294.

.... of rst order logic with a transitive closure operator in [Talanov 81] for the more expressive extension of rst order logic with a xed point operator in [Blass, Gurevich and Kozen 85] later subsumed by the 0 1 law for the in nitary logic over bounded number of variables L 1; proved in [Kolaitis and Vardi 92] for some pre x de ned fragments of monadic second order logic in [Kolaitis and Vardi 90] who also established strong relations between decidability and 0 1 laws of such fragments; for modal logic in [Halpern and Kapron 94] The Modal Logic of the Countable Random Frame 3 In general, however, ....

Ph. G. Kolaitis and M. Y. Vardi, 01 laws for innitary logics, Information and Computation, 98, 1992, pp. 250{ 294.

....the query language datalog, and practically every aspect of it expressive power, optimization, adding negation, implementation techniques was the subject of numerous papers. For the study of expressive power of query languages, which will interest us most in this paper, a very nice result of [28] showed that the infinitary logic with finitely many variables, L 1 , has a 0 1 law over finite structures. As many fixpoint logics can be embedded into it, this result gives many expressivity bounds for datalog like languages. In the presence of an order relation, it is again a classical ....

Ph. Kolaitis, M. Vardi. Infinitary logic and 0-1 laws. Information and Computation, 98 (1992), 258--294.

....of variables and varying quantifierdepth. Although these early results from descriptive complexity theory have been put in much more elegant forms later using so called fixed point logics, the importance of the number of variables as a complexity measure remained. In 1990, Kolaitis and Vardi [51] proved a 0 1 law for the infinitary finite variable logics. As a corollary, they re proved a result of Blass, Gurevich, and Kozen [7] that there is a 0 1 law for least fixed point logic. The fact that makes Kolaitis and Vardi s paper so remarkable is that it uses finite variable logics as a ....

....that for all graphs A; B 2 C we have A = B ( A C 2 B: Remark 2.6. There is no analogous result without counting. On the contrary, if C is a class of graphs such that for all A; B 2 C we have A = B ( A L k B then (C) 0. This follows easily from results of Kolaitis and Vardi [51]. 2.3. Model checking. For any logic L, we let MODEL CHECKING FOR L be the following problem: Input: A sentence 2 L and an fEg structure A. Question: Does A satisfy We could actually define a separate MODEL CHECKING problem for each vocabulary , but this would not give us any further ....

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Ph. G. Kolaitis and M. Y. Vardi. Infinitary logic and 0-1 laws. Information and Computation, 98:258--294, 1992.

....is a notion intermediate between finite axiomatisability and full axiomatisability. Finite variable axiomatisability does not in itself capture any interesting complexity class. On the one hand there are undecidable classes of finite structures that are axiomatisable with just two variables (see [41] for examples) while on the other hand, some classes of very low complexity are not axiomatisable with a finite number of variables. Nonetheless, the notion is still interesting from two points of view: 1) many complexity theoretic questions can be reproduced here, as we shall see; and 2) many ....

.... is equivalent, over C to a formula of L 1 . Thus, in particular when C is the class of all finite structures (or indeed any class of structures with bounded cardinality) then every formula of LFP is equivalent over C to a formula of L 1 . This was pointed out by Kolaitis and Vardi in [41]. A variant of fixed point logic, known as PFP (for partial fixed point logic) was introduced in [3] Here, we allow the formation of formulas pfp R;x ( t) even when is not R positive. Since may not be monotone, it may not have a least fixed point (or any fixed point, for that matter) ....

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Ph.G. Kolaitis and M.Y. Vardi, Infinitary logics and 0-1 laws, Information and Computation, 98(2):258--294, 1992.

....languages rely on the socalled extension axioms. Suppose we have atomic types s(x 1 ; x k ) and t(x 1 ; x k ; x k 1 ) such that s ae t. Then, in a random structure A, we can almost surely extend every realization u 2 s A to a realization (u; u k 1 ) 2 t A . Kolaitis and Vardi [10] established a 0 1 law for the infinitary logic L 1 . In fact they proved a stronger result: Every formula in L 1 is almost surely equivalent to a (finite) disjunction of atomic types. Theorem 3.2 For every formula (x) in the infinitary logic L 1 there exist atomic types t i (x) ....

Ph. Kolaitis and M. Vardi, Infinitary logic and 0-1 laws, Information and Computation 98 (1992), 258-294.

....infinitary logic and fixpoint logic. Using these games, Kolaitis and Vardi [27] have established, e.g. the 0 1 law for infinitary logic. Also, variants of Ehrenfeucht Fraiss e games have been designed and used for monadic second order logic [12] firstorder logic with counting [6] Datalog [29, 28], fragments of fixpoint logic [31] and transitive closure logic [14] We will be using a variant of the game in [31] Definition 8 Suppose we have two structures A and B of the same vocabulary oe. Let c 1 ; c s and d 1 ; d s be the interpretations of the constants of oe in A and ....

....in a paper by Barwise [3] although pebble games are not explicitly mentioned there. In more explicit form it is proved by Immerman in [19] but only the first order fragment of L k 1 is mentioned. The most accessible proof of Theorem 4 is probably the one given by Kolaitis and Vardi in [28]. 2 As usual we denote the Spoiler by male and the Duplicator by female pronouns. 9 By imposing restrictions on the admissible sequences of 9 and 8 moves one defines games that characterize the expressive power of quantifier classes in L k 1 : Let P f9; 8g be closed under subwords. ....

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Ph. Kolaitis and M. Vardi, Infinitary logics and 0-1 laws, Information and Computation 98 (1992), 258--294.

....indexable, then P 6= NP. 13 Chapter 2 Infinitary Logic and Element Types The logics LFP and PFP, introduced in Section 1.3, can be viewed as fragments of an infinitary logic in which each formula has only finitely many distinct variables. This was the view taken by Kolaitis and Vardi [Kolaitis and Vardi, 1992b] who established some results about the expressive power of this logic on finite structures. Among other things, they gave a characterization of equivalence of finite structures in this logic. Inspired by this work, we undertook a study of the application of notions from classical model theory ....

....note that there are only finitely many maps from subsets of A into B. Thus, one of the sets in the chain must be repeated, and hence, can be repeated indefinitely. Writing L k for the fragment of first order logic with at most k variables, we obtain the following corollary: Corollary 2. 6 ( Kolaitis and Vardi, 1992b] For finite structures A and B , the following are equivalent: ffl For every sentence OE 2 L k 1 , A j= OE, if and only if, B j= OE ffl For every sentence OE 2 L k , A j= OE, if and only if, B j= OE When the sequence of sets of partial isomorphisms is finite, we can view it in terms of ....

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Ph. G. Kolaitis and M. Y. Vardi. Infinitary logics and 0-1 laws. Information and Computation, 98(2):258--294, 1992.

....of equalities x i = x j and inequalities x i 6= x j , where 1 6 i j 6 k, which defines on every structure A the set e A = fu 2 A k : A j= e(u)g. For each atomic type t let r t be the unique natural number of pairwise distinct components of each tuple of type t. Theorem 2. 5 (Kolaitis, Vardi [9]) For every formula (x) in the infinitary logic L 1 there exist atomic types t i (x) such that A j= 8x( x) i t i (x) for all but an exponentially decreasing fraction of structures A when the size of A tends to infinity. Lemma 2.6 For each n 2 N, let G = V; E) be a random graph of ....

Ph. Kolaitis and M. Vardi. Infinitary logics and 0--1 laws. Information and Computation, 98:258--294, 1992.

....the query language datalog, and practically every aspect of it expressive power, optimization, adding negation, implementation techniques was the subject of numerous papers. For the study of expressive power of query languages, which will interest us most in this paper, a very nice result of [19] showed that the infinitary logic with finitely many variables, L 1 , has a 0 1 law over finite structures. As many fixpoint logics can be embedded into it, this result gives many expressivity bounds for datalog like languages. In the presence of an order relation, it is again a clas1 sical ....

Ph. Kolaitis, M. Vardi. Infinitary logic and 0-1 laws. Information and Computation, 98 (1992), 258--294.

....graph or a well behaved depth function. m = 9 =1 x 0 (x) 8x i6m i (x) 9x m (x) axiomatizes those of these graphs that have overall depth m. For any set U of natural numbers, the disjunction W m2U m defines the class of those layered rooted directed graphs whose depth is in U . Also cf. [11] for similar examples using, however, 3 variables. 6 Outlook and open problems The major question that remains open is, whether similar results can be obtained for arbitrary k instead of the still rather special case k = 2. More precisely, we have the following problems for arbitrary k: 1) Do ....

Ph. Kolaitis, M. Vardi, Infinitary Logic and 0-1 Laws, Information and Computation 98 (1992), 258-294.

.... of rst order logic with a transitive closure operator in [Talanov 96] for the more expressive extension of rst order logic with a xed point operator in [Blass, Gurevich and Kozen 85] later subsumed by the 0 1 law for the in nitary logic over bounded number of variables L 1; proved in [Kolaitis and Vardi 90] for some pre x de ned fragments of monadic second order logic in [Kolaitis and Vardi 90] who also established strong relations between decidability and 0 1 laws of such fragments; for modal logic in [Halpern and Kapron 94] However, in general, the 0 1 law turns out to be rather a rare ....

.... extension of rst order logic with a xed point operator in [Blass, Gurevich and Kozen 85] later subsumed by the 0 1 law for the in nitary logic over bounded number of variables L 1; proved in [Kolaitis and Vardi 90] for some pre x de ned fragments of monadic second order logic in [Kolaitis and Vardi 90] who also established strong relations between decidability and 0 1 laws of such fragments; for modal logic in [Halpern and Kapron 94] However, in general, the 0 1 law turns out to be rather a rare phenomenon than a rule. It can be easily seen that the presence of a single constant in the ....

[Article contains additional citation context not shown here]

Ph. G. Kolaitis and M. Y. Vardi, 0-1 laws for innitary logics, , Information and Computation, 98, 1992, 250294.

....been introduced, including least, inflationary and partial fixpoint logics, as well as transitive closure logics, cf. 1; 8] Fixpoint logics can all be embedded into L 1 , infinitary logic with finitely many variables, which is much easier to analyze. In particular, L 1 has a 0 1 law [23], which gives a uniform derivation of the 0 1 law for all fixpoint logics. It follows that L 1 cannot express most counting properties, such as parity of cardinality. The theory extends nicely to the ordered setting, where transitive closure and fixpoint logics capture familiar complexity ....

Ph. Kolaitis, M. Vardi. Infinitary logic and 0-1 laws. Information and Computation, 98 (1992), 258--294. 24 \Delta

....been introduced, including least, inflationary and partial fixpoint logics, as well as transitive closure logics, cf. 1, 8] Fixpoint logics can all be embedded into L 1 , infinitary logic with finitely many variables, which is much easier to analyze. In particular, L 1 has a 0 1 law [21], which gives a uniform derivation of the 0 1 law for all fixpoint logics. It follows that L 1 cannot express most counting properties, such as parity of cardinality. The theory extends nicely to the ordered setting, where transitive closure and fixpoint logics capture familiar complexity ....

Ph. Kolaitis, M. Vardi. Infinitary logic and 0-1 laws. Information and Computation, 98 (1992), 258--294.

.... same is true (see Corollary 11) While all these logics are relevant to database query languages (as we shall see shortly) we shall also consider infinitary logic, which is of interest in finite model theory, as logic which subsumes fixpoint logics and possesses nice properties, such as 0 1 law [39]. It is defined exactly as first order logic, except that arbitrary disjunctions and conjunctions are allowed. That is, if f i ( x)g is an arbitrary collection of formulae, then W i i ( x) and V i i ( x) are formulae. We use L1 to denote infinitary logic. Suppose L is one of the ....

....with fi i ( z) Now a straightforward induction on the structure of shows that for any D 2 Inst(SC; X) and for any c over X, D j= c) 0 ( c) This shows that L 1 has the total Ramsey property over M, and thus it has generic collapse over M. 2 Using the 0 1 law for infinitary logic [39], we obtain: Corollary 4 The parity test is not definable as a L 1 (hR; i) query. 2 3.3 Equivalence results and extensions to algebra We now want to extend the results above to the setting of relational algebras. In this section we show that a number of well known results on equivalence ....

Ph. Kolaitis, M. Vardi. Infinitary logic and 0-1 laws. Information and Computation, 98 (1992), 258--294.

....the query language datalog, and practically every aspect of it expressive power, optimization, adding negation, implementation techniques was the subject of numerous papers. For the study of expressive power of query languages, which will interest us most in this paper, a very nice result of [19] showed that the in nitary logic with nitely many variables, L 1 , has a 0 1 law over nite structures. As many xpoint logics can be embedded into it, this result gives many expressivity bounds for datalog like languages. In the presence of an order relation, it is again a clas1 sical ....

Ph. Kolaitis, M. Vardi. Innitary logic and 0-1 laws. Information and Computation, 98 (1992), 258-294.

....clear if the same is true. While all these logics are relevant to database query languages (as we shall see shortly) we shall also consider infinitary logic, which is of interest in finitemodel theory, as logic which subsumes fixpoint logics and possesses nice properties, such as 0 1 law [27]. It is defined exactly as first order logic, except that arbitrary disjunctions and conjunctions are allowed. That is, if f i ( x)g is an arbitrary collection of formulae, then W i i ( x) and V i i ( x) are formulae. We use L1 to denote infinitary logic. Suppose L is one of the logics ....

....indiscernibles [13] with respect to a countable collection of formulae, if the signature is analytic. This gives us: Proposition 5 Let M = hR; Omega i where Omega is analytic, and has countably many symbols. Then L 1 has generic collapse over M. 2 Using the 0 1 law for infinitary logic [27], we obtain: Corollary 4 The parity test is not definable as a L 1 (hR; i) query. 2 3.4 When interpreted structure matters: query safety We now want to give the reader first indication that the kind of an interpreted structure one adds, can make a difference, even in the active case. ....

Ph. Kolaitis, M. Vardi. Infinitary logic and 0-1 laws. Information and Computation, 98 (1992), 258--294.

....the query language datalog, and practically every aspect of it expressive power, optimization, adding negation, implementation techniques was the subject of numerous papers. For the study of expressive power of query languages, which will interest us most in this paper, a very nice result of [27] showed that the infinitary logic with finitely many variables, L 1 , has a 0 1 law over finite structures. As many fixpoint logics can be embedded into it, this result gives many expressivity bounds for datalog like languages. In the presence of an order relation, it is again a classical ....

Ph. Kolaitis, M. Vardi. Infinitary logic and 0-1 laws. Information and Computation, 98 (1992), 258--294.

....and bound, are among x 1 ; x k : Let L k 1 be the closure of L k under the operations of conjunction and disjunction applied to arbitrary (finite or infinite) sets of formulas. Let L 1 = S k2 L k 1 : The logic L 1 was introduced by Barwise in [2] Kolaitis and Vardi [17] showed that LFP and PFP are fragments of L 1 on the class of all finite structures. 2.2 Generalized Quantifiers Let C be any collection of structures over the signature oe = hR 1 ; Rm i (where R i has arity n i ) that is closed under isomorphism. We associate with C the ....

....2 F . Corollary 3.4 For every formula of PFP(Q) there exists a formula of FO(Q) in flat normal form such that and are equivalent on the class of pure sets. In particular, PFP(Q) collapses to FO(Q) on pure sets. Proof. A straightforward modification of the proof that PFP L 1 (see [17]) shows that PFP(Q) L 1 (Q) Since each formula of PFP(Q) contains only finitely many different quantifiers, we actually get the inclusion PFP(Q) L (Q) Hence the claim follows from Theorem 3.3. Note that Corollary 3.4 implies the same flat normal form also for formulas of IFP(Q) ....

Ph. G. Kolaitis and M. Y. Vardi. Infinitary logics and 0-1 laws. Information and Computation, 98(2):258--294, 1992.

....structures, and hence is of no interest. However, the restriction L 1 , consisting of all formulas that involve only a finite number of distinct variables, has interesting model theoretic properties, and is still strong enough to simulate fixed point operations. Indeed, Kolaitis and Vardi [16] proved that PFP L 1 . In accordance with Definition 1, L 1 (Q 1 ) denotes the extension of this logic with all cardinality quantifiers and their vectorized versions. However, it makes no difference whether we allow vectorization or not, as we will see below. To adapt this infinitary ....

Ph. G. Kolaitis and M. Y. Vardi. Infinitary logics and 0--1 laws. Information and Computation, 98:258--294, 1992.

....of Vardi and Wolper extends as well, and with a PLTL formula #, one can associate a Bchi automaton with 2 states (see [LPZ85, VW86] so that PLTL model checking can be done in time O( S ) as for LTL and CTL # . The interested, or unconvinced, reader will nd a longer argumentation in [LS00a] Proposals with branching past can be found e.g. in [Rei89, Wol89, Sti92, Kam94, KP95] These de nitions for PCTL and PCTL # are from [LS95] These logics are equivalent to the CTL lp and CTL # lp (lp for linear past ) from [KP95] Our PCTL # further coincides with the OCL logic from [ZC93] ....

.... for PCTL and PCTL # are from [LS95] These logics are equivalent to the CTL lp and CTL # lp (lp for linear past ) from [KP95] Our PCTL # further coincides with the OCL logic from [ZC93] The PCTL # from [HT87] di ers from our PCTL # since its path quanti ers always forget the past (see [LS95, LS00a] for a formalism allowing both cumulative and forgettable past) An empirical observation is that, in most cases, linear time logic with past is not more di cult than its pure future fragment. For example, Theorem 4.4 can be extended to: Theorem 5.6 [Mar02] The model checking problem for L(F, F ....

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F. Laroussinie and Ph. Schnoebelen. Specication in CTL+Past for veri- cation in CTL. Information and Computation, 156(1/2):236263, 2000.

....that (Spec1) holds after any reset, the formula G reset ) G(alarm ) F is not exactly what we aim at. With (Spec2) a problem that occurred before the reset may account for the alarm ringing, which is probably not what we had in mind. For this kind of situations, Laroussinie and Schnoebelen [15, 16] proposed to use a new modality, N (read Now , or from now on ) that forgets all the past moments (see below for the formal semantics) With N, one can state the intended property of the alarm example via e.g. G reset ) NG(alarm ) F Not much is known about N, except that the ....

....for the formal semantics) With N, one can state the intended property of the alarm example via e.g. G reset ) NG(alarm ) F Not much is known about N, except that the translations of LTL Past and CTL Past into (resp. LTL and CTL carry over to LTL Past Now and CTL Past Now [16]. Our contribution. In this paper, we investigate NLTL, i.e. LTL Past Now, and give automata theoretic decision procedures for model checking and satisfiability. These algorithms run in EXPSPACE, which is optimal: we show that both satisfiability and model checking are EXPSPACE complete for ....

[Article contains additional citation context not shown here]

F. Laroussinie and Ph. Schnoebelen. Specification in CTL+Past for verification in CTL. Information and Computation, 156(1/2):236--263, 2000.

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Ph. Kolaitis and M. Vardi. Infinitary logics and 0--1 laws. Information and Computation, 98:258--294, 1992.

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Ph. Kolaitis, M. Vardi, Infinitary Logic and 0-1 Laws, Information and Computation 98 (1992), 258-294.

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