Ajtai M. and Gurevich Y. [1987], `Monotone versus positive', Journal of the ACM 34, 1004{ 1015.  Home/Search   Document Not in Database   Summary   ACM   TOC   Related Articles   Check

....are known to fail, including, for example, the L osTarski Theorem [6] It is not known, however, whether Lyndon s Theorem holds with respect to finite structures. It is also not known whether the L os Tarski Lyndon Theorem holds with respect to finite structures (for a closely related failure, see [2]) Resolving the latter problem is an important and well known open question in finitemodel theory (see Problem 1.9 on the finite model theory website and an incorrect claim in [11] On one hand, the class of existential positive 1st order formulas corresponds to the class of ....

M. Ajtai and and Y. Gurevich, "Monotone versus positive," J. ACM 34(1987), 1004--1015.

....nes a class that is monotone in . Then is equivalent to a sentence in which every relation symbol in occurs positively. Corollary 2.2.2. Let be a sentence that de nes a class that is closed under surjective homomorphisms. Then is equivalent to a positive sentence. Ajtai and Gurevich [1] showed that the rst statement fails over the class of nite structures. Stolboushkin [68] gave a simpler counterexample, which can be modi ed to show that the second statement also fails (see [60] 2.3 Preservation under homomorphisms In this section we discuss what is certainly the most ....

M. Ajtai and Y. Gurevich. Monotone versus positive. Journal of the ACM, 34:1004-1015, 1987.

.... superset closed DML sentence is equivalent to some X positive DML sentence. The proof relies on Stolboushkin s refutation [13] of Lyndon s Lemma (that every monotone first order property is expressible positively) for finite models. Although Lyndon s Lemma was first refuted for finite models in [2], we rely on Stolboushkin s construction of a FO sentence Omega in the signature hH; i, where H and are two binary predicate symbols, such that Omega is finitely monotone, 3 but Omega is finitely equivalent to no positive FO sentence. In general, the failure of Lyndon s Lemma for DML ....

M. Ajtai and Y. Gurevich. Monotone versus positive. Journal of the ACM, 34(4):1004--1015, 1987.

....oe has the property that every substructure of a model of oe is a model of oe, then oe is equivalent to a universal sentence. 23 This theorem fails for finite structures [Tai59] see also [Gur84] Still other such results are much harder to prove. A nice example is Ajtai and Gurevich s result [AG87] that Lyndon s Theorem (which says that monotone and positive classes coincide) fails for finite structures. The second line of research could be called preservative; here we consider theorems of model theory that continue to hold for finite model theory. Again, some such results are easy (such ....

M. Ajtai and Y. Gurevich. Monotone versus positive. Journal of the ACM, 34:1004--1015, 1987.

....perspective it is fairly rich. We note that C d strictly includes the class of depth d; size 2 O( log n) 1= d 1) circuits on 2 O( log n) 1= d 1) variables which contain only unbounded fanin AND and OR gates. This follows from results of Okol nishnikova [28] and Ajtai and Gurevich [1] (see also [8] Section 3.6) which show that there are monotone functions which can be computed by AC 0 circuits but are not computable by AC 0 circuits which have no negations. 4 Product Distributions A product distribution over f0; 1g n is characterized by parameters 1 ; n ....

M. Ajtai and Y. Gurevich. Monotone versus positive, J. ACM 34(4) (1987), 10041015.

....theory perspective it is fairly rich. We note that C strictly includes the class of depth d; size 2 O( log n) 1= d 1) circuits on 2 O( log n) 1= d 1) variables which contain only unbounded fanin AND OR gates. This follows from results of Okol nishnikova [24] and Ajtai and Gurevich [1] (see also [7] Section 3.6) which show that there are monotone functions which can be computed by AC 0 circuits but are not computable by AC 0 circuits which have no negations. 4 Learning under Product Distributions A product distribution over f0; 1g n is characterized by parameters 1 ; ....

M. Ajtai and Y. Gurevich. Monotone versus positive, J. ACM 34(4) (1987), 1004-1015.

....fixed points can be expressed with a single least fixed point application. One problem is that fixed point operations can be computed most easily for monotone formula, but monotonicity is undecidable [32] Also, not every formula monotone in its parameter P is equivalent to a formula positive in P [5]. Since positivity is decidable, it is fortunate that Gurevich and Shelah [34] proved that, for every monotone formula y, there is a positive formula y such that y and y have the same least fixed point. In the pragmatic direction, database researchers have been looking for efficient ways of ....

Ajtai, M and Gurevich, Yuri. "Monotone versus Positive". JACM 34, 4 (Oct. 1987), 1004-1015.

....example, 18] exhibits a monotone function in NC 1 that is not in monotone NC. In [16] Razborov shows that monotone Boolean circuits cannot compute the Boolean permanent function within polynomial size. But this function is exactly BPM and is known to be in P [11] Theorem 5. 2 AC 0 6= AC 0 [2], NC k 6= NC k [18] NL 6= NL [18] SAC k 6= SAC k [18] AC k 6= AC k [18] P 6= P [16, 11] 14 Even though NC 1 6= NC 1 it is perhaps natural to ask how much resources are necessary for functions in NC 1 to be computable non cancellatively. So for example, is NC 1 P Since ....

M. Ajtai and Y. Gurevich, Monotone versus positive, J. Assoc. Comput. Mach., 34:4 (1987), 1004-1015.

....of 92 93, I taught a graduate course in Finite Model Theory at the Mathematics Department of UCLA. Although the experience was very satisfying, at least for me, I continued to feel somewhat unsatisfied about a few things, among them the most important for me was the result by Ajtai and Gurevich [2] that Lyndon s Lemma fails for finite models. The classical result by Roger Lyndon [9] is that any first order formula, monotone in a certain predicate, is equivalent to some formula of the same signature that is positive in the predicate. Apart from the fact that the lemma is used in proofs of ....

....mentioned that, unlike most other classical results that discontinue to hold for finite models, and whose refutation is kind of immediate (Compactness, Craig Interpolation, Godel Completeness, etc. Lyndon s Lemma resisted all attempts to refute it for many years. And the proof Ajtai and Gurevich [2] finally came up with is hairy indeed. Not only that the proof is hard and extremely involved (which one might expect, given that among immediate corollaries to the refutation was the result of Furst, Saxe, and Sipser [6] see also Ajtai [1] that, basically, constantdepth positive simulation of ....

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M. Ajtai and Y. Gurevich. Monotone versus positive. Journal of the ACM, 34:1004--1015, 1987.

....and B = A;R 0 1 : R 0 m on the same universe it holds that if R 1 R 0 1 ; Rm R 0 m then A 2 S implies B 2 S. 1 A formula 2 FO(oe) is positive, if all predicates from oe occur only positively, i.e. within the scope of an even number of nested negation symbols. In [1], Ajtai and Gurevich constructed an example of a monotone, finitely) first order definable set of finite structures over some signature oe which they showed could not be defined by a positive first order formula. An alternative, considerably simpler example was recently given by Stolboushkin, ....

....[7] that no polynomial size family of Boolean circuits can solve BPM, the problem of deciding whether a bipartite graph has a perfect matching, which is well known to be solvable in polynomial time. This gives us an analogue to Ajtai and Gurevich s result for positive vs. monotone first order [1]. Let monP be the class of monotone problems in P. 4.1 Corollary. posPaemonP. 2 So none of the positive definitions in the previous section can capture monP. But that does not rule out the existence of a syntactic definition of monP. Since BPM is not contained in posP, the first idea is the ....

M. Ajtai and Y. Gurevich. Monotone versus positive. Journal of the ACM, 34:1004--1015, 1987.

....[25, 14, 24, 29] padding this implies that NC 1 ffl mono 6ae mP . This separation of circuit classes was strengthened to a properly exponential gap by Tardos [43] by showing that a good enough monotone approximation to clique is computable in P . mAC 0 6= AC 0 mono: Ajtai and Gurevich [2] showed this result for a counting function which is in nonmonotone AC 0 by a Chinese remaindering argument. mNC 1 6= mNL: Karchmer and Wigderson [23] showed that the ustconn function, which decides whether two nodes are connected in the given undirected input graph, requires monotone ....

....programs, or formulas (over any monotone basis) where each variable may appear only once. In particular there is an analogous argument for the circuit version of mNP : guess an input z, compute using z, and also check that z x where x is the real input. Finally we note the triviality of AC [2] circuits (polynomial size CNF or DNF formulas) Theorem 2.5. mAC [2] is trivial. Proof: For an AND or ORs computing a monotone function, it suffices to replace each negated input by a 0. Similarly for an OR of ANDs, it suffices to replace each negated input by a 1. 2 This argument fails for ....

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M. Ajtai and Y. Gurevich. Monotone versus positive. J. ACM, 34(4):1004--1015, 1987.

....negated input by a 1. 2 This method fails for depth three; we do not even know if AC [3] mono ae mP . 2.6. Rephrasing Known Results We may rephrase some known results on monotone complexity as follows: Theorem 2.4 ( Raz85a] mP 6= mNP. Theorem 2.5 ( Raz85b] mP 6= P mono. Theorem 2. 6 ( AG87] mAC 0 6= AC 0 mono. Theorem 2.7 ( KW90] mNC 1 6= mNL. Theorem 2.8 ( Yao89] mTC 0 6= mNC 1 . Theorem 2.9 ( RW90] mNC 1 mono 6ae mNC. GRIGNI AND SIPSER: MONOTONE COMPLEXITY 2.7. Monotone Separations As listed above, there are many separations of monotone classes with no ....

Ajtai M., Gurevich Y., Monotone versus positive. J. ACM, Vol. 34 (1987), pp. 1004--1015.

....0 ) 0 ( x Gamma x 0 ) 2 (y Gamma y 0 ) 2 S(x 0 ; y 0 ) Hence, it is monotone (as already observed before) The converse of Proposition 9, on the other hand, is not obvious. Ajtai and Gurevich have shown that Lyndon s theorem fails when only finite models are considered [2]. We will show that Lyndon s theorem also fails in our setting: Theorem 10. There is a monotonic spatial transformation, expressible in the spatial calculus, that is not expressible by a positive spatial calculus formula. For a sketch of the proof we refer to the Appendix. We conclude this ....

M. Ajtai and Y. Gurevich. Monotone versus Positive. Journal of the Association for Computing Machinery, Vol. 34, pages 1004--1015, October 1987.

.... Sigma 1 1 and SO; but let s look at this logic more closely. It is reasonable to expect that any bona fide logic should be such that its formulae can be recursively enumerated; that is, we should be able to systematically list all well formed formulae of the logic. However, Ajtai and Gurevich [4] showed that it is undecidable as to whether an arbitrary formula of FO is monotone or not (even when we allow infinite structures) and consequently the formulae of the logic ( SigmaLFP 0 ) FO] are not recursively enumerable. But this observation does not force us to dispense with our basic ....

....[28] showed that the logics ( SigmaLFP) FO] and ( SigmaLFP 0 ) FO] are equally expressive, i.e. that restricting the syntax of ( SigmaLFP 0 ) FO] as above does not restrict the expressive power. This is especially interesting when one compares it with Ajtai and Gurevich s result [4] that there are monotone formulae of FO that are not logically equivalent to any positive formula (this result does not hold when we consider the class of all structures, finite and infinite) But we are straying from our goal of examining the logical characterization of complexity classes. We ....

M. Ajtai and Y. Gurevich, Monotone versus positive, J. Assoc. Comput. Mach. 34 (1987) 1004--1015.

.... R( x) where arity ( x) arity (R) 32 If an occurrence of in is in the scope of n negations and in the negative scope of i implications, then is positive in if n i is even. 33 However, a first order operator may be monotone over all finite structures while failing to be positive [Ajtai and Gurevich, 1987 ] The restriction to positive operators is therefore less natural in the context of Computer Science [ Livchak, 1983; Gurevich, 1984] Higher Order Logic 15 3 Canonical semantics of higher order logic 3.1 Tarskian semantics of second order logic Let us consider the semantics of second ....

M. Ajtai and Y. Gurevich. Monotone versus positive. Journal of the ACM, 34:1004--1015, 1987.

.... proofs of classical results are invalid over F : Furthermore, it has been shown that, when relativized to the class F , many of these results actually become false, including the LosTarksi theorem, the Beth definability theorem, Craig s interpolation theorem (see [14] and Lyndon s lemma (see [2]) In addition, many natural and computationally simple properties, such as parity and graph connectedness, are not expressible in FO. As a consequence, first order logic (FO) is not as natural and attractive, over F , as it is in the general case. A central motivation for the investigation of ....

.... in which each relation in R occurs only positively, i.e. in the scope of no negations. It is still an open problem whether the Homomorphism preservation theorem fails over F . We discuss this question in depth in Section 1, in which we present some partial positive results. Ajtai and Gurevich [2] showed that Lyndon s lemma fails over the class of finite models. More recently, Stolboushkin [22] has constructed a simpler counterexample. Below, we give a slight simplification of Stolboushkin s example that is also monotone in every relation symbol. This result, and generalized preservation ....

M. Ajtai and Y. Gurevich. Monotone versus positive. Journal of the ACM, 34:1004-- 1015, 1987.

....with the caveat that nondeterministic bits are allowed to be negated. These definitions are made precise in [4] In this framework many theorems about monotone complexity may be conveniently restated. For example: Theorem 1.1 [10] mP 6= mNP. Theorem 1.2 [11] mP 6= P mono. Theorem 1. 3 [1] mAC 0 6= AC 0 mono. Theorem 1.4 [7] mNC 1 6= mNL. Theorem 1.5 [13] mTC 0 6= mNC 1 . Theorem 1.6 [9] NC 1 mono 6ae mNC. This classification scheme for monotone functions inherits much of the naturalness and robustness of the more familiar nonmonotone scheme. Most of the familiar ....

M. Ajtai and Y. Gurevich, Monotone versus positive, Journal of the ACM 34 (1987), 1004--1015.

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M. Ajtai and Y. Gurevich. Monotone versus positive. In Journal of ACM, 34 (1987), pages 1004 -- 1015.

.... is EPF. Proof. Analogously to Theorem 30. ut As before, the theorem also holds for extensions of FO ;S IFP . It is known that Lyndon s theorem fails on finite structures; there is a first order sentence monotone in a given predicate P which is not equivalent to any formula positive in P ([1]) A question arises, is there any alternative characterization of monotonicity on finite structures We do not know the answer in the context of first order logic; here we give a characterization in monadic second order logic and partial fixed point logic on ordered structures using the same ....

M. Ajtai and Y. Gurevich. Monotone versus positive. Journal of ACM, 34:1004 -- 1015, 1987.

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Ajtai M. and Gurevich Y. [1987], `Monotone versus positive', Journal of the ACM 34, 1004{ 1015.

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A. Ajtai and Y. Gurevich, Monotone versus positive, Journal of ACM, 34:5 (1987), pp. 1004-1015

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M. Ajtai and Y. Gurevich (1987): Monotone versus positive, J. of the ACM 34, 1004-1015.

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M. Ajtai and Y. Gurevich. Monotone versus positive. Journal of the ACM, 34:1004--1015, 1987.

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M. Ajtai and Y. Gurevich. Monotone versus positive. Journal of the ACM, 34:1004-- 1015, 1987.

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A. Ajtai and Y. Gurevich, Monotone versus positive, Journal of ACM, 34:5 (1987), pp. 1004-1015

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