M. Ajtai, "\Sigma 1 1 formulae on finite structures." Ann. Pure Appl. Logic 24 (1983) 1--48.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....0 ae ACC 0 [2] ae ACC 0 TC 0 NC 1 ACC 1 Delta Delta Delta NC P It is believed, but not known, that P 6= NC: in other words, that there are inherently sequential problems in P that cannot be efficiently parallelized. Some small progress has been made towards proving this [1, 8, 20, 22]: parity is in ACC 0 [2] but not AC 0 , ACC 0 [p] and ACC 0 [q] are incomparable if p and q are distinct primes, and majority is in TC 0 but not ACC 0 [2] Thus the first two inclusions in this series are proper, but ACC 0 [6] and P (or even NP) could be identical for all anyone has ....

M. Ajtai, "\Sigma 1 1 formulae on finite structures." Ann. Pure Appl. Logic 24 (1983) 1--48.

....t(n) less than n, where the latter is the set of problems computable dynamically on a RAM (with word size O(log n) in time t(n) Example 3.2 Consider the simple boolean query: Parity, which is true iff the input binary string has an odd number of one s. This is well known not to be in static FO [A83, FSS84]. The dynamic algorithm for Parity maintains a bit b which is toggled after any change to the string. We also remember the input string so that we can tell if a request has actually changed the string. The vocabulary of the Parity problem is oe = hMi consisting of a single monadic relation ....

M. Ajtai (1983), "\Sigma 1 1 Formulae on Finite Structures," Annals of Pure and Applied Logic 24, 1-48.

....be ordinary AND and OR gates (and the inputs to be variables or negated variables) we get the class AC 0 . As we shall see below, we already understand this class because we can take a relatively simple language (the binary strings with an odd number of ones) and prove that it is not in AC 0 [FSS84, Aj83]. By allowing new operations as well we get the AC 0 closure of those operations. To be specific, a language is in the AC 0 closure of a family F of functions if it can be computed by a constant depth, polynomial size family of unbounded fan in circuits of AND gates, OR gates, and gates for ....

....non uniform (with arbitrary added predicates) versions. More detailed definitions and proofs can be found in [BIS88] 5 4. The Basic Result: Parity Here we outline the proof of Furst Saxe Sipser [FSS84] that the parity language is not in AC 0 . The independent proof of this result by Ajtai [Aj83], in a very different conceptual framework, uses very similar methods. Though [FSS84] is usually phrased as an argument by contradiction, the proof can be thought of as demonstrating a property of all AC 0 languages not posessed by parity. Consider first the special case of AC 0 where the ....

M. Ajtai, "\Sigma 1 1 formulae on finite structures", Annals of Pure and Applied Logic 24 (1983), 1-48.

....for each input size. AC 0 is a more interesting class, but we still know how to prove natural languages to be outside of it. The simple language fx 2 f0; 1g : x has an odd number of ones g is not in non uniform AC 0 , as shown by Furst, Saxe, and Sipser [FSS84] and independently Ajtai [Aj83]. Curiously, although most interesting languages in non uniform AC 0 are known to be in log time uniform AC 0 , we do not know how to prove a language outside the latter without proving it outside the former. We get further interesting subclasses of NC 1 by looking at circuit families with ....

M. Ajtai, "\Sigma 1 1 formulae on finite structures", Annals of Pure and Applied Logic 24 (1983), 1-48.

....to polynomial size F . 5 This theory also does not prove that every formula can be evaluated over any evaluation of its atoms. This is because the witnessing of such a proof would allow to express size n formulas by size n O(1) constant depth circuits (cf. Corollary 3. 3) which is impossible by [1, 20]. extending F by new (infinitely many) connectives C n;k (OE 1 ; OE n ) for 1 n and k n, with intended meaning that C n;k (OE 1 ; OE n ) is true iff the number of true OE i s is k, and by adding new axioms: 1. A j C 1;1 (A) 2. C n;0 (A 1 ; An ) j ( A 1 : An ) ....

M. Ajtai: "\Sigma 1 1 -formulae in finite structures", Annals of Pure and Applied Logic, 24, (1983), pp.1-48.

....perform more complicated computations such as binary multiplication, or determining whether the number of bits in the input is even. One of the most important achievements of computational complexity theory is the following result, due to Furst, Saxe and Sipser [11] and, independently, to Ajtai [1]. Theorem 10. Let q 1: MOD q = 2 AC 0 : This implies, by a relatively simple reduction, that multiplication cannot be performed by polynomial size constant depth circuit families. See [11] The connection to finite monoids is given by the following theorem, due to Barrington and Th erien ....

M. Ajtai, "\Sigma 1 1 formulae on finite structures", Annals of Pure and Applied Logic 24 (1983) 1--48.

....input binary string has an odd number of one s. This is well known not to be in static FO 1 This expects that the complexity class C is closed under polynomial increases in the input size. For more restricted classes C, such as linear time, we insist that jjf n ( r)jj = O(jjeval n;oe ( r)jj) A83, FSS84] The dynamic algorithm for PARITY maintains a bit b which is toggled after any change to the string. We also remember the input string so that we can tell if a request has actually changed the string. The vocabulary of the PARITY problem is oe = hM 1 i consisting of a single monadic ....

M. Ajtai (1983), "\Sigma 1 1 Formulae on Finite Structures," Annals of Pure and Applied Logic 24, 1-48.

....see that NC P, but like P NP this inclusion is believed, but not known, to be proper. From the definitions we have AC 0 ae ACC 0 [2] ae ACC 0 NC 1 AC 1 ACC 1 NC 2 Delta Delta Delta NC P The parity function is clearly in ACC 0 [2] and Ajtai and Furst et al. [1, 12] have shown that it is not in AC 0 . Razborov [30] has shown that majority is in NC 1 but not in ACC 0 [2] and Smolensky [33] has shown that ACC 0 [p] and ACC 0 [q] are incomparable if p and q are distinct primes. Thus the first and second inclusions are proper. However, for all anyone ....

M. Ajtai, "\Sigma 1 1 formulae on finite structures." Ann. Pure Appl. Logic 24 (1983) 1--48.

....we can determine whether the number of 1 s in the input string is at least k; where k is a constant. We can even determine whether the number of 1 s in the input is at least log n. This last fact is far from obvious see Fagin, et al. 9] Furst, Saxe and Sipser [10] and, independently, Ajtai [1] showed that if k 1 then the regular language MOD k = fa 1 Delta Delta Delta an 2 f0; 1g : n X i=1 a i j 0 (mod k)g is not in AC 0 : It follows from this that one cannot perform binary multiplication in AC 0 ; or determine whether the majority of the input bits are on. What happens ....

....of semigroup theory and Ramsey style combinatorics. The important connection to semigroup theory is discussed in the next subsection. 1. 4 Connections to algebra We remarked above that if we had a direct proof of the equality FO Reg = FO[Reg] then we could prove the circuit lower bounds of [1] and [10] directly. This is because we can give a precise characterization of the languages in FO[Reg] in terms of semigroup theoretic invariants of regular languages. Similarly, we can precisely characterize the classes Mod(s,q) Reg] in semigroup theoretic terms, and thus we possess an effective ....

M. Ajtai, "\Sigma 1 1 formulae on finite structures", Annals of Pure and Applied Logic 24 (1983) 1--48.

....to develop new techniques and new understanding. There is a subclass of NC 1 for which separation results are known. AC 0 is the class of problems which have circuits of polynomial size and constant depth in a model with unbounded fan in. Furst, Saxe, and Sipser [FSS84] and independently Ajtai [Aj83] proved that the exclusive OR function is not in AC 0 , separating this class from NC 1 . Later work has attempted to extend the frontier upward from AC 0 by proving lower bounds for more powerful subclasses. Razborov [Ra87] considered the extension of AC 0 obtained by also allowing ....

....The class AC 0 is defined as those languages recognized by families of circuits of AND and OR gates with arbitrary fan in, size n O(1) and depth O(1) See, e.g. Co85] for more on the classes NC i and AC i . It is easy to show AC 0 NC 1 , and the inclusion is known to be strict [FSS84, Aj83]. Both NC 1 and AC 0 have equivalent definitions (in their non uniform versions) by families of expressions. NC 1 is the class of languages recognized by families of Boolean formulas of polynomial length [Sp71] or by families of polynomial length and depth O(log n) AC 0 is the class of ....

M. Ajtai, "\Sigma 1 1 formulae on finite structures", Annals of Pure and Applied Logic 24 (1983), 1-48.

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M. Ajtai, "\Sigma 1 1 Formulae on Finite Structures," Annals of Pure and Applied Logic 24, 1983, (1-48).

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