K. Reinhardt and E. Allender. Making nondeterminism unambiguous. SIAM Journal on Computing, 29:1118--1131, 2000. Preliminary version in Proceedings of 38th IEEE Conference on Foundations of Computer Science, 1997, pp 244--253.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....between a circuit based language class and its functional counterpart. A morphism h : Sigma Delta is isometric iff h applied to each a 2 Sigma yields a word of the same length. An important part of our results, motivated by recent interest in classes intermediate between L and NL [2, 12, 14], is the investigation of the problem range in the variable case. Restricting the underlying morphism affects the complexity of the range problem, e.g. the range problem for prefix codes is in L and is complete for this class. Now, one might expect that imposing the code property on h( Sigma) ....

K. Reinhardt and E. Allender, Making nondeterminism unambiguous, Proc. of the 38th IEEE FOCS , pp. 244--253, 1997.

....inputs (i.e. to count the number of 1 s) FSS84] #AC 0 is a strictly more powerful class. The two remaining classes, #L and #SAC 1 , are more problematic. If every language in NL has its characteristic function in #L, then NL = UL. However, since it was recently shown that NL poly = UL poly [RA97], it is no longer clear if this should be considered unlikely. In fact, the results of [RA97] show that, in the nonuniform setting, every function that can be computed by Boolean NL circuits is in #L, and thus the arithmetic circuits are at least as powerful as the Boolean circuits. Analogous ....

....The two remaining classes, #L and #SAC 1 , are more problematic. If every language in NL has its characteristic function in #L, then NL = UL. However, since it was recently shown that NL poly = UL poly [RA97] it is no longer clear if this should be considered unlikely. In fact, the results of [RA97] show that, in the nonuniform setting, every function that can be computed by Boolean NL circuits is in #L, and thus the arithmetic circuits are at least as powerful as the Boolean circuits. Analogous results hold for #SAC 1 . 4.1 Arithmetic Boolean Circuits Another model that has received ....

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K. Reinhardt and E. Allender. Making nondeterminism unambiguous. To appear in Proc. 38th IEEE Conference on Foundations of Computer Science (FOCS), 1997.

....programs (e.g. NL poly PhiL poly) The construction of Wigderson [25] directly generalizes to arithmetic programs over arbitrary fields. The simulation of nondeterministic branching programs by arithmetic programs over any field also follows from a recent result of Reinhardt and Allender [22] who proved, using the construction of [25] that NL poly = UL poly. However, we give an alternative construction which is more efficient, i.e. the resulting arithmetic program is smaller. The models considered in this paper are closely related to complexity classes defined by logspace ....

K. Reinhardt and E. Allender. Making nondeterminism unambiguous. In Proc. of the 38th Annu. IEEE Symp. on Foundations of Computer Science, pages 244--253, 1997.

....to believe that NL is a subset of GapL( DET) We mean subset in the following sense: a language is in GapL if its characteristic function is in GapL. This is because the 0 1 valued functions in DET must di#er in at most one accepting path. However, a recent result of Reinhardt and Allender [RA97] shows that the inclusion is true in a nonuniform setting. On the other hand, notice that NL is contained in DET # . In fact, Allender and Ogihara [AO96] consider AC 0 (Det) and show 2 Here, Perm denotes the function which, given an n dimensional matrix A with integer entries, evaluates to ....

K. Reinhardt and E. Allender. Making nondeterminism unambiguous. In<F4.621e+05> Proceedings of the 38th IEEE Symposium on Foundations of Computer<F5.291e+05> Science, Los Alamitos, CA, 1997. IEEE.

....(e.g. NL poly PhiL poly) The construction of Wigderson [19] directly generalizes to arithmetic programs over arbitrary fields. The efficient simulation of nondeterministic branching programs by arithmetic programs over any field also follows from a recent result of Reinhardt and Allender [16] who proved, using the construction of [19] that NL poly = UL poly. However, we give an alternative construction which is more efficient, i.e. the resulting arithmetic program is smaller. Remark: Nisan [13] considered a similar complexity model called algebraic branching programs. These ....

K. Reinhardt and E. Allender. Making nondeterminism unambiguous. In Proc. of the 38th Annu. IEEE Symp. on Foundations of Computer Science, pages 244--253, 1977.

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K. Reinhardt and E. Allender. Making nondeterminism unambiguous. SIAM Journal on Computing, 29:1118--1131, 2000. Preliminary version in Proceedings of 38th IEEE Conference on Foundations of Computer Science, 1997, pp 244--253.

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K. Reinhardt and E. Allender. Making nondeterminism unambiguous. SIAM J. Comput., 29:1118--1131, 2000.

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K. Reinhardt and E. Allender. Making nondeterminism unambiguous. SIAM Journal of Computing, 29:1118--1131, 2000.

....some Motivation Many of the observations in this paper are motivated by the desire to prove a collapse of some complexity classes between NL and UL. UL is unambiguous logspace; more formal definitions appear below. It was observed in [ARZ99] that the nonuniform collapse NL poly = UL poly of [RA00] holds also in the uniform case under a very plausible hypothesis. Namely, NL = UL if there is a set in DSPACE(n) that has exponential hardness in the sense of [NW94] More recently, it has been pointed out by [KvM02] that this same conclusion can be weakened to a worst case circuit lower ....

....defined in [AJ93] to be the class of functions f such that there is an NL transducer M with the property that f(x) is the lexicographically largest string produced by M along any accepting computation path on input x. It is known that OptL is contained in AC AJ95] and the question is raised in [RA00] if perhaps OptL is equal to FNL (the class of functions computable in NL) The following takes care of an easy special case. Theorem 6.1 Let f beafunctioninOptL with the property that there is an NL transducer realizing f that produces at most n distinct outputs for any string x of length n. ....

K. Reinhardt and E. Allender. Making nondeterminism unambiguous. SIAM J. Comput., 29:1118--1131, 2000.

....in theoretical computer science. It says that we do not need one algorithm, Turing machine, grammar or whatever to recognize a language but we may use a hole family of them, where each is used only for words of one special size. Connections of nonuniformity and counting can for example be found in [RA97] and [AR98] A common characterization of nonuniformity is by advice strings. One major observation is that most lower bounds of problems or statements saying that a problem does not belong to a certain class also hold for the nonuniform version of the measure or class. This also holds for Lemma ....

K. Reinhardt and E. Allender. Making nondeterminism unambiguous. In 38 th IEEE Symposium on Foundations of Computer Science (FOCS), pages 244--253, 1997.

....Shiyu Zhou y Abstract We show that the complexity class LogFew is contained in NL SPL. Previously, this was known only to hold in the nonuniform setting. Key Words: Nondeterministic Logspace Computation, Nonuniform Complexity, Derandomization, ffl biased Sample Space. 1 Introduction In [RA97], a probabilistic construction was used to show that the complexity classes NL poly and UL poly coincide. That is, in the context of nonuniform complexity, nonuniform logspace is no more powerful than unambiguous logspace. It was observed in [AR98] that the equality NL=UL holds also in the uniform ....

....in [AR98] that the equality NL=UL holds also in the uniform setting, under a plausible hypothesis concerning pseudorandom number generators. However, it remains an important open question whether NL=UL can be established without resorting to unproved assumptions. The results and techniques of [RA97] were extended in [AR98] in a number of ways. One extension involves the class LogFew, defined in [BDHM92] Formal definitions appear below. No inclusion relation was known between NL and LogFew in the uniform setting, although UL is trivially contained in LogFew, and thus NL poly LogFew poly. ....

[Article contains additional citation context not shown here]

K. Reinhardt and E. Allender. Making nondeterminism unambiguous. In 38 th IEEE Symposium on Foundations of Computer Science (FOCS), pages 244--253, 1997.

....of det(M ) is the number of perfect matchings in G. Sketch: Given the proper advice strings, a GapL algorithm can take as input the matrix M , compute its planar embedding (since this is in L poly) then compute its normal form embedding along a unique computation path (since NL # UL poly [24]) and then use the algorithm in [17] to compute a number whose absolute value is the number of perfect matchings in M . Since the determinant is complete for GapL under projections, the result follows. The paper is organized as follows. In Section 2 we present our hardness result for planarity. ....

K. Reinhardt and E. Allender. Making nondeterminism unambiguous. In 38 th IEEE Symposium on Foundations of Computer Science (FOCS), pages 244--253, 1997. to appear in SIAM J. Comput.

....and CCR 9734918. z Supported in part by the DFG Project La 618 3 1 KOMET. important open question, although we provide evidence that they do. More precisely, if there are problems in DSPACE(n) having exponential hardness, then all of our results hold in the uniform setting. 1 Introduction In [RA97], the authors presented new results concerning NL, UL, and #L. The current paper builds on this earlier work, in an attempt to better understand these complexity classes, as well as some related classes. In the process, we present a new upper bound on some problems related to matchings in graphs. ....

....D ae ae ae ae a a a a a a a a Gamma Gamma Gamma Theta Theta Theta Theta Theta Theta Figure 3: Uniform inclusions among these classes. the oracle queries and answers at the start of the computation, but that instead these can be guessed one by one as needed. Since UL poly = NL poly [RA97], it follows that, in the nonuniform setting, NL is contained in SPL. However, it needs to be noted at this point that it is not quite clear what the nonuniform version of SPL should be. Here are two natural candidates: ffl SPL poly = fA : 9B 2 SPL 9k9(ff n )jff n j n k and x 2 A , x; ff n ....

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K. Reinhardt and E. Allender. Making nondeterminism unambiguous. In 38 th IEEE Symposium on Foundations of Computer Science (FOCS), pages 244--253, 1997.

.... concerns the class USPACE(logn) Is it possible to show that it is contained in SC 2 or to give an o(log 2 n) space algorithm, as has been possible for symmetric logspace and the class RUSPACE(logn) In this regard, it is interesting to call the reader s attention to a very recent result in [22]: USPACE(logn) poly = NSPACE(logn) poly. Thus any improved upper bound on the complexity of USPACE(logn) will have strong implications on the complexity of NSPACE(logn) A third line of investigation is to consider these questions for polynomial time bounded auxiliary pushdown automata where ....

K. Reinhardt and E. Allender. Making nondeterminism unambiguous. In Proc. 38th Annual IEEE Symposium on Foundations of Computer Science, pages 244--253, 1997.

....is computable in GapLogCFL. b) If A # LogCFL the formal power series # 1 A is computable in GapLogCFL poly. The proof of part (a) is immediate once it is observed that the coe#cients of the matrix presented in Lemma 4.1 can be computed in GapLogCFL. Part (b) follows because of the results of [RA97]. It is not clear how to make the second inclusion uniform, although it is pointed out in [AR98] that this inclusion does hold in the uniform setting if there are sets in DSPACE(n) with su#ciently high circuit complexity. It is open if there are corresponding hardness results. As mentioned in the ....

K. Reinhardt and E. Allender. Making nondeterminism unambiguous. In Proceedings of 38th IEEE Conference on Foundations of Computer Science, 1997.

....of CFL s. It turns out that formal power series which are characteristic functions of LogDCFL languages are invertible in GapLogCFL. For LogCFL languages, the same inclusion holds in a non uniform setting; we do not know how to make it uniform, as it relies on the disambiguating construction of [RA97]. It is open if there is a corresponding hardness result. ffl Finally, for formal power series which are characteristic functions of (deterministic) context sensitive languages, the inverses and roots can be computed in DSPACE(n) and NSPACE(n) respectively. ....

K. Reinhardt and E. Allender. Making Nondeterminism Unambiguous. to appear in Proc. 38th IEEE Conference on Foundations of Computer Science 97.

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