E. Allender, R. Beals, and M. Ogihara. The complexity of matrix rank and feasible systems of linear equations. In ACM Symposium on Theory of Computing (STOC), 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....than their corresponding polynomial time counterparts. For example the logspace alternating hierarchy collapses to NL, as follows from the celebrated closure of NL under complementation [5, 8] The PL hierarchy collapses to PL, as shown by Ogihara [6] The C=L hierarchy collapses to its 2 level [1]. Interestingly, nothing was known so far for the MOD q L hierarchy, though here the polynomial time analogue is known to collapse. As an immediate consequence of our above result we obtain that the logspace hierarchy collapses as well. 2 Logspace MOD Classes The basic class underlying all ....

Allender, E., Beals, R., and Ogihara, M. The complexity of matrix rank and feasible systems of linear equations. Computational Complexity 8 (1999), 99-126.

....improve the efficiency of the result from [15] and to extend it to presumably wider log space language classes such as mod p L [11, 16] and to log space counting classes, such as #L. These extensions provide efficient PSM solutions for natural linear algebraic and graph theoretic problems (cf. [23, 2, 1, 11]) not covered by previous solutions. But perhaps most of the added value is in the simple and efficient solutions obtained for problems which are naturally represented by small Branching Programs (see Section 2) For instance, for every problem having linear size deterministic Branching ....

....define the counting logspace classes #L (resp. diffL) to include any integer valued function f for which there exists a nondeterministic logspace TM such that on every input x the number of accepting computation paths (resp. accepting minus rejecting) on x is equal to f(x) The reader may refer to [11, 16, 1] for further discussion of the structure and importance of these log space classes. We use the terms protocol and function to denote parameterized families of protocols and functions. RANK SUM(A1 ; A2 ; An) Input: a Theta a matrices over GF (q) A1 ; A2 ; An . Computes: ....

E. Allender, R. Beals, and M. Ogihara. The complexity of matrix rank and feasible systems of linear equations (extended abstract). In Proc. of 28nd STOC, pages 503--513, 1996.

.... counting the number of operations in an abstract domain D we refer to Baur and Strassen about the link between matrix multiplication and determinant computation [52, 53, 7] See also the link with matrix powering and the complexity class GapL following Toda, Vinay, Damm and Valiant as explained in [3], for example. We may also mention Valiant s theorem that the determinant is universal for formulas [54] For integer matrices, computing the sign of the determinant is a priori an easier problem than computing its value. We will try to identify the di erences between these two problems even if ....

E. Allender, R. Beals, and M. Ogihara. The complexity of matrix rank and feasible systems of linear equations. Computational Complexity, 8(2):99126, 1999.

.... counting the number of operations in an abstract domain D we refer to Baur and Strassen about the link between matrix multiplication and determinant computation [52,53,7] See also the link with matrix powering and the complexity class GapL following Toda, Vinay, Damm and Valiant as explained in [3], for example. We may also mention Valiant s theorem that the determinant is universal for formulas [54] For integer matrices, computing the sign of the determinant is a priori an easier problem than computing its value. We will try to identify the di#erences between these two problems even ....

E. Allender, R. Beals, and M. Ogihara. The complexity of matrix rank and feasible systems of linear equations. Computational Complexity, 8(2):99--126, 1999.

....via a simple affine transformation into an # # matrix M x , 7 such that the output value f(x) directly corresponds to the rank of M x . This is formalized by the following lemma, which is implicit in [22] a similar, though less efficient, transformation can be derived from Lemma 2. 3 in [1]) A proof of the lemma appears in Appendix A. Lemma 3.2 [22] Let K = GF(p) where p is prime, and suppose that BP is a mod p branching program of size # 1 computing a Boolean function f : 0, 1 n # 0, 1 . Then, there exists an affine (degree 1) transformation L : K n # K #,# such ....

E. Allender, R. Beals, and M. Ogihara. The complexity of matrix rank and feasible systems of linear equations. In Proc. of 28nd STOC, pages 503--513, 1996.

....open the question whether there exists a language in AC 0 whose characteristic function is not in LDiffAC 0 . It is well known that PARITY is not in AC 0 . Therefore with Corollary 2 we have separated AC 0 from LDiffAC 0 . In their paper [1] Agrawal et al. showed that the class AC 0 [2] is exactly the class of languages whose characteristic function is in GapAC 0 . Combining this result with our result will give AC 0 [2] LDiffAC 0 . Because of the result in [12] showing that the MAJORITY is not computable in AC 0 [2] we have: AC 0 ( AC 0 [2] LDiffAC 0 ( TC ....

....is not in AC 0 . Therefore with Corollary 2 we have separated AC 0 from LDiffAC 0 . In their paper [1] Agrawal et al. showed that the class AC 0 [2] is exactly the class of languages whose characteristic function is in GapAC 0 . Combining this result with our result will give AC 0 [2] = LDiffAC 0 . Because of the result in [12] showing that the MAJORITY is not computable in AC 0 [2] we have: AC 0 ( AC 0 [2] LDiffAC 0 ( TC 0 : Note that the modulus 2 is special. If p is any odd prime number, the MOD p function is provably not in GapAC 0 , because (as noted in ....

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E. Allender, R. Beals, M. Ogihara, The complexity of matrix rank and feasible systems of linear equations. In Proceedings of the 28th ACM Symposium on Theory of Computing (STOC), pp:161--167, 1996.

....polynomial time counterparts. For example the logspace alternating hierarchy collapses to NL, as follows from the celebrated closure of NL under complementation [Imm88, Sze87] The PL hierarchy collapses to PL, as shown by Ogihara [Ogi98] The C=L hierarchy collapses to its Delta 2 level [ABO96] Interestingly, nothing was known so far for the MOD q L hierarchy, though here the polynomial time analogue is known to collapse. As an immediate consequence of our above result we obtain that the logspace hierarchy collapses as well. 2 Logspace MOD Classes The basic class underlying all ....

E. Allender, R. Beals, and M. Ogihara. The complexity of matrix rank and feasible systems of linear equations (extended abstract). In Proceedings 28th Symposium on Theory of Computing, pages 161--167. ACM Press, 1996.

.... linear functions, i.e. f(x 1 ; x n ) k j=1 i a j 1 x 1 Phi a j 2 x 2 Phi : Phi a j n x n Phi b j j a j i ; b j 2 f0; 1g; k n : 1) The complexity and the properties of boolean linear functions have been the subject of several studies over the past few years [1, 6, 7, 10]. Informally speaking, the main interest in linear functions is motivated by the fact that they have small (i.e. polynomial) circuit size complexity [13] but they have a rather rich behavior recently used in [6] to approximate general boolean functions within a good trade off between ....

Allender E, Beals R, and Ogihara M. (1996), "The complexity of matrix rank and feasible systems of linear equations", in Proc. of 28-th ACM STOC, to appear. Also available by ftp/www in ECCC (Tech. Rep. 1996).

....matrices. ffl The #L hierarchy = L #L : Delta :#L = AC 0 (#L) the class of problems AC 0 reducible to computing the determinant of integer matrices. The first two of these hierarchies collapse, and they coincide with NC 1 reducibility. ffl AC 0 (C= L) L C=L = NC 1 (C= L) [ABO96]. ffl AC 0 (PL) PL = NC 1 (PL) O96, BF97] These hierarchies are defined using Ruzzo Simon Tompa reducibility [RST] which is the usual notion of oracle access for space bounded nondeterministic Turing machines. It seems natural to conjecture that AC 0 and NC 1 reducibility ....

....of some important and natural problems. For instance, the set of singular matrices (matrices with determinant zero) is complete for C= L, and a variety of other problems regarding computation of the rank and determining if a system of linear equations is feasible are complete for L C=L [ABO96]. Some other problems in linear algebra and problems involving Markov decision processes were shown to be complete for PL in [J84, MGA97] One important problem whose complexity remains unresolved is the perfect matching problem. No deterministic NC algorithm is known for matching at all, but the ....

[Article contains additional citation context not shown here]

E. Allender, R. Beals, and M. Ogihara. The complexity of matrix rank and feasible systems of linear equations. DIMACS Technical report 97-40 (submitted for publication). A preliminary version appears in ACM Symposium on Theory of Computing (STOC), 1996.

....p span programs over the finite fields GF(p) have the same power as modular branching programs. That is, a function can be computed by a polynomial size mod p branching program if and only if it can be computed by a polynomial size span program over GF(p) Span programs were considered also in [5, 3, 4, 2, 12, 20]. Pudl ak and Sgall [20] defined a similar model called dependency programs. They proved that span programs are at least as strong as dependency programs over every field, and for fixed finite fields the reverse also holds, that is over fixed finite fields span programs and dependency programs are ....

....prime p the three models (arithmetic programs, dependency programs and span programs) over GF(p) are equivalent in power to mod p branching programs, and the class of functions computable by polynomial size programs in these models over GF(p) is equal to the class Mod p L poly. It is proved in [2] that logspace uniform polynomial size span programs over the rationals characterize the class L C=L . Similarly, logspace uniform polynomial size dependency programs over the rationals characterize the class C=L. Our results imply that logspace uniform polynomial size arithmetic programs over ....

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E. Allender, R. Beals, and M. Ogihara. The complexity of matrix rank and feasible systems of linear equations. In Proc. of the 28th Annu. ACM Symp. on the Theory of Computing, pages 161--167, 1996.

....proof) that if AC 0 (Det) and DET # coincide, LH collapses. Two nice applications of the GapL characterization in Theorem 4 have 6. 2 15 been the drastic simplification of Jung s theorem on PL (see [AO96] and the characterization of the complexity of computing the rank of a matrix (see [ABO96] Acknowledgements We thank Eric Allender for setting us on the track of pursuing the determinant problem, and for supplying us with several interesting references. We thank an anonymous referee for pointing out that our EREW algorithm from Section 6.1 is even an OROW algorithm. Acknowledgement ....

E. Allender, R. Beals, and M. Ogihara. The complexity of matrix rank and feasible systems of linear equations. In<F4.621e+05> Proceedings of the 28th ACM Symposium on Theory of<F5.291e+05> Computing, pages 161-- 167, New York, 1996. ACM.

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E. Allender, R. Beals, and M. Ogihara. The complexity of matrix rank and feasible systems of linear equations. In ACM Symposium on Theory of Computing (STOC), 1996.

No context found.

E. Allender, R. Beals, and M. Ogihara. The complexity of matrix rank and feasible systems of linear equations. In Proc. 28th ACM Symposium on Theory of Computing (STOC), pages 161--167, 1996.

No context found.

E. Allender, R. Beals, M. Ogihara, The complexity of matrix rank and feasible systems of linear equations. In Proceedings of the 28th ACM Symposium on Theory of Computing (STOC), pp. 161--167, 1996.

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E. Allender, R. Beals, and M. Ogihara. The complexity of matrix rank and feasible systems of linear equations. Computational Complexity, 8:99-126, 1999.

....and it is well known that TC NC . So, the Santha Tan result in light of the observation of Allender and Ogihara o ers an alternative proof of the Zalcstein Garzon Theorem. Actually, the exact complexity theoretic characterization of the problem has been known. Allender, Beals, and Ogihara [ABO99] show that verifying the rank of a matrix is complete for the second level of the boolean hierarchy [CGH 88] over C= L, both over the integers and over the eld of rational numbers. They also show that the class AC (C= L) is equal to the logspace disjunctive truth table reducibility closure ....

E. Allender, R. Beals, and M. Ogihara. The complexity of matrix rank and feasible systems of linear equations. Computational Complexity, 8:99-126, 1999.

....counting, and circuits, and AC 0 reducibility helps elucidate this structure. Below is a short list of some important complexity classes, along with some standard complete problems. We won t present definitions or complete references here. For more details, you can consult [GHR95, ABO99, CM87, Ete97, Bus93] Basic Complexity 17 Complexity Class Complete Problems under # AC 0 m P linear programming circuit evaluation least fixed point evaluation C=L matrix singularity many questions in linear algebra (rank etc. NL finding shortest paths transitive closure L graph ....

E. Allender, R. Beals, and M. Ogihara. The complexity of matrix rank and feasible systems of linear equations. Computational Complexity, 8:99--126, 1999.

No context found.

E. Allender, R. Beals, M. Ogihara, The complexity of matrix rank and feasible systems of linear equations. In Proceedings of the 28th ACM Symposium on Theory of Computing (STOC), pp. 161--167, 1996.

.... ffl maximum flow with unary weights All of these problems were previously known to be hard for NL, and were known to be (nonuniformly) reducible to the determinant [KUW86, MVV87] It was observed in [BGW] that the perfect matching problem is in (nonuniform) ModmL for every m, and as reported in [ABO97], Vinay has pointed out that a similar argument shows that the matching problem is in (nonuniform) co C= L. A different argument seems to be necessary to show that the matching problem is itself in (nonuniform) C= L. Since SPL is contained in C=L co C= L, this follows from our new bound on ....

....classes by means of GapL functions. We mention in particular the following three complexity classes, of which the first two have been studied previously. ffl PL = fA : 9f 2 GapL; x 2 A , f(x) 0g (See, e.g. Gil77, RST84, BCP83, Ogi96, BF97] ffl C=L = fA : 9f 2 GapL; x 2 A , f(x) 0g [AO96, ABO97, ST94]. ffl SPL = fA : A 2 GapLg. It seems that this is the first time that SPL has been singled out for study. In the remainder of this section, we state some of the basic properties of SPL. Proposition 2.1 UL SPL C=L co C=L. The second inclusion holds because SPL is easily seen to be closed ....

E. Allender, R. Beals, and M. Ogihara. The complexity of matrix rank and feasible systems of linear equations. DIMACS technical report 97-40, submitted for publication, a preliminary version appeared in STOC 96, 1997.

....f(x) 0g. 1 Tomo Yamakami [Yam96] has recently defined #AC 0 somewhat differently, and his definition does not appear comparable to ours. 3 PP and PL were first studied in [Gil77] and have been considered in many papers; C=P was studied in [Wag86] and elsewhere, and C=L was studied in [ABO96] (see also [ST] PNC 1 and C=NC 1 were defined and studied in [CMTV96] see also [Mac] A main result of this paper is that PAC 0 and C=AC 0 coincide with TC 0 . However, there are two difficulties that must be overcome before we can even state this theorem. We must deal with (a) ....

E. Allender, R. Beals, and M. Ogihara. The complexity of matrix rank and feasible systems of linear equations. In Proc. 28th ACM Symposium on Theory of Computing (STOC), pages 161--167, 1996.

....the proof below that #LH i is actually 3 Note that this corrects a mis statement in the version of this paper that appeared in Proc. 9th IEEE Structure in Complexity Theory Conference, 1994. It is also worth noting that it has recently been shown that this entire hieararchy collapses to L C=L [2]. equal to the class of functions computed by AC 0 i (#L) circuits, with the added restriction that the output gate be an oracle gate. Proof. The first inclusion is obvious when i = 1; thus assume the induction hypothesis for i and let f be in #L g for some g 2 #LH i . Thus there is some ....

.... a bounded error probabilistic logspace algorithm for PL [54] ffl Can anything more be said about the relationship between PLH and NC 1 (PL) or #LH and NC 1 (#L) Note that for many classes C of interest (including AC k , NC k , NL, L, NP [4, 34] AC 0 (C) is equal to NC 1 (C) In [2] it is shown that this equality also holds for C = C=L. Acknowledgments The first author thanks Birgit Jenner for several motivating and informative discussions. Dave Barrington, Michelangelo Grigni, Sanjeev Saluja, Heribert Vollmer, and Pierre McKenzie also provided helpful information. We ....

E. Allender, R. Beals, and M. Ogihara, The complexity of matrix rank and feasible systems of linear equations, manuscript.

.... logspace) and C=L (which characterizes the complexity of singular matrices, as well as questions about computing the rank) It is known that some natural hierarchies defined using these complexity classes collapse: ffl AC 0 (C= L) C=L C=L : Delta :C=L = NC 1 (C= L) L C=L [AO96, ABO96] ffl AC 0 (PL) PL PL : Delta :PL = NC 1 (PL) PL [AO96, Ogi96, BF] In contrast, the corresponding #L hierarchy (equal to the class of problems AC 0 reducible to computing the determinant) AC 0 (#L) FL #L : Delta :#L is not known to collapse to any fixed level. Does the ....

E. Allender, R. Beals, and M. Ogihara. The complexity of matrix rank and feasible systems of linear equations. In ACM Symposium on Theory of Computing (STOC), 1996.

.... logspace) and C=L (which characterizes the complexity of singular matrices, as well as questions about computing the rank) It is known that some natural hierarchies defined using these complexity classes collapse: ffl AC 0 (C= L) C=L C=L : Delta :C=L = NC 1 (C= L) L C=L [AO96, ABO96] ffl AC 0 (PL) PL PL : Delta :PL = NC 1 (PL) PL [AO96, Ogi96, BF97] In contrast, the corresponding #L hierarchy (equal to the class of problems AC 0 reducible to computing the determinant) AC 0 (#L) FL #L : Delta :#L is not known to collapse to any fixed level. Does the ....

E. Allender, R. Beals, and M. Ogihara. The complexity of matrix rank and feasible systems of linear equations. In ACM Symposium on Theory of Computing (STOC), 1996.

....inclusion in the preceding theorem. 3 Note that this corrects a mis statement in the version of this paper that appeared in Proc. 9th IEEE Structure in Complexity Theory Conference, 1994. It is also worth noting that it has recently been shown that this entire hieararchy collapses to L C=L [2]. Corollary 23 ffl #LH = AC 0 (#L) AC 0 (DET) ffl PLH = AC 0 (PL) ffl C=LH = AC 0 (C= L) Part 2 of this corollary was previously observed by Carsten Damm and Peter Rossmanith [24, 38] The reader should not be alarmed by the fact that L PL is contained in AC 0 (PL) and in fact ....

E. Allender, R. Beals, and M. Ogihara, The complexity of matrix rank and feasible systems of linear equations, manuscript.

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E. Allender, R. Beals, and M. Ogihara. The complexity of matrix rank and feasible systems of linear equations. Computational Complexity, 8:99-126, 1999.

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