H. R. Lewis and C. H. Papadimitriou. Symmetric space-bounded computation. Theoretical Computer Science, 19, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....The join A B of two languages is de ned by A B : f0x j x 2 Ag[ f1x j x 2 B g. For de nitions of standard complexity classes see [48] The class CFL contains the context free languages, DCFL contains the deterministic context free languages. The class SL of symmetric logspace is de ned in [38]. The class UL was introduced by Alvarez and Jenner [4] and FewL and Mod k L are due to Buntrock et al. 18] The class SAC consists of all languages decided by logarithmically depth bounded polynomially sized semi unbounded fan in circuits. When discussing circuit classes, selection of ....

....k th prime number is O(k ) the primes needed in the Chinese remainder representation of X are at most n . They can thus be calculated in logspace. So B is hard for #L and thus complete for L . This shows that the rst statement is true for L . Case 6: C = SL. It is known [38, 45] that the complement of the undirected graph accessibility problem UGAP is m complete for SL. We show that UGAP has a prover in FL . It maps each input (G; s; t) to the set of all vertices reachable from s in G. Clearly, this prover is in FL . On input hG; s; ti ; I the veri er ....

H. Lewis and C. Papadimitriou. Symmetric space-bounded computation. Theoretical Comput. Sci., 19:161-187, 1982.

....b GI lies in fact in NC. We focus in this paper in the family of colored graphs with color multiplicities bounded by the constants 2 and 3 proving that 2 GI and 3 GI are many one complete for symmetric logarithmic space SL under logarithmic space reductions. The complexity class SL introduced in [11] has di erent characterizations, but the easiest way to de ne it is the following: SL is the class of problems that are logarithmic space reducible to the reachability problem for undirected graphs, UGAP. Our results improve on the one hand the upper bound for the problem given by Luks from NC to ....

Lewis, H., Papadimitriou, C.: Symmetric space-bounded computation. Theoretical Computer Science 19 (1982) 161-187

....proved in [9] that b GI lies in fact in NC. We focus here on the family of colored graphs with color multiplicities bounded by the constants 2 and 3 proving that 2 GI and 3 GI are many one complete for symmetric logarithmic space SL under AC reductions. The complexity class SL introduced in [17] has di erent characterizations, but the easiest way to de ne it is precisely as the class of problems logarithmic space reducible to the reachability problem for undirected graphs, UGAP. Our results improve on the one hand the upper bound for the problem given by Luks from NC to SL (SL is ....

H. Lewis, C. Papadimitriou, Symmetric space-bounded computation, Theoretical Computer Science 19 (1982) 161-187.

....others as the non halting classes. From the definitions it is clear that the containments indicated in the figure hold. In addition to these classes and the standard classes DSPACE(s) and NSPACE(s) we will also refer to the class SSPACE(s) of languages computed by a symmetric Turing machine [25] running in space s. For our purposes it su#ces to know that any language in SSPACE(s) can be reduced in DSPACE(s) to an undirected (s, t) connectivity (USTCON ) problem for a graph on 2 vertices. For the special case where s(n) log n, we write BPL,RL,SL,BPHL, etc. for the various classes. ....

H. Lewis and C. Papadimitiou. Symmetric spacebounded computation. Theoretical Computer Science, 19:161--187, 1982.

....September 28, 1994 Abstract We present a Logspace, many one reduction from the undirected st connectivity problem to its complement. This shows that SL = co Gamma SL. 1 Introduction This paper deals with the complexity class symmetric Logspace, SL, defined by Lewis and Papadimitriou in [LP82] This class can be defined in several equivalent ways: 1. Languages which can be recognised by symmetric nondeterministic Turing Machines that run within logarithmic space. See [LP82] 2. Languages that can be accepted by a uniform family of polynomial size contact schemes (also sometimes ....

....This paper deals with the complexity class symmetric Logspace, SL, defined by Lewis and Papadimitriou in [LP82] This class can be defined in several equivalent ways: 1. Languages which can be recognised by symmetric nondeterministic Turing Machines that run within logarithmic space. See [LP82] 2. Languages that can be accepted by a uniform family of polynomial size contact schemes (also sometimes called switching networks. See [Raz91] 3. Languages which can be reduced in Logspace via a many one reduction to USTCON , the undirected st connectivity problem. A major reason for the ....

Lewis and Papadimitriou. Symmetric space-bounded computation. Theoretical Computer Science, 19, 1982.

....familiarity with computational complexity, see e.g. 21] in particular with circuit complexity, see e.g. 27] L denotes deterministic logarithmic space. SL (symmetric log space) is the class of all problems that can be solved in log space on a symmetric (nondeterministic) Turing machine, see [14] for more details. Important results for SL are the closure of SL under log space bounded Turing reductions, i.e. SL = L SL [19] and the fact that problems in SL can be solved in deterministic space O(log(n) 4 3 ) 3] A collection of SL complete problems can be found in [2] For the de ....

.... y 0 (R(x; x 0 ) R(y 0 ; y) R(x 0 ; x) R(y; y 0 ) By Lemma 11, a R b if and only if A j= STCu; v T (u; v) 0; max) Thus containment in SL follows from [12] In order to show SL hardness we use the SL complete undirected graph accessibility problem (UGAP) see also [14]: INPUT: An undirected graph G = V; E) and two nodes a; b 2 V . QUESTION: Does there exist a path in G from a to b Let G = V; E) a; b be an instance of UGAP, where E ffv; wg j v; w 2 V g and of course V E = We de ne a 2 homogeneous STS R over V [ E by R = f(ce; ec; j c 2 V; e ....

H. R. Lewis and C. H. Papadimitriou. Symmetric space-bounded computation. Theoretical Computer Science, 19(2):161-187, 1982.

....directed graph with two designated nodes s and t decide whether there is a path from s to t) is known to be complete for NL, even in the case of acyclic graphs with in degree at most 2. The accessibility problem for undirected graphs, UGAP, is complete for the class SL, symmetric logarithmic space [21]. This class has different characterizations, but the easiest way to define it is precisely as the class of problems logarithmic space reducible to UGAP. #L defined by [4] analogously to Valiant s class #P, is the class of functions f : Sigma IN that count the number of accepting paths of a ....

H. Lewis and C. Papadimitriou. Symmetric space-bounded computation. Theor. Computer Science 19:161--187, 1982.

....while the author was at the University of Wisconsin Madison Computer Sciences Department under the support of NSF grant CCR 95 10244. canbesolvedinR HL (sometimes written RL poly or simply RL) Since this problem is complete for symmetric logspace (SL) with respect to logspace reductions [8], the relation SL # RHL follows. The most space e#cient deterministic algorithm for USTCON requires space O( log n) 4 3 ) 2] Wefocuson the variant of this problem in which the graph in question is regular of a fixed degree d: d Regular Undirected Graph Connectivity (d USTCON) Instance: A ....

H. Lewis and C. Papadimitriou. Symmetric space-bounded computation. Theoretical Computer Science, 19:161--187, 1982.

....with the label matching the next symbol of ; in case of no match, stay put during that step and continue with the next symbol of . If we can construct universal traversal sequences in logspace, then we can solve undirected graph connectivity in logspace, and symmetric logspace equals logspace [LP82] However, we do not know how to generate universal traversal sequences in logspace or even in polynomial time. Aleliunas et al. AKL 79] showed that most sequences of length O(n 3 ) over f1; 2; n 1g are universal traversal sequences for size n, but as of now the best explicit ....

H. Lewis and C. Papadimitriou. Symmetric space-bounded computation. Theoretical Computer Science, 19(2):161-187, 1982.

....There is an analogous problem, called UGAP, for undirected graphs. It follows at once that UGAP # NL. However, it is an open question whether UGAP is complete for NL. The complexity class SL (symmetric log space) is defined in such a way that UGAP is complete for SL. See Lewis and Papadimitriou[16] for details. The completeness of UGAP for NL is equivalent to the assertion that SL = NL. A set A can be viewed as an algebra in which the set of basic operations is empty. In that case, for any subset # of A 2 , Cg A (#) is nothing but the smallest equivalence relation on A containing #. ....

H. R. Lewis and C. H. Papadimitriou, Symmetric space-bounded computation, Theoret. Comput. Sci. 19 (1982), no. 2, 161--187.

....P complete. When restricted to undirected graphs with only existential nodes, this problem is equivalent to the undirected graph accessibility problem, called UGAP or USTCON, which is known to be complete for the special case of nondeterministic log space known as symmetric log space (SL) [LP82]. A.2.4 Hierarchical Graph Accessibility Problem (HGAP) Given: A hierarchical graph G = V; E) and two designated vertices s and t. A hierarchical graph Gamma = G 1 ; G k ) consists of k subcells G i , 1 i k. Each subcell is a graph that 52 ffl A Compendium of Problems Complete for P ....

H. R. Lewis and C. H. Papadimitriou. Symmetric space-bounded computation. Theoretical Computer Science, 19:161--187, 1982.

....can be simulated by a deterministic algorithm that is simultaneously in polynomial time and O(log 2 n) space. Using Nisan s generator, Nisan, Szemer edi, Wigderson [NSW92] showed that undirected (s; t) connectivity, which is in RL [AKLLR79] and is complete for the complexity class SL [LP82], can be computed in DSPACE(log 3=2 n) It was this result that motivated our research that we present in this work. As with these other results, Nisan s pseudorandom generator is a major component of our simulation. One key observation that enables us to apply Nisan s generator to ....

H. Lewis and C. Papadimitiou. Symmetric space-bounded computation. Theoretical Computer Science, 19:161-187, 1982.

....in log space is solvable deterministically in log space. In addition, proving stcon 2 DL would, by a padding argument, show that NSPACE(f(n) DSPACE(f(n) for all space constructible f(n) Omega# log n) Sav70, Theorem 3] ustcon is complete for symmetric logarithmic space (SL) LP82] a seemingly weaker variant of NL where a configuration A derives a configuration B if and only if B derives A. ustcon is known to be solvable by randomized logarithmic space, polynomial expected time algorithms, even errorless ones [AKL 79, BCD 89] De1 The choice of logarithmic here is ....

H. R. Lewis and C. H. Papadimitriou. Symmetric space-bounded computation. Theoretical Computer Science, 19(2):161--187, August 1982.

.... application is due to Aleliunas, Karp, Lipton, Lov asz and Rackoff [2] who used random walks to show that the undirected graph connectivity (USTCON) problem is in RHL (sometimes denoted RL poly or just RL) Since USTCON is complete for symmetric logspace (SL) with respect to logspace reductions [10], the relation SL RHL follows. The most space efficient known deterministic algorithm for USTCON requires space O( log n) 4=3 ) 3] We define d regular undirected graph connectivity (d USTCON) to be the variant of this problem in which the graph in question is regular of a fixed degree d: ....

H. Lewis and C. Papadimitriou. Symmetric space-bounded computation. Theoretical Computer Science, 19:161--187, 1982.

....the graph nodes in polynomial time, thus showing that USTCON can be solved in RL a result we already mentioned in section 1.1.1. So STCON is as hard as NL, while USTCON is not harder than RL which looks easier than NL. That s the time for a name for a new Class Indeed Lewis and Papadimitriou [LP82] defined a class SL, Symmetric Logspace, and showed that the following definitions are equivalent: 1. Languages which can be reduced in Logspace via a many one reduction to USTCON, the undirected st connectivity problem. 2. Languages which can be recognized by symmetric nondeterministic Turing ....

....that the following definitions are equivalent: 1. Languages which can be reduced in Logspace via a many one reduction to USTCON, the undirected st connectivity problem. 2. Languages which can be recognized by symmetric nondeterministic Turing Machines that run within logarithmic space. See [LP82] 3. Languages that can be accepted by a uniform family of polynomial size contact schemes (also sometimes called switching networks. See [Raz91] In particular, the Aleliunas et al. result shows that SL RL. Adding this to the former inclusions we get: L SL RL NL. If we have to guess ....

Lewis and Papadimitriou. Symmetric space-bounded computation. Theoretical Computer Science, 19, 1982.

....(USTCON) Given a graph G with n vertices and m edges, and given two vertices s and t of G, we are to decide if s and t are in the same connected component. We are interested in space bounded algorithms for USTCON, which is an important problem in the study of space bounded complexity classes [3, 9]. Throughout this paper, we assume that our workspace takes the form of p registers, each capable of storing a log n bit number. There are two well known approaches to solving USTCON: via a deterministic graph search on G (e.g. depth first search) and via a simulation of a random walk on G [1] ....

H. R. Lewis and C. H. Papadimitriou. Symmetric space-bounded computation. Theoretical Computer Science, 19:161--187, 1982. 14

....the label matching the next symbol of oe; in case of no match, stay put during that step and continue with the next symbol of oe. If we can construct universal traversal sequences in logspace, then we can solve undirected graph connectivity in logspace, and symmetric logspace equals logspace [LP82] However, we do not know how to generate universal traversal sequences in logspace or even in polynomial time. Aleliunas et al. AKL 79] showed that most sequences of length O(n 3 ) over f1; 2; n Gamma 1g are universal traversal sequences for size n, but as of now the best ....

H. Lewis and Ch. Papadimitriou. Symmetric space-bounded computation. Theoretical Computer Science, 19(2):161--187, 1982.

....it is possible to keep track of the complete history of a computation without increasing the running time. It is remarkable, that this is precisely the same construction which shows the equivalence of nondeterminism with symmetry and of determinism with reversability in the time bounded case ([14,3]) The main result of this work is to show that the unambiguous logspace class RUSPACE(logn) possesses a complete problem. The proof makes intensive use of the space specific possibility to cycle through all configurations of a machine without increasing the resource bound. The proof doesn t seem ....

P. Lewis and C.H. Papadimitriou. Symmetric space-bounded computation. Theoret. Comput. Sci., 19:161--187, 1982.

....in polynomial time. Also note that this property can be generalized (in the obvious way) to operators having arbitrarily many postconditions. 4.4 Complexity of Symmetric Planning In this Subsection, we prove Theorem 18. First, we introduce the concept of symmetric Turing machines (as defined by Lewis and Papadimitriou [1982]) with the aid of peeking Turing machines. Then, the acceptance problem for polynomially space bounded symmetric TMs is shown to be Pspace hard and reduced to the plan existence problem for symmetric PSN instance. The reduction is similar to (but considerably more complex than) the reduction used ....

....t Gamma1 1 ; t Gamma1 k ; p) where for 1 i k. If t i = ff i ; D i ; fi i ) then t Gamma1 i = fi i ; GammaD i ; ff i ) The PTM M is symmetric iff ffi Gamma1 2 Delta whenever ffi 2 Delta. This implies that if C M C 0 , then C 0 M C for all C 2 C(M ) Theorem 27 [Lewis and Papadimitriou, 1982] Let S be any function from N to N . If a language L is accepted in space S by a k tape symmetric PTM, k 2, then L is accepted in space S by a 2 tape symmetric PTM. Lemma 28 The class of languages accepted by symmetric PTMs operating in polynomial space is Pspace hard to recognize. Proof: ....

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H. R. Lewis, C. H. Papadimitriou, Symmetric space-bounded computation, Theoret. Comput. Sci. 19 (1982) 161-- 187.

....Ta Shma z November 9, 1995 Abstract We present a Logspace, many one reduction from the undirected st connectivity problem to its complement. This shows that SL = coSL. 1 Introduction This paper deals with the complexity class symmetric Logspace, SL, defined by Lewis and Papadimitriou in [LP82] This class can be defined in several equivalent ways: 1. Languages which can be recognised by symmetric nondeterministic Turing Machines that run within logarithmic space. See [LP82] 2. Languages that can be accepted by a uniform family of polynomial size contact schemes (also sometimes ....

....This paper deals with the complexity class symmetric Logspace, SL, defined by Lewis and Papadimitriou in [LP82] This class can be defined in several equivalent ways: 1. Languages which can be recognised by symmetric nondeterministic Turing Machines that run within logarithmic space. See [LP82] 2. Languages that can be accepted by a uniform family of polynomial size contact schemes (also sometimes called switching networks. See [Raz91] 3. Languages which can be reduced in Logspace via a many one reduction to USTCON, the undirected st connectivity problem. A major reason for the ....

Lewis and Papadimitriou. Symmetric space-bounded computation. Theoretical Computer Science, 19, 1982.

....Szemer edi and Wigderson [NSW92] 1 Introduction Undirected st connectivity (USTCON) is a fundamental computational problem, and algorithms for it serve as basic subroutines for more complex graph problems. It is complete for the class SL of symmetric nondeterministic log space computations [LP82] and is a subproblem of Directed st connectivity, which captures the class NL of general nondeterministic computation. The combinatorics of USTCON, as well as its time complexity, are extremely well understood. However, its space complexity is still a mystery, which was a source of some ....

Harry R. Lewis and Cristos H. Papadimitriou. Symmetric space-bounded computation. Theoretical Computer Science, 19:161-- 187, 1982.

.... problem for directed graphs (stcon) is the prototypical complete problem for nondeterministic logarithmic space [7] Both stcon and the undirected version of the problem, ustcon, are DLOG hard any problem solvable deterministically in logarithmic space can be reduced to either problem [4, 7]. Establishing the deterministic space complexity of stcon would tell us a great deal about the relationship between deterministic and nondeterministic space bounded complexity classes. For example, showing a deterministic log space algorithm for directed connectivity would prove that DSPACE(f(n) ....

H. R. Lewis and C. H. Papadimitriou. Symmetric space-bounded computation. Theoretical Computer Science, 19(2):161--187, Aug. 1982.

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H. R. Lewis and C. H. Papadimitriou. Symmetric space-bounded computation. Theoretical Computer Science, 19, 1982.

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Lewis and Papadimitriou. Symmetric space-bounded computation. Theoretical Computer Science, 19, 1982.

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Lewis, C. Papadimitriou, Symmetric space-bounded computation, Theoret. Comput. Sci. 19 (1982) 161-187.

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