J. H. Reif. Symmetric complementation. In Proceedings of the 14th ACM Symposium on Theory of Computing, pages 201--214. ACM SIGACT, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....be accepted if s and t are connected via a path in G. The similar problem, STCON , where the graph G is allowed to be directed is complete for NL, non deterministic Logspace. Several combinatorial problems are known to be in SL or co Gamma SL, e.g. 2 colourability is complete in co Gamma SL [Rei82] The following facts are known regarding SL relative to other complexity classes in the vicinity : L SL RL NL: Here, L is the class deterministic Logspace and RL is the class of problems that can be accepted with one sided error by a randomized Logspace machine running in polynomial ....

.... theorem, we get that L = SL where L is the class of languages accepted by Logspace oracle Turing machines with oracle from SL, and is defined similarly, being careful with the way we allow queries (see [RST82] This also shows that the symmetric Logspace hierarchy defined in [Rei82] collapses to SL. 2 Proof of Theorem 2.1 Overview of proof. We show that we can upper and lower bound the number of connected components of a graph, using connectivity problems. We upper bound this number using a transitive closure method, which can be easily done since we are allowed to ....

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J. H. Reif. Symmetric complementation. In Proc. 14th ACM Symposium on Theory of Computing (STOC), pages 201--214, 1982. 7

....at all, they show that RL is very close to L, which turns the L 6= NL question into RL 6= NL, which looks wide open. 19 show that SL coRL. However, this technique failed to solve the problem whether the class SL is closed under complement. As a consequence, an SL hierarchy was built [Rei82, BCD 89] and turned out to contain many interesting problems, such as 2 colorability [Rei82] In a result co authored with my advisor Noam Nisan, we develop a new technique and show that SL = coSL, collapsing, in particular, the SL hierarchy. 20 Chapter 2 Explicit Extractors In this ....

....6= NL, which looks wide open. 19 show that SL coRL. However, this technique failed to solve the problem whether the class SL is closed under complement. As a consequence, an SL hierarchy was built [Rei82, BCD 89] and turned out to contain many interesting problems, such as 2 colorability [Rei82] In a result co authored with my advisor Noam Nisan, we develop a new technique and show that SL = coSL, collapsing, in particular, the SL hierarchy. 20 Chapter 2 Explicit Extractors In this chapter we present the currently known techniques for building explicit extractors. In section 2.2 ....

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J. H. Reif. Symmetric complementation. In Proc. 14th ACM Symposium on Theory of Computing (STOC), pages 201--214, 1982.

....of Computer Science, Hebrew University of Jerusalem. email: am cs.huji.ac. il 1 similar problem STCON, where the graph G has directed edges, is complete for nondeterministic Logspace (NL) Several combinatorial problems are known to be in SL or coSL, e.g. 2 colourability is complete for coSL [Rei82] The following facts are known regarding SL relative to other complexity classes in the vicinity : L SL RL NL: Here, L is the class deterministic Logspace and RL is the class of problems that can be accepted with one sided error by a randomized Logspace machine running in polynomial ....

....where L SL is the class of languages accepted by Logspace oracle Turing machines with oracle from SL, being careful with the way we allow queries (see [RST84] Corollary 1. 1 L SL = SL In particular we show that both symmetric Logspace hierarchies , the one defined by alternation in [Rei82] and the one defined by oracle queries in [BPS92] collapse to SL. 2 Proof of Theorem 2.1 Overview of proof. We design a many one reduction from coUSTCON to USTCON. We start by developing, in subsection 2.2, simple tools for combining reductions. In particular these tools 2 will allow us to ....

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J. H. Reif. Symmetric complementation. In Proc. 14th ACM Symposium on Theory of Computing (STOC), pages 201--214, 1982.

....Similarly, we show that a planar embedding, when one exists, can be found in FL SL . Previously, these problems were known to reside in the complexity class AC 1 , via a O(log n) time CRCW PRAM algorithm [22] although planarity checking for degree three graphs had been shown to be in SL [23, 20]. 1 Introduction The problem of determining if a graph is planar has been studied from several perspectives of algorithmic research. From most perspectives, optimal algorithms are already known. Linear time sequential algorithms were presented by Hopcroft and Tarjan [10] and (via another ....

.... so far is the bound of AC 1 that follows from the logarithmic time CRCW PRAM algorithm of Ramachandran and Reif [22] In a recent survey of problems in the complexity class SL [2] the planarity problem for graphs of bounded degree is listed as belonging to SL, but this is based on the claim in [23] that checking planarity for bounded degree graphs is in the Symmetric Complementation Hierarchy , and on the fact that SL is closed under complement [20] and thus this hierarchy collapses to SL) However, the algorithm presented in [23] actually works only for graphs of degree 3, and no ....

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J. Reif. Symmetric complementation. Journal of the ACM, 31(2):401--421, 1984.

....a solution to P . Since the set of relations is bipartite and partition equivalent we name the partitions X and Y and construct a function f : V f0; 1g, as follows: f(v) 0 if h(v) 2 X; 1 if h(v) 2 Y . Clearly, f is a 2 coloring of G V . It is known that 2 coloring is co Gamma SL complete [11], that is, it is complete for the complement of the class of symmetric logspace problems. Moreover, Nisan and Ta Schma have shown that SL = co Gamma SL [13] which gives us the following: Corollary 1. CSP( Gamma ) is SL complete. Proof. In the proof of Theorem 4, CSP( Gamma ) is trivially ....

....Moreover, Nisan and Ta Schma have shown that SL = co Gamma SL [13] which gives us the following: Corollary 1. CSP( Gamma ) is SL complete. Proof. In the proof of Theorem 4, CSP( Gamma ) is trivially reduced to 2 colorability (and vice versa) Thus we can immediately apply Reif s result [11] followed by Nisan and Ta Schma s result [13] 4.1 Non closure properties of lpn In this section we will show that some sets of relations in lpn are not closed under constant, majority, ACI, or affine functions. Consider the graph C 6 in Fig. 2 representing a bipartite relation i j k a b c ....

Reif. J.H. Symmetric complementation. In Proceedings of the 14th ACM Symposium on Theory of Computing, pages 210 -- 214, 1982.

....machines, where each processor may follow a pointer in unit time, pointer jumping is easily possible in O(lg n) time by pointer doubling. In fact, using randomization, such pointer machines can even evaluate a constant depth polynomial size circuit of ustconn and negation gates in O(lg n) time [34]. We use the communication game method of Karchmer and Wigderson, although since we must work with a weaker function than ustconn, the proof will differ in a number of essential ways. Note that we already know that mL is strictly weaker than mNL, simply because mL is closed under complementation. ....

J. H. Reif. Symmetric complementation. J. ACM, 31(2):401--421, 1984.

....Remarks: This problem is also called UGAP by many authors. See Problem 5.1 for additional comments. 5.3 k Vertex Disjoint Paths (k PATHS) Given: An undirected graph G = V; E) and two designated vertices s and t. Problem: Are there k vertex disjoint paths from s to t Reference: Reif [24]. Hint: Observe 1 PATH is USTCON, Problem 5.1. To show the problem is in SL, Reif notes that for any graph G = V; E) and vertices s; t 2 V , the k PATHS instance G, s, and t has a yes answer if and only if for all v 1 ; v k Gamma1 2 V Gamma fs; tg, the USTCON instances G 0 , s, and ....

....; v k . Problem: Is v in the k connected component determined by v 1 ; v k A k connected component is a maximal k connected subgraph. A graph H = W; F ) is k connected if for all distinct vertices w 1 ; w 2 2 W , there exist k vertex disjoint paths from w 1 to w 2 . Reference: Reif [24]. Hint: MemkCC is in SL since instance G, v, and v 1 ; v k is yes if and only if 1ik k PATHS G; v; and v i ; and Lemma 2.3 applies. For hardness reduce USTCON, Problem 5.1, to MemkCC. Given an instance G, s, and t ask whether G, s, and t is an instance of Mem1CC. Remarks: The ....

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John H. Reif. Symmetric complementation. Journal of the ACM, 31(2):401--421, April 1984.

....tell us anything new about nondeterministic time. Soon after we proved Theorem 1, Tompa et al. 1] gave two extensions: they proved that LOG(CFL) the set of problems log space reducible to a context free language is closed under complementation, and they showed that Symmetric Log Space (cf. [11, 13]) is contained in ZPLP, the class of errorless probabilistic Turing machines running in O[log n] space and polynomial expected time. We suggest the following open problems: 1. Is (FO without pos TC) closed under complementation 2. Is Symmetric Log Space, equivalently (FO pos STC) ....

J. Reif, "Symmetric Complementation," JACM 31, No. 2, April (1984), 401-421.

....be accepted if s and t are connected via a path in G. The similar problem, STCON , where the graph G is allowed to be directed is complete for NL, non deterministic Logspace. Several combinatorial problems are known to be in SL or co Gamma SL, e.g. 2 colourability is complete in co Gamma SL [Rei82] The following facts are known regarding SL relative to other complexity classes in the vicinity : L SL RL NL: Here, L is the class deterministic Logspace and RL is the class of problems that can be accepted with one sided error by a randomized Logspace machine running in polynomial ....

....by Logspace oracle Turing machines with oracle from SL, and SL SL is defined similarly, being careful with the way we allow queries (see [RST84] Corollary 1. 1 L SL = SL SL = SL In particular this shows that both symmetric Logspace hierarchies , the one defined by alternation in [Rei82] and the one defined by oracle queries in [BALPS94] collapse to SL. 2 Proof of Theorem 2.1 Overview of proof. We design a many one reduction from co Gamma USTCON to USTCON . We start by developing, in subsection2.2, simple tools for combining reductions. In particular these tools will allow ....

[Article contains additional citation context not shown here]

J. H. Reif. Symmetric complementation. In Proc. 14th ACM Symposium on Theory of Computing (STOC), pages 201--214, 1982.

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Reif,J.H., `Symmetric Complementation,' Journal of the ACM, vol.31, no.2, 1984, pp.401-421.

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J.H. REIF, Symmetric Complementation, J. ACM, 31(2), 1984a, pp. 401-421.

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J. H. Reif. Symmetric complementation. In Proceedings of the 14th ACM Symposium on Theory of Computing, pages 201--214. ACM SIGACT, 1982.

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J. H. Reif. Symmetric complementation. In Proc. 14th ACM Symposium on Theory of Computing (STOC), pages 201--214, 1982.

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John Reif. Symmetric complementation. Journal of the ACM, 31(2):401--421, 1984. 20

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J. Reif, Symmetric Complementation, Journal of the ACM, vol. 31(2), pp 401-421, 1984.

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John H. Reif, Symmetric complementation. Journal of the ACM 31(2) (1984), 401--421.

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John Reif, "Symmetric Complementation," JACM 31, No. 2, April, 1984, (401-421).

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