Chicago Journal of Theoretical Computer Science

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Volume 1995, Article 1 30 June, 1995

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• anonymous ftp at cs.uchicago.edu Nisan and Ta-Shma Symmetric Logspace (Info)

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[ii]

Symmetric Logspaceis Closed Under Complement

Noam Nisan  Amnon Ta-Shma

30 June, 1995

Abstract

We present a Logspace, many-one reduction from the undirected

Abstract-1

s–t connectivity problem to its complement. This shows that SL = coSL.

1 Introduction

This paper deals with the complexity class symmetric Logspace, SL, defined

1-1

by Lewis and Papadimitriou in [LP82]. This class can be defined in several equivalent ways:

1. Languages that can be recognized by symmetric, nondeterministic Turing Machines that run within logarithmic space. See [LP82].

2. Languages that can be accepted by a uniform family of polynomialsize contact schemes (these may also be called switching networks). See [Raz91].

3. Languages that can be reduced in Logspace via a many-one reduction to USTCON, the undirected s–t connectivity problem.

A major reason for the interest in this class is that it captures the com-

1-2

plexity of USTCON. The input to USTCON is an undirected graph G, and two vertices in it, s and t. The input should be accepted if s and t are connected via a path in G. The similar problem STCON, where the graph G has directed edges, is complete for non-deterministic Logspace (NL). Several

Nisan and Ta-Shma Symmetric Logspace §2.1

1 combinatorial problems are known to be in SL or coSL (e.g., 2-colorability is complete for coSL [Rei82]).

The following facts are known regarding SL relative to other complexity

1-3

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 time. The containment SL ⊆ RL is the only non-trivial one in the line above and follows directly from the randomized Logspace algorithm for USTCON of [AKL^{+}79]. It is also known that SL ⊆ SC [Nis92], SL ⊆ ⨁ L [KW93], and SL ⊆ DSPACE(log^{1.5} n) [NSW92].

After the surprising proofs that NL is closed under complement were

1-4

found [Imm88, Sze88], Borodin et al. [BCD^{+}89] asked whether the same is true for SL. They could prove only the weaker statement, namely that SL ⊆ coRL, and left “SL = coSL?” as an open problem. In this paper we solve the problem in the affirmative by exhibiting a Logspace, many-one reduction from USTCON to its complement. Quite surprisingly, the proof of our theorem does not use inductive counting, as do the proofs of NL = coNL. This is a simpler proof than Borodin’s, et al.; however, it uses the [AKS83] sorting networks.

Theorem 1 SL = coSL

It should be noted that the monotone analogs (see [GS91]) of SL and coSL are known to be different [KW88].

As a direct corollary of our theorem, we obtain L^{⟨SL⟩} = SL where L^{⟨SL⟩}

1-5

is the class of languages accepted by Logspace oracle Turing machines with oracle from SL, using the oracle model of [RST84].

Corollary 1.1 L^{⟨SL⟩} = SL

In particular, we show that both “symmetric Logspace hierarchies” (the

1-6

one defined by alternation in [Rei82] and the one defined by oracle queries in [BPS92]) collapse to SL.

2

2 Proof of Theorem

2.1 Overview of proof

We design a many-one reduction from coUSTCON to USTCON. We start

2.1-1

by developing simple tools for combining reductions in subsection 2.2. In particular, these tools will allow us to use the AKS sorting networks in order to “count.” At this point, the main ingredient of the reduction will be the calculation of the number of the connected components of a graph. An upper bound to this number is easily obtained using transitive closure, while the main idea of the proof is to obtain a lower bound by computing a spanning forest of the graph. We do this in subsection 2.3. Everything is put together in subsection 2.4.

2.2 Projections to USTCON

In this paper we will use only the simplest kind of reductions (i.e., Logspace-

2.2-1

uniform projection reductions [SV85]). Moreover, we will be interested only in reductions to USTCON. We define this kind of reduction and we show some of its basic properties in this subsection.

Notation 2.1 Given f: {0, 1}^{∗} → {0, 1}^{∗}, denote by f_{n}: {0, 1}^{n} → {0, 1}^{∗} the restriction of f to inputs of length n. Denote by f_{n,k} the kth bit function of f_{n} (i.e., if f_{n}: {0, 1}^{n} → {0, 1}^{m}, then f_{n}(x⃗) = (f_{n,1}(x⃗), …, f_{n,m}(x⃗))).

Notation 2.2 We represent an n-node undirected graph G using \binom{n}{2} variables x⃗ = (x_{i,j})_{1≤i<j≤n} s.t. x_{i,j} is 1 iff (i, j) ∈ E(G). If f(x⃗) operates on graphs, we will write f(G), meaning that the input to f is a binary vector of length \binom{n}{2} representing G.

We say that f: {0, 1}^{∗} → {0, 1}^{∗} reduces to USTCON(m) if we can,

2.2-2

uniformly and in Logspace, label the edges of a graph of size m with { 0, 1, x_{i}, ¬x_{i} | 1 ≤ i ≤ n }, s.t. f_{n,k}(x⃗) = 1 if and only if there is a path from 1 to k in the corresponding graph. We may formalize this with a definition.

Definition 2.1 We say that f: {0, 1}^{∗} → {0, 1}^{∗} reduces to USTCON(m), m = m(n), if there is a uniform family of Space(log(n)) functions {σ_{n,k}} s.t. for all n and k:

3 • σ_{n,k} is a projection (i.e., σ_{n,k} is a mapping from { {i, j} | 1 ≤ i < j ≤ m }

to { 0, 1, x_{i}, ¬x_{i} | 1 ≤ i ≤ n }). We abuse the notation by writing σ_{n,k}(i, j) instead of σ_{n,k}({i, j}).

• Given x⃗ define G_{x⃗,k} to be the graph G_{x⃗,k} = ({1, …, m}, E) where

E = { (i, j) | σ_{n,k}(i, j) = 1 or (σ_{n,k}(i, j) = x_{i} and x_{i} = 1)
or (σ_{n,k}(i, j) = ¬x_{i} and x_{i} = 0) }

• f_{n,k}(x⃗) = 1 ⟺ there is a path from 1 to m in G_{x⃗,k}.

If σ is restricted to the set { 0, 1, x_{i} | 1 ≤ i ≤ n }, we say that f monotonically reduces to USTCON(m).

Lemma 2.1 If f has uniform monotone formulae of size s(n), then f is monotonically reducible to USTCON(O(s(n))).

Proof of Lemma 2.1 Given a formula ϕ, recursively build (G, s, t) as follows:

• If ϕ = x_{i}, then build a graph with two vertices s and t and one edge between them labeled x_{i}.

• If ϕ = ϕ_{1} ∧ ϕ_{2}, and (G_{i}, s_{i}, t_{i}), then the graphs for ϕ_{i}, i = 1, 2 identify s_{2} with t_{1}, and define s = s_{1}, t = t_{2}.

• If ϕ = ϕ_{1} ∨ ϕ_{2}, and (G_{i}, s_{i}, t_{i}), then the graphs for ϕ_{i}, i = 1, 2 identify s_{1} with t_{1} and s_{2} with t_{2}, and define s = s_{1} = t_{1} and t = s_{2} = t_{2}.

Proof of Lemma 2.1 □

2.2-3

Definition 2.2 Sort: {0, 1}^{n} → {0, 1}^{n} is the Boolean sorting function (i.e., it moves all the zeros to the beginning of the string).

Using the AKS sorting networks [AKS83], which belong to NC^{1}, we derive the following corollary.

Corollary 2.2 Sort is monotonically reducible to USTCON(poly).

4 Lemma 2.3 If f monotonically reduces to USTCON(m_{1}), and g reduces to USTCON(m_{2}), then f ∘ g reduces to USTCON(m_{1}^{2} · m_{2}), where ∘ is the standard function composition operator.

Proof of Lemma 2.3 The function f monotonically reduces to a graph with m_{1} vertices, where each edge is labeled with one of {0, 1, x_{i}}. In the composition function f ∘ g, each x_{i} is replaced by x_{i} = g_{i}(y⃗), which can be reduced to a connectivity problem of size m_{2}. Replace each edge labeled x_{i} with its corresponding connectivity problem. There can be m_{1}^{2} edges, each replaced by a graph with m_{2} vertices. The resulting new graph has m_{1}^{2} · m_{2} vertices.

Proof of Lemma 2.3 □

2.3 Finding a spanning forest

We show how to build a spanning forest using USTCON in this section. Reif

2.3-1

[Rei82], and independently, Cook, have previously noted this idea. Given a graph G, index the edges from 1 to m. We can view the indices

2.3-2

as weights to the edges, and, as no two edges have the same weight, we know that there is a unique minimal spanning forest F. In our case, where the edges are indexed, this minimal forest is the lexicographically first spanning forest.

It is well known that the greedy algorithm finds a minimal spanning forest.

2.3-3

Let us recall how the greedy algorithm works in our case. The algorithm builds a spanning forest F which is empty at the beginning, F = ∅. Then the algorithm checks the edges one by one according to their order. For each edge e, if e does not close a cycle in F, then e is added to the forest. This may be stated as F = F ∪ {e}.

At first glance, the algorithm looks sequential. However, claim 2.4.1 shows

2.3-4

that the greedy algorithm is actually highly parallel. To check whether an edge participates in the forest, we need only one s–t connectivity problem over an easily-obtainable graph.

Definition 2.3 For an undirected graph G denote by LFF(G) the lexicographically first spanning forest of G. Let SF(G) ∈ \{0, 1\}^{\binom{n}{2}} be:

SF_{i,j}(G) = \{ \begin{matrix}0 & (i, j) ∈ LFF(G) \\ 1 & otherwise \end{matrix}

5 Lemma 2.4 SF reduces to USTCON(poly).

Proof of Lemma 2.4 Let F be the lexicographically first spanning forest of G. For e ∈ E, define G_{e} to be the subgraph of G containing only the edges { e′ ∈ E | index(e′) < index(e) }.

Claim 2.4.1 e = (i, j) ∈ F ⟺ e ∈ E and i is not connected to j in G_{e}.

Proof of Claim 2.4.1 Let e = (i, j) ∈ E. Denote by F_{e} the forest which the greedy algorithm built at the time it was checking e. So e ∈ F if and only if e does not close a cycle in F_{e}.

(⟹) e ∈ F and therefore, e does not close a cycle in F_{e}, but then e does not close a cycle in the transitive closure of F_{e}, and in particular e does not close a cycle in G_{e}.

(⟸) e does not close a cycle in G_{e} therefore, e does not close a cycle in F_{e} and e ∈ F.

Proof of Claim 2.4.1 □

Therefore, SF_{i,j}(G) = ¬x_{i,j} or i is connected to j in G_{(i,j)}. Because ¬x_{i,j} can be viewed as the connectivity problem over the graph with two vertices and one edge labeled ¬x_{i,j}, it follows from lemmas 2.1 and 2.3 that SF reduces to USTCON. Notice, however, that the reduction is not monotone.

Proof of Lemma 2.4 □

2.4 Putting it together

First, we want to build a function that takes one representative from each

2.4-1

connected component. We define LI_{i}(G) to be 0 if and only if the vertex i has the largest index in its connected component.

Definition 2.4 LI(G) ∈ {0, 1}^{n}

LI_{i}(G) = \{ \begin{matrix}0 & i has the largest index in its connected component \\ 1 & otherwise \end{matrix}

Lemma 2.5 LI reduces to USTCON(poly).

6 Proof of Lemma 2.5 LI_{i}(G) = ⋁_{j=i+1}^{n}(i is connected to j in G). So LI is a simple monotone formula over connectivity problems, and LI reduces to USTCON by lemmas 2.1 and 2.3. This is, actually, a monotone reduction.

Proof of Lemma 2.5 □

Using the spanning forest and the LI function, we can exactly compute

2.4-2

the number of connected components of G (i.e., given G, we can compute a function NCC_{i} which is 1 iff there are exactly i connected components in G).

Definition 2.5 NCC(G) ∈ {0, 1}^{n}

NCC_{i}(G) = \{ \begin{matrix}1 & there are exactly i connected components in G \\ 0 & otherwise \end{matrix}

Lemma 2.6 NCC reduces to USTCON(poly).

Proof of Lemma 2.6 Let F be a spanning forest of G. It is easy to see that if G has k connected components, then |F | = n − k. Define:

f(G) = Sort ∘ LI(G)
g(G) = Sort ∘ SF(G)

Then:

\begin{aligned}f_{i}(G) = 1 & ⟹ & k < i \\ g_{i}(G) = 1 & ⟹ & n − k < i & ⟹ & k > n − i, \end{aligned}

and thus:

NCC_{i}(G) = f_{i+1}(G) ∧ g_{n−i+1}(G)

Therefore, applying lemmas 2.1, 2.2, 2.3, 2.4, and 2.5 proves the lemma.

Proof of Lemma 2.6 □

Finally, we can reduce the non-connectivity problem to the connectivity

2.4-3

problem, thus proving that SL = coSL.

Lemma 2.7 coUSTCON reduces to USTCON(poly).

7 Nisan and Ta-Shma Symmetric Logspace §3

Proof of Lemma 2.7 Given (G, s, t), define G^{+} to be the graph G∪{(s, t)}. Denote by #CC(H) the number of connected components in the undirected graph H.

s is not connected to t in G ⟺ #CC(G^{+}) = #CC(G) − 1
⟺ ⋁_{i=2}^{n} NCC_{i}(G) ∧ NCC_{i−1}(G^{+})

Therefore, applying lemmas 2.1, 2.3, and 2.6 proves the lemma.

Proof of Lemma 2.7 □

3 Extensions

Denote by L^{⟨SL⟩} the class of languages accepted by Logspace oracle Turing

3-1

machines with an oracle from SL. An oracle Turing machine has a work tape and a write-only query tape (with unlimited length) which is initialized after every query. We get: Corollary 3.1 L^{⟨SL⟩} = SL.

Proof of Corollary 3.1 Let M be an oracle Turing Machine running in

Corollary 3.1 Proof-1

L^{⟨SL⟩}, and fix an input x⃗ to M. We build the “configuration” graph G(V, E) of M using the following process:

• Let V contain all possible configurations.

• Then, (v, w) ∈ E with the label “q is (not) s–t connected,” if, starting from configuration v, the next query is q. If the oracle answers that “q is (not) connected,” then the machine moves to configuration w.

Notice that we can ignore the direction of the edges, as backward edges

Corollary 3.1 Proof-2

do not benefit us. The reason is that from any vertex v there is only one forward edge leaving v that can be traversed (i.e., whose label matches the oracle’s answer). Therefore, if we reach v using a “backward edge” w→v, then the only forward edge leaving v that can be traversed is v→w.

Now we can replace query edges labeled “q is connected” with the s–t

Corollary 3.1 Proof-3

connectivity problem q, and edges labeled “q is not connected” with the s–t connectivity problem obtained using our theorem that SL = coSL, resulting in one, not too big, s–t connectivity problem. It is also clear that this can be done in Logspace, completing the proof.

Proof of Corollary 3.1 □

Nisan and Ta-Shma Symmetric Logspace (Ref)

8 As the symmetric Logspace hierarchy defined in [Rei82] is known to be

3-2

within L^{⟨SL⟩}, this hierarchy collapses to SL.

As can be easily seen, the above argument holds for any undirected graph

3-3

with undirected query edges, which is exactly the definition of SL^{⟨SL⟩} given by [BPS92]. Thus, SL^{⟨SL⟩} = SL, and, by induction, the SL hierarchy defined in [BPS92] collapses to SL.

4 Acknowledgments

We would like to thank Amos Beimel, Allan Borodin, Assaf Schuster, Robert

4-1

Szelepcsényi, and Avi Wigderson for helpful discussions.

Acknowledgement of support: This work was supported by BSF Grant 92-00043 and by a Wolfeson Award administered by the Israeli Academy of Sciences. The work was revised while visiting BRICS, Basic Research in Computer Science, Centre of the Danish National Research Foundation. A preliminary version of this paper appeared in the proceedings of the 27th Annual ACM Symposium on the Theory of Computing, 1995 .

References for CJTCS Volume 1995, Article 1

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