Skip to main content

Questions tagged [logspace]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
31 votes
1 answer
910 views

Treewidth and the NL vs L Problem

ST-Connectivity is the problem of determining whether there exists a directed path between two distinguished vertices $s$ and $t$ in a directed graph $G(V,E)$. Whether this problem can be solved in ...
Shiva Kintali's user avatar
26 votes
2 answers
983 views

What are the consequences of $L = \oplus L$?

Shiva Kintali has just announced a (cool!) result that graph isomorphism for bounded treewidth graphs of width $\geq 4$ is $\oplus L$-hard. Informally, my question is, "How hard is that?" We know ...
Aaron Sterling's user avatar
22 votes
0 answers
790 views

What is the power of general poly-size permutation branching programs?

Call $\mathsf{PPBP}$ the class of languages decided by poly-size families of permutation branching programs, which are layered branching programs (i.e., the ones defined here) whose transitions ...
Damiano Mazza's user avatar
17 votes
0 answers
512 views

What if an $\mathsf L$-complete problem has $\mathsf{NC}^1$ circuits? More generally, what evidence is there against $\mathsf{NC}^1=\mathsf{L}$?

Edit: let me reformulate the question in a more specific way (and change the title accordingly). A slightly edited version of the original question follows. Is there a result comparable to the Karp-...
Damiano Mazza's user avatar
15 votes
1 answer
483 views

Complexity of the search version of 2-SAT assuming $\mathsf{L = NL}$

If $\mathsf{L = NL}$, then there is a logspace algorithm that solves the decision version of 2-SAT. Is $\mathsf{L = NL}$ known to imply that there is a logspace algorithm to obtain a satisfying ...
user avatar
15 votes
1 answer
821 views

Hardness of Computing Weisfeiler-Lehman labels

The 1-dim Weisfeiler-Lehman algorithm (WL) is commonly known as canonical labeling or color refinement algorithm. It works as follows : The initial coloring $C_0$ is uniform, $C_0(v) = 1$ for all ...
Shiva Kintali's user avatar
15 votes
1 answer
307 views

Does ${\bf AC^0PAD} = {\bf PPAD}$?

What happens if we define ${\bf PPAD}$ such that instead of a polytime Turing-machine/polysize circuit, a logspace Turing-machine or an ${\bf AC^0}$ circuit encodes the problem? Recently giving ...
domotorp's user avatar
  • 14.1k
15 votes
1 answer
642 views

Log-space reduction from Parity-L to CNOT circuits?

Question. In their paper Improved simulation of stabilizer circuits, Aaronson and Gottesman claim that simulating a CNOT circuit is ⊕L-complete (under logspace reductions). It is clear that it ...
Niel de Beaudrap's user avatar
14 votes
1 answer
388 views

What are the consequences of $P \subseteq L/poly$?

A language is in $L/poly$ if there exists a logspace Turing machine that decides the language with polynomial amount of advice. See here for more info: https://en.wikipedia.org/wiki/L/poly ...
Michael Wehar's user avatar
13 votes
1 answer
524 views

What are the obstructions to extending $L=SL$ to $L=NL$?

Omer Reingold's proof that $L=SL$ gives an algorithm for USTCON (In an Undirected graph with special vertices $s$ and $t$, are they Connected?) using only logspace. The basic idea is to build an ...
Alex Meiburg's user avatar
13 votes
1 answer
302 views

Large classes which contain LOGSPACE for which strict inclusions are unknown

The wikipedia page on PSPACE mentions that the inclusion $NL\subset PH$ is not known to be strict (unfortunately without references). Q1: What about $L\subset PH$ and $L\subset P^{\#P}$ - are these ...
Łukasz Grabowski's user avatar
12 votes
1 answer
373 views

A super-linear time problem in NL

It is a well-known fact that $ \mathsf{NL} = \cup_{k>0} \mathsf{2NFA[k]} $, where $ \mathsf{2NFA[k]} $ is the class of languages recognized by two-way nondeterministic finite automata with $ k>0 ...
Abuzer Yakaryilmaz's user avatar
10 votes
1 answer
243 views

$BPL$ with polylog random bits is in $L$

Consider a $BPL$ machine (namely, a probabilistic algorithm that uses logspace and polynomially many random bits). It is known (Saks-Zhou) that $BPL \subseteq DSPACE(log^{1.5}(n))$. My question is ...
BharatRam's user avatar
  • 393
10 votes
1 answer
241 views

On sparse complete sets and P vs L

Mahaney's Theorem tells us that if there is a sparse $NP$-complete set under polynomial-time many-one reductions, then $P = NP$. (See "Sparse complete sets for NP: Solution of a conjecture of Berman ...
Michael Wehar's user avatar
10 votes
1 answer
186 views

It is known that $L \subsetneq PH$?

Is it known whether $Logspace$ is strictly contained in the polynomial time hierarchy ? Are there oracles relative to which these classes are equal / distinct ?
GBathie's user avatar
  • 296
10 votes
0 answers
178 views

Alternative proofs of Savitch's theorem?

Question: Are there any known proofs of Savitch's theorem that $NL \subseteq L^2$ besides the usual one? By the usual one I mean the proof based on recursively querying whether there is a midpoint. ...
Elle Najt's user avatar
  • 1,469
9 votes
1 answer
405 views

Two way deterministic multihead counter automata or logspace TM with counter

Is that known something about languages recognized by two-way deterministic multihead counter automaton or logspace TM with counter (equivalent model)? This class called Aux2DC in my advisor's paper. ...
Alexander Rubtsov's user avatar
9 votes
1 answer
883 views

Is it known if $\mathrm{CFL} \subseteq\mathrm{ NSPACE}(o(log^2(n)))$?

$\mathrm{CFL}$ is the class of context-free languages. Question Is $\mathrm{CFL}$ known to be solvable in $o(log^{2}(n))$ non-deterministic space? What about $\mathrm{DCFL}$?
Michael Wehar's user avatar
7 votes
1 answer
216 views

Is Circuit Minimization $P$-hard under logspace reductions?

By Circuit Minimization, I am referring to the following decision problem. Circuit Minimization Input: A bit string $x$ and a number $k$. Question: Does there exist a Boolean Circuit $C$...
Michael Wehar's user avatar
6 votes
1 answer
803 views

What are the consequences of solving XOR 3-SAT in Logspace?

XOR Formulas Consider boolean formulas with connectives $\wedge$ (AND) and $\oplus$ (XOR). Such a boolean formula is a valid instance for XOR SAT if it is a conjunction of $\oplus$-clauses. An $\...
Michael Wehar's user avatar
6 votes
1 answer
4k views

k-Vertex Cover problem is in parameterized Log space

$k$-Vertex Cover: Given a graph $G = (V, E)$ where $V$ is a set of vertices and $E$ a set of edges, and an integer $k$, the $k$-Vertex Cover problem determines if there exists a subset of vertices $...
GOLD's user avatar
  • 175
6 votes
2 answers
664 views

Complexity of comparison unary>binary

What is the smallest widely-known complexity class to which $$\left\{\langle i,j\rangle\middle|\begin{array}{@{}l@{\ }l@{}} & i\ \text{is a unary encoding of a positive integer}\ \hat\imath\\\...
user avatar
6 votes
0 answers
241 views

What is the space complexity of computing the eigenvectors of a matrix?

By the answer to this question, computing the eigenvalues of a matrix to within $2^{-n}$ precision can be done in polylogarithmic space. Is it also possible to compute the eigenvectors of a matrix to ...
jschnei's user avatar
  • 161
6 votes
0 answers
471 views

Can we show that $\mathsf{NL}^\mathsf{NL} = \mathsf{NL}$? [closed]

We know by Immerman–Szelepcsényi theorem that $\mathsf{NL}=\mathsf{coNL}$? Does it follow from this theorem that $\mathsf{NL}^\mathsf{NL} = \mathsf{NL}$? Here, $\mathsf{NL}^\mathsf{NL}$ denotes the ...
albert's user avatar
  • 87
5 votes
1 answer
227 views

What circuit depth is enough to compute a log-space complete problem?

To the best of my knowledge it is unknown that $\mathsf{L}$ is subset of $\mathsf{NC}^1$. (Here $\mathsf{NC}^1$ is the class of decision problems solvable by a family of Boolean circuits, with ...
Alexey Milovanov's user avatar
5 votes
1 answer
491 views

Can Lexicographic BFS be implemented in logspace?

Input: Given graph $G=(V,E)$ with vertices labeled in some order Output: Change the labeling of vertices such that the labeling starts $v_1$ as $u_1$. Next, label the neighbors of $v_1$ as $u_2,u_3,...
Diamond's user avatar
  • 51
5 votes
0 answers
235 views

On space complexity of permanent modulo $2^t$?

We know from here that permanent of $0/1$ matrix modulo $2^t$ is in $DTIME(n^{t+3})$ and hence in $P$. My question is whether permanent of $0/1$ matrix modulo $2^t$ is in $L$ as well or is the current ...
Turbo's user avatar
  • 13.1k
5 votes
0 answers
1k views

Any known connections between open problems for time and space: P vs L, NP vs NL, BPP vs BPL, ⊕P vs ⊕L

It would be nice to show that $P=L$ implies $NP=NL$. Or, $NP=NL$ implies $UP=UL$. Or maybe, $⊕P = ⊕L$ implies $PP = PL$. Are there any known connections between the problems: P vs L, UP vs UL, NP ...
Michael Wehar's user avatar
4 votes
1 answer
233 views

Can $L=SL$ be shown with the replacement product instead of the zig-zag product?

(This is a bit of follow-up to https://cstheory.stackexchange.com/posts/comments/93266 but is a distinct enough question I though it should be on its own.) In Omer Reingold's logspace USTCON ...
Alex Meiburg's user avatar
4 votes
0 answers
110 views

$\mathsf{NL}$ vs. $\mathsf{AC}^1$

It is known that $\mathsf{NL} \subseteq \mathsf{AC}^1$ (because $\mathsf{NL}$-complete problem PATH belongs to $\mathsf{AC}^1$). Are there problems in $\mathsf{AC}^1$ that are unknown to be in $\...
Alexey Milovanov's user avatar
4 votes
0 answers
633 views

Is there any NC-complete problem with respect to logspace reduction?

The question is on the title. We all know that $\text{NL}$ and $\text{P}$ have such problems. So I wonder the same thing about $\text{NC}$. More interestingly, is there any $k \ge 2$ and any $\text{...
Vincent J. Ruan's user avatar
4 votes
0 answers
98 views

Are there any non-relativized separations between $L$ and $PH$?

In one sense, $P$ vs. $PSPACE$ is the "easiest" first step to showing $P \neq NP$... and this is one you hear often about. In a different sense, you could take $L$ at one end and then $PH$ at the ...
Alex Meiburg's user avatar
3 votes
1 answer
481 views

NFA to 2DFA: what are the upper and lower bounds?

Suppose that one has an NFA (from, say, a regular expression). What is the state complexity of turning it into a 2DFA?
Demi's user avatar
  • 528
3 votes
0 answers
245 views

What is consequence of $PH\subseteq NSPACE((\log n)^2)$?

What is consequences of $PH\subseteq NSPACE((\log n)^2)$? We don't even know PH is equals to L or not. I am wondering what will be happened when $PH\subseteq NSPACE((\log n)^2)$?
Mohsen Ghorbani's user avatar
3 votes
0 answers
119 views

Restricted k-set cover is in NL or L

Restricted $k$-set cover: Input: $(U,S_1,S_2,\cdots, S_n, k)$, $U=[n]$ and $S_i\subseteq U$ for all $1\leq i \leq n$. Output: $\bigcap_{i\in I}S_i$ where $I=\{1,i_1,i_2\cdots,i_k\}, i_1=min(S_1),i_2=...
GOLD's user avatar
  • 175
3 votes
0 answers
116 views

Number theoretic problems complete for $\mathsf{RL}$

Are there number theoretic problems (such as those related to $\mathsf{gcd}$) that are in $\mathsf{RL}$? Can these also be $\mathsf{RL}$-complete problems (is there any $\mathsf{RL}$-complete ...
user avatar
2 votes
1 answer
2k views

Detecting undirected cycles in logarithmic space [closed]

I have a lot of difficulties with constructing algorithms that use $O(\log n)$ space, as I am unsure about how much can be stored on the worktape. I am trying to figure out an algorithm for the ...
user16655's user avatar
  • 131
2 votes
0 answers
158 views

Searching for a proof that non-deterministic logspace with errors is contained in $PL$

In this 10 year old question Non-deterministic logspace with two-sided error the author asked for a complexity class related to $NL$. Namely $NL$ but we are allowed to have two-sided error for the ...
user0001's user avatar
2 votes
0 answers
123 views

Gurevich's theorem on primitive recursive functions being logspace-computable

I recently came across the following result attributed to Gurevich, according to which I understood that the class of problems solvable by primitive recursive functions is precisely the class L of ...
Noel Arteche's user avatar
2 votes
0 answers
214 views

Simultaneous evidence for $L\neq NL$ and $P\neq NP$

We believe $L\neq NL$ and $P\neq NP$. Is there any evidence which simultaneously imply $L\neq NL$ and $P\neq NP$?
Turbo's user avatar
  • 13.1k
1 vote
2 answers
2k views

Why NL is not L

I'm a beginner in learning complexity and get confused at NL. NL is the class of languages that are decidable in logarithmic space on a nondeterministic Turing machine. In other words, NL = NSPACE($\...
Zachary HUANG's user avatar
1 vote
0 answers
177 views

Perm and Det mod $2^k$ - II

Given a $0/1$ square matrix, the permanent and determinant modulo $2^k$ is in $\oplus P$ and $\oplus L$ respectively for any fixed $k$. In fact both are in $\oplus L$ (in fact in $\oplus SPACE(k^2\log ...
Turbo's user avatar
  • 13.1k
1 vote
0 answers
74 views

One way analogues of Logspace

When we say a function is one-way we typically mean a function is encodable in $P$ but its decryption is not in $P$ but in $UP$. Likewise we say a function is logspace one-way if the function is ...
Turbo's user avatar
  • 13.1k
1 vote
0 answers
64 views

Reference to "compressibility" of logarithmic space [closed]

Is there a reference somewhere for the result SPACE($O(\log n)$) = SPACE($\log n$)? i.e. Big-O doesn't matter in logspace since you can compress the space. I feel like this is an elementary result but ...
user59894's user avatar
1 vote
0 answers
74 views

parametrized logspace algorithm for k-dominating set for planar graphs

$k$-Dominating set: Given a graph $G=(V,E)$ where $V$ is a set of vertices and $E$ a set of edges, and an integer $k$, the $k$-Dominating set problem determines if there exists a subset of vertices $...
GOLD's user avatar
  • 175
0 votes
1 answer
103 views

Complexity status of restricted k-clique [closed]

Restricted $k$-clique: Input: $(G,v,k)$ where $v$ is vertex in $V$ Output: k-clique containing vertex $v$. What is the space and time complexity status of this Restricted $k$-clique problem? Is ...
GOLD's user avatar
  • 175
0 votes
1 answer
194 views

Comparing SAT to MCSP reduction class separations and faster SAT class separations?

Assume $SAT$ is in $QuasiP$. We immediately infer $NQuasiP=QuasiP$ and $EXP=NEXP$. From https://people.csail.mit.edu/rrw/easy-witness-nqp.pdf we infer $NQuasiP$ is not in $P/poly$ which implies $...
Turbo's user avatar
  • 13.1k
0 votes
0 answers
69 views

Is a problem, that is $L$-complete under non-uniform $AC^0$ reductions, necessarily outside of (non-uniform or uniform) $NC^1$?

I don't have much intuition about non-uniformity so the question may be quite naive.
A. G.'s user avatar
  • 43
0 votes
0 answers
61 views

Is there a construction which multiplies and adds spanning trees in Logspace?

I.1 Suppose we have two planar graphs $G_1$ and $G_2$ with spanning tree count $C_1$ and $C_2$ respectively then is there a graph construction in Logspace to get a planar graph from $G_1$ and $G_2$ ...
Turbo's user avatar
  • 13.1k
0 votes
0 answers
224 views

$CH=UL$ and partial breaking of transitive closure bottleneck problem and Savitch's theorem?

Let $L^t=DSPACE[O(\log n)^t]$, $NL^t=NSPACE[O(\log n)^t]$ and $UL^t=USPACE[O(\log n)^t$. Savitch provides $NL\subseteq L^{2}$. If $CH=UL$ we clearly got rid of the transitive closure bottleneck ...
Turbo's user avatar
  • 13.1k