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9
votes
1answer
130 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 ...
13
votes
0answers
174 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 ...
12
votes
1answer
208 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 ...
1
vote
1answer
83 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 ...
6
votes
1answer
181 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. ...
7
votes
0answers
187 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 ...
10
votes
1answer
218 views

Ruzzo-Simon-Tompa oracle access mechanism

In a paper on relativizing logspace computations, Ladner and Lynch construct an oracle relative to which $\mathsf{NL} \nsubseteq \mathsf{P}$. There are some more pathological examples in this vein in ...
14
votes
1answer
529 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 ...
29
votes
1answer
739 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 ...
5
votes
0answers
392 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 ...
13
votes
1answer
301 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 ...
25
votes
2answers
704 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 ...