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Given $k$ deterministic FSAs of $n$ states the intersection of their languages is empty is decidable in $n^{o(k)}$ time is an open problem.

  1. For unbounded $k$ it is known the problem is $PSPACE$ complete. What is the scale of $k$ for the problem to be PSPACE complete? Is it $k=\Omega(n)$?

  2. What is the consequence the problem is in $FPT$ having $f(k)poly(n)$ complexity for $f(k)=2^{poly(k)}$?

  3. Is there a result on intersection of deterministic DFSAs which is consistent to $CH$ is $NL$ or $L$ hypothesis (which provides $P=CH\neq PSPACE$)?

How about if the $FSA$s are all non deterministic?

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Regarding question (1):

As long as $k$ is $\Omega(n^c)$ for some $c > 0$, then the problem is $PSPACE$-complete. See this paper by Klaus-Jörn Lange and Peter Rossmanith (1992) for some related results.

https://doi.org/10.1007/3-540-55808-X_33

Regarding question (2):

Say that the intersection problem is parameterized by the number of FSA's. If this parameterized problem is in $FPT$, then $PTIME \neq NLOGSPACE$.

In fact, the result is quite a bit stronger than that. Essentially, it says that if for every $\alpha > 0$, there exists $k$ sufficiently large such that the $k$th slice of the parameterized problem is solvable in $O(n^{\alpha \cdot k})$ time, then $PTIME \neq NLOGSPACE$.

See here and here.

Regarding question (3):

I am still trying to understand this question.

Regarding last question:

Both the DFA and NFA parameterized intersection problems (when parameterized by the number of automata) are fpt-equivalent. In fact, the fpt-equivalence is very strong and preserves the parameter (I called it an LBL-equivalence). Regarding your question, this implies that if one of them is in $FPT$, then the other is in $FPT$ too.

Additional information:

Part of the reason why we have these hardness results for the parameterized FSA intersection problem is because it is $XNL$-complete under a strong form of fpt-reduction. $XNL$ is a parameterized complexity class that is related to $k \cdot log(n)$ binary space nondeterministic Turing machines.

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    $\begingroup$ 1. All these are one directional implications. Correct? 2. What is the canonical representation of the k dfa intersection problem? Are we given the grammar? Or something else? Grammar representation seems tiny compared to the number of states $n$. $\endgroup$
    – Mr.
    Mar 13 at 23:41
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    $\begingroup$ The answer is not referencing sat. So k dfa intersection in fpt has no implication to eth? $\endgroup$
    – Mr.
    Mar 13 at 23:43
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    $\begingroup$ @1.. Yes, whether you are talking about automata, grammars, or expressions, that could make a difference in the complexity. Please feel welcome to list different ways of representing regular languages and we can look at the complexity in each case. :) $\endgroup$ Mar 13 at 23:43
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    $\begingroup$ Sorry I thought the unary case is np complete and speed up of unary case leads to eth failure. Am I wrong? $\endgroup$
    – Mr.
    Mar 14 at 0:56
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    $\begingroup$ @1.. Yes, the unary case is $NP$-complete and the parameterized version of the unary case is $W[1]$-complete through an fpt-equivalence with $k$-Clique. I have some slides on the $W[1]$-completeness on my website, but it is not formally published. $\endgroup$ Mar 14 at 1:09

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