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My decision problem deals with synthesis from register automata:

  • input: a register automaton $A$, a number $k$;
  • return YES if there exists $k$-register transducer that realizes the automaton, else NO.

(The meaning of terms 'synthesis', 'register automata', 'transducer', 'realizes' is not important.)

The problem upper complexity bound is $O(2^{k}\cdot 2^{|A|})$. Q1: What complexity class is this? Something from "parameterized complexity"? (which I am not familiar with)

Now to the second question. My problem has the monotonicity property: if the answer is YES for given $A$ and $k$, then the answer is YES for every larger $k$. This makes me believe that the problem is easier than EXP-hard in $k$. The not-quite working argument is the following (Q2: have you seen something similar?). Fix an arbitrary language L in APSPACE-hard (and recall that APSPACE=EXPTIME). For the sake of contradiction, suppose there exists a poly-time reduction to our problem that produces exactly-same$^?$ $A$, but varies $k$. With this reduction, we create an algorithm that decides L in polynomial time, for almost-all$^?$ words in L (contradicting to $L$ being EXPTIME-hard). The idea is to run the reduction, but before running our-problem-solver on the produced instance, check if the result follows from applying the monotonicity to the current value of $k$ and its previously computed values $k_{no}$ and $k_{yes}$:

  • $k_{no}$ is the largest $k$ that we computed so far such that our-problem-solver returned (hence if $k\leq k_{no}$ we can immediately return NO, based on the monotonicity);

  • $k_{yes}$ is the smallest $k$ that we computed so far such that our-problem-solver returned YES (hence if $k\geq k_{yes}$, return YES);

The above argument does not quite work, because (1) it assumes that the produced automaton $A$ is exactly the same for every word in the language $L$ (while by reduction definition, those $A$s shall only have the same size), and (2) not all words in $L$ are decided in polynomial time: a finite number of words requires exponential time. Q2: Have you seen something similar? ("monotonicity"? "almost-all poly-time reduction"? "exactly-same rather than same-size parameter in the reduction"?)

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  • $\begingroup$ hm, interesting why downvote. $\endgroup$ – Ayrat Feb 10 at 16:40

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