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Small Hazard-free Transducers

MPS-Authors
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Bund,  Johannes
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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Lenzen,  Christoph
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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Fulltext (public)

arXiv:1811.12369.pdf
(Preprint), 417KB

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Citation

Bund, J., Lenzen, C., & Medina, M. (2018). Small Hazard-free Transducers. Retrieved from http://arxiv.org/abs/1811.12369.


Cite as: http://hdl.handle.net/21.11116/0000-0002-9FAD-9
Abstract
Recently, an unconditional exponential separation between the hazard-free complexity and (standard) circuit complexity of explicit functions has been shown. This raises the question: which classes of functions permit efficient hazard-free circuits? Our main result is as follows. A \emph{transducer} is a finite state machine that transcribes, symbol by symbol, an input string of length $n$ into an output string of length $n$. We prove that any function arising from a transducer with $s$ states, that is input symbols which are encoded by $\ell$ bits, has a hazard-free circuit of size $2^{\BO(s+\ell)}\cdot n$ and depth $\BO(\ell+ s\cdot \log n)$; in particular, if $s, \ell\in \BO(1)$, size and depth are asymptotically optimal. We utilize our main result to derive efficient circuits for \emph{$k$-recoverable addition}. Informally speaking, a code is \emph{$k$-recoverable} if it does not increase uncertainty regarding the encoded value, so long as it is guaranteed that it is from $\{x,x+1,\ldots,x+k\}$ for some $x\in \NN_0$. We provide an asymptotically optimal $k$-recoverable code. We also realize a transducer with $\BO(k)$ states that adds two codewords from this $k$-recoverable code. Combined with our main result, we obtain a hazard-free adder circuit of size $2^{\BO(k)}n$ and depth $\BO(k\log n)$ with respect to this code, i.e., a $k$-recoverable adder circuit that adds two codewords of $n$ bits each. In other words, $k$-recoverable addition is fixed-parameter tractable with respect to $k$.