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#### The circuit subfunction relations are $sum^P_2$-complete

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##### Fulltext (public)

MPI-I-93-121.pdf

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##### Supplementary Material (public)

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##### Citation

Borchert, B., & Ranjan, D.(1993). *The circuit subfunction
relations are $sum^P_2$-complete* (MPI-I-93-121). Saarbrücken: Max-Planck-Institut für Informatik.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0014-B74C-4

##### Abstract

We show that given two Boolean
circuits $f$ and $g$ the following three problems are $\Sigma^p_2$-complete:
(1) Is $f$ a c-subfunction of $g$, i.e.\ can one set some of the variables
of $g$ to 0 or 1 so that the remaining circuit computes the same function
as $f$?
(2) Is $f$ a v-subfunction of $g$, i.e. can one change the names of the
variables of $g$ so that the resulting circuit computes the same function
as $f$?
(3) Is $f$ a cv-subfunction of $g$, i.e.\ can one
set some variables of $g$ to 0 or 1 and simultanously
change some names of the other variables of $g$ so that the new circuit
computes the same function as $f$?
Additionally we give some bounds for the complexity of the following
problem: Is $f$ isomorphic to $g$, i.e. can one change the names of the
variables bijectively so that the circuit resulting from $g$ computes the
same function as $f$?