# MPI-I-93-121

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

### Borchert, Bernd and Ranjan, Desh

#### May 1993, 14 pages.

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##### Status: available - back from printing

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 $f$?

**URL to this document: **https://domino.mpi-inf.mpg.de/internet/reports.nsf/NumberView/1993-121

**BibTeX**
`@TECHREPORT{``BorchertRanjan93``,`

` AUTHOR = {Borchert, Bernd and Ranjan, Desh},`

` TITLE = {The circuit subfunction relations are $sum^P_2$-complete},`

` TYPE = {Research Report},`

` INSTITUTION = {Max-Planck-Institut f{\"u}r Informatik},`

` ADDRESS = {Im Stadtwald, D-66123 Saarbr{\"u}cken, Germany},`

` NUMBER = {MPI-I-93-121},`

` MONTH = {May},`

` YEAR = {1993},`

` ISSN = {0946-011X},`

`}`