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

Borchert, Bernd; Ranjan, Desh

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

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Borchert, Bernd
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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Ranjan, Desh
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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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: https://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 $f$?