eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2014-03-05
41
52
10.4230/LIPIcs.STACS.2014.41
article
On Symmetric Circuits and Fixed-Point Logics
Anderson, Matthew
Dawar, Anuj
We study properties of relational structures such as graphs that are decided by families of Boolean circuits. Circuits that decide such properties are necessarily invariant to permutations of the elements of the input structures. We focus on families of circuits that are symmetric, i.e., circuits whose invariance is witnessed by automorphisms of the circuit induced by the permutation of the input structure. We show that the expressive power of such families is closely tied to definability in logic. In particular, we show that the queries defined on structures by uniform families of symmetric Boolean circuits with majority gates are exactly those definable in fixed-point logic with counting. This shows that inexpressibility results in the latter logic lead to lower bounds against polynomial-size families of symmetric circuits.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol025-stacs2014/LIPIcs.STACS.2014.41/LIPIcs.STACS.2014.41.pdf
symmetric circuit
fixed-point logic
majority
counting
uniformity