LIPIcs.STACS.2014.41.pdf
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On Symmetric Circuits and Fixed-Point Logics
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31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Symposium on Theoretical Aspects of Computer Science (STACS) - License:
Creative Commons Attribution 3.0 Unported license
- Publication Date: 2014-03-05
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Matthew Anderson and Anuj Dawar. On Symmetric Circuits and Fixed-Point Logics. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 41-52, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)
https://doi.org/10.4230/LIPIcs.STACS.2014.41
Abstract
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.
Subject Classification
Keywords
- symmetric circuit
- fixed-point logic
- majority
- counting
- uniformity
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