On Symmetric Circuits and FixedPoint Logics
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 fixedpoint logic with counting. This shows that inexpressibility results in the latter logic lead to lower bounds against polynomialsize families of symmetric circuits.
 Publication:

arXiv eprints
 Pub Date:
 January 2014
 DOI:
 10.48550/arXiv.1401.1125
 arXiv:
 arXiv:1401.1125
 Bibcode:
 2014arXiv1401.1125A
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Logic in Computer Science;
 F.1.3;
 F.4.1;
 F.1.1
 EPrint:
 22 pages. Full version of a paper to appear in STACS 2014