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.

Highlights

  • A property of graphs on n vertices can be seen as a Boolean function which takes as inputs the n 2 potential edges and outputs either 0 or 1

  • That is, permuting the n 2 inputs according to some permutation of [n] leaves the value of the function unchanged. We call such Boolean functions invariant. Note that this does not require the Boolean function to be invariant under all permutations of its inputs, which would mean that it was entirely determined by the number of inputs that are set to 1

  • Theorem 2 A graph property is decided by a polynomial-time uniform family of polynomial-size symmetric Boolean circuits if, and only if, it is defined by a fixedpoint sentence interpreted in G ⊕ [n], ≤, i.e., the structure that is the disjoint union of an n-vertex graph G with a linear order of length n

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Summary

Introduction

A property of graphs on n vertices can be seen as a Boolean function which takes as inputs the n 2 potential edges (each of which can be 0 or 1) and outputs either 0 or 1. An extended abstract of this paper appeared in STACS 2014 [1]

Present address
Preliminaries
Symmetric and Uniform Circuits
Symmetry and Support
Supporting Partitions
Support Theorem
Translating Symmetric Circuits to Formulas
Rigid Circuits
Computing Supports
Succinctly Evaluating Symmetric Circuits
Translating to Formulas of FP
Consequences
Coherent and Locally Polynomial Circuits
Future Directions
Full Text
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