# Guarded Second-Order Logic, Spanning Trees, and Network Flows

## TL;DR

This paper explores the expressive power of guarded second-order logic over hypergraphs, extending known results to larger cardinalities and introducing encoding methods for sets of vertices.

## Contribution

It generalizes the equivalence of guarded second-order logic and monadic second-order logic to hypergraphs of arbitrary size and develops encoding techniques for vertex sets.

## Key findings

- Guarded second-order logic matches monadic second-order logic over all countable hypergraphs.
- Extended the expressive power equivalence to hypergraphs of any size.
- Introduced encoding methods for sets of vertices using single vertices.

## Abstract

According to a theorem of Courcelle monadic second-order logic and guarded second-order logic (where one can also quantify over sets of edges) have the same expressive power over the class of all countable $k$-sparse hypergraphs. In the first part of the present paper we extend this result to hypergraphs of arbitrary cardinality. In the second part, we present a generalisation dealing with methods to encode sets of vertices by single vertices.

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Source: https://tomesphere.com/paper/0910.3085