Flipping and Forking

TL;DR

This paper introduces flips and flip independence in relational structures, linking combinatorial graph operations to logical stability concepts, and extends characterizations of monadic stability to broader structures, with implications for model checking.

Contribution

It generalizes flip-based characterizations of monadic stability from graphs to arbitrary relational structures and connects flip independence with forking independence.

Findings

Flip independence is equivalent to forking independence in monadically stable structures.

Provides combinatorial characterizations of monadically stable classes of relational structures.

Characterizes forking independence in monadically stable graphs and structures with nowhere dense Gaifman graphs.

Abstract

Monadic stability and the more general monadic dependence (or NIP) are tameness conditions for classes of logical structures, studied in the 80's in Shelah's classification program in model theory. They recently emerged in algorithmic and structural graph theory and finite model theory as central notions in relation with the model checking problem for first-order logic: the problem was shown to be fixed-parameter tractable for inputs which come from a fixed class of graphs which is monadically stable, and is conjectured to be tractable in all monadically dependent classes. Several combinatorial characterizations of such graph classes turned out to be essential in their algorithmic treatment; they are all based on the fundamental operation of "flipping" a graph. We introduce the notions of $\textit{flips}$ and $\textit{flip independence}$ in arbitrary relational structures. We lift prior combinatorial characterizations of monadically stable graph classes to monadically stable classes of relational structures. We show the equivalence of flip independence with $\textit{forking independence}$ (over models) -- a logical notion of paramount importance in stability theory -- in monadically stable structures, shedding new light on the relevance of flips, also characterizing forking independence (over models) combinatorially. We give more precise descriptions of forking independence in the case of monadically stable graphs, and relational structures with a nowhere dense Gaifman graph.