First-Order Logic with Connectivity Operators (2107.05928v2)

Abstract: First-order logic (FO) can express many algorithmic problems on graphs, such as the independent set and dominating set problem, parameterized by solution size. On the other hand, FO cannot express the very simple algorithmic question of whether two vertices are connected. We enrich FO with connectivity predicates that are tailored to express algorithmic graph properties that are commonly studied in parameterized algorithmics. By adding the atomic predicates $conn_k (x, y, z_1 ,\ldots, z_k)$ that hold true in a graph if there exists a path between (the valuations of) $x$ and $y$ after (the valuations of) $z_1,\ldots,z_k$ have been deleted, we obtain separator logic $FO + conn$. We show that separator logic can express many interesting problems such as the feedback vertex set problem and elimination distance problems to first-order definable classes. We then study the limitations of separator logic and prove that it cannot express planarity, and, in particular, not the disjoint paths problem. We obtain the stronger disjoint-paths logic $FO + DP$ by adding the atomic predicates $disjoint-paths_k [(x_1, y_1 ),\ldots , (x_k , y_k )]$ that evaluate to true if there are internally vertex disjoint paths between (the valuations of) $x_i$ and $y_i$ for all $1 \le i \le k$. Disjoint-paths logic can express the disjoint paths problem, the problem of (topological) minor containment, the problem of hitting (topological) minors, and many more. Finally, we compare the expressive power of the new logics with that of transitive closure logics and monadic second-order logic.

• The paper introduces two logical systems—separator logic (FO) and disjoint-paths logic (FO+DP)—to effectively capture graph connectivity.
• It establishes strict expressiveness hierarchies, uncovering limitations like the inexpressibility of global properties in these frameworks.
• The study bridges classic FO and more complex logics such as MSO, paving the way for fixed-parameter tractable approaches in graph algorithms.

Overview of "First-Order Logic with Connectivity Operators"

In the paper "First-Order Logic with Connectivity Operators," the authors Nicole Schirrmacher, Sebastian Siebertz, and Alexandre Vigny have undertaken the task of extending the expressive capabilities of first-order logic (FO) for graph-related problems by integrating specialized connectivity predicates. Their main contribution is the introduction and analysis of two new logical systems: separator logic ($FO$) and disjoint-paths logic ($FO+DP$).

Summary of the Work

The motivation for this work stems from the limitations of classical first-order logic, which is unable to express certain fundamental graph-theoretic properties, such as connectivity. Addressing this gap, the authors propose first-order extensions augmented with connectivity operators:

1. Separator Logic ($FO$): This logic enriches traditional FO with predicates $conn_k(x,y,z_1,\ldots,z_k)$, which indicate connectivity between vertices $x$ and $y$ after deleting the vertices $z_1,\ldots,z_k$. This logical framework enables the expression of various parameterized problems such as the feedback vertex set and specific elimination distance problems.
2. Disjoint-Paths Logic ($FO+DP$): By extending $FO$ with disjoint path predicates, $disjoint_k[(x_1,y_1),\ldots,(x_k,y_k)]$, the logic becomes capable of expressing problems that involve internal vertex-disjoint paths between multiple pairs of vertices. This provides the expressive power to depict complex problems like the disjoint paths problem and (topological) minor containment.

Expressiveness and Hierarchy

The paper establishes a structured hierarchy of expressiveness among the logic fragments:

• Separator logic fragments $FO_k$ demonstrate a strict hierarchy in terms of expressiveness due to the inability of $FO_k$ to express $(k+2)$-connectivity.
• Disjoint-paths logic $FO+DP_k$ also forms a strict hierarchy of expressiveness, where $FO+DP_{k+1}$ can express properties that $FO+DP_k$ cannot, recognizing more complex connectivity patterns.

Implications and Future Directions

From a theoretical standpoint, these logics broaden the capabilities of descriptive complexity in parameterized algorithmics. Practically, separator logic provides a framework that is potentially applicable to a broader class of algorithms, especially in scenarios where fixed-parameter tractability is a concern.

The paper highlights significant findings about the limits of these logics, such as the incapability of separator logic to express global graph properties like planarity or disjoint paths, and the inexpressibility of bipartiteness in disjoint-paths logic. This has practical ramifications, indicating the trade-off between expressiveness and tractability.

Connections to Established Logics

The authors compare these new logical systems with existing frameworks, notably MSO and transitive-closure logics. Separator logic and disjoint-paths logic are part of a continuum between the expressive monadic second-order logic (MSO), known for capturing a wide array of properties but being algorithmically intractable for general graphs, and transitive-closure logics, which offer alternative expressivity at different computational complexities.

Conclusion

The authors provide a foundational expansion of first-order logical frameworks with tailored connectivity predicates that manage to retain fixed-parameter tractability while enabling the expression of robust graph-theoretic properties. This paper paves the way for further explorations into logical systems that balance expressiveness with computational feasibility in the domain of parameterized complexity and algorithm design. Future research will likely explore additional logical extensions and their potential algorithmic efficiencies on various classes of graphs.