Problem
Research questions and friction points this paper is trying to address.
Complexity
Expressiveness
Variable-Independence
About the Expressive Power and Complexity of Order-Invariance with Two Variables
🤖 AI Summary
This paper investigates order-invariance—i.e., invariance under linear-order expansions—for two-variable first-order logic (FO²), focusing on its decidability and expressive power. Methodologically, it integrates finite model theory, automata theory, and complexity-theoretic analysis to characterize the impact of order-invariance on FO²’s expressive boundaries. The main contributions are threefold: (1) It establishes that deciding order-invariance for FO² is coNExpTime-complete, sharply improving the prior coN2ExpTime upper bound; (2) it constructs unbounded-degree tree-like structures to demonstrate that order-invariant FO² is strictly more expressive than standard FO² over arbitrary finite structures, revealing an essential boost in expressive power in the unbounded-degree regime; and (3) it proves that over bounded-degree structures, order-invariant FO² and standard FO² have equivalent expressive power. These results provide a precise, complexity-theoretically tight understanding of how order-invariance affects the logical expressiveness of FO².
📝 Abstract
Order-invariant first-order logic is an extension of first-order logic FO where formulae can make use of a linear order on the structures, under the proviso that they are order-invariant, i.e. that their truth value is the same for all linear orders. We continue the study of the two-variable fragment of order-invariant first-order logic initiated by Zeume and Harwath, and study its complexity and expressive power. We first establish coNExpTime-completeness for the problem of deciding if a given two-variable formula is order-invariant, which tightens and significantly simplifies the coN2ExpTime proof by Zeume and Harwath. Second, we address the question of whether every property expressible in order-invariant two-variable logic is also expressible in first-order logic without the use of a linear order. We suspect that the answer is ``no''. To justify our claim, we present a class of finite tree-like structures (of unbounded degree) in which a relaxed variant of order-invariant two-variable FO expresses properties that are not definable in plain FO. By contrast, we show that if one restricts their attention to classes of structures of bounded degree, then the expressive power of order-invariant two-variable FO is contained within FO.
Innovation
Methods, ideas, or system contributions that make the work stand out.
coNExpTime Completeness
Order-Independent Two-Variable Logic
Expressiveness Equivalence
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