🤖 AI Summary
This paper addresses the expressive gap between polynomial-time algorithms and isomorphism-invariant logics, focusing on the expressive boundaries of Fixed-Point Logic with Counting (IFPC). Methodologically, it systematically investigates the logical strength arising from combining Witnessed Symmetric Choice (WSC) with first-order interpretations (I). The main contributions are threefold: (1) It separates IFPC+WSC from IFPC+WSCI, proving that IFPC+WSC is not closed under first-order interpretations—resolving the Dawar–Richerby open problem; (2) it demonstrates that nesting WSC operators strictly increases expressive power; and (3) it establishes a new sufficient condition for canonization of CFI graphs and shows that IFPC+WSC+I canonizes CFI graphs beyond the capability of all previously known mainstream logics. Technically, the work integrates fixed-point logic, group actions and orbit decompositions, structural interpretations, and automata-based witnessing techniques.
📝 Abstract
At the core of the quest for a logic for PTime is a mismatch between algorithms making arbitrary choices and isomorphism-invariant logics. One approach to overcome this problem is witnessed symmetric choice. It allows for choices from definable orbits which are certified by definable witnessing automorphisms. We consider the extension of fixed-point logic with counting (IFPC) with witnessed symmetric choice (IFPC+WSC) and a further extension with an interpretation operator (IFPC+WSC+I). The latter operator evaluates a subformula in the structure defined by an interpretation. This structure possibly has other automorphisms exploitable by the WSC-operator. For similar extensions of pure fixed-point logic (IFP) it is known that IFP+WSCI simulates counting which IFP+WSC fails to do. For IFPC+WSC it is unknown whether the interpretation operator increases expressiveness and thus allows studying the relation between WSC and interpretations beyond counting. We separate IFPC+WSC from IFPC+WSCI by showing that IFPC+WSC is not closed under FO-interpretations. By the same argument, we answer an open question of Dawar and Richerby regarding non-witnessed symmetric choice in IFP. Additionally, we prove that nesting WSC-operators increases the expressiveness using the so-called CFI graphs. We show that if IFPC+WSC+I canonizes a particular class of base graphs, then it also canonizes the corresponding CFI graphs. This differs from various other logics, where CFI graphs provide difficult instances.