[Paper Review] Algorithmic Correspondence and Canonicity for Possibility Semantics

Zhiguang Zhao|arXiv (Cornell University)|Dec 15, 2016

Logic, Reasoning, and Knowledge23 references4 citations

TL;DR

This paper extends the unified correspondence theory to possibility semantics by adapting the ALBA algorithm to compute first-order correspondents for inductive formulas over both full and filter-descriptive possibility frames. It establishes canonicity and correspondence for inductive formulas in this setting, showing that nominals are interpreted as regular open closures of singletons, and proves soundness over dual algebras of both frame types, thereby completing the Sahlqvist-type correspondence and canonicity result for possibility semantics.

ABSTRACT

The present paper develops a unified correspondence treatment of the Sahlqvist theory for possibility semantics, extending the results in \cite{Ya16} from Sahlqvist formulas to the strictly larger class of inductive formulas, and from the full possibility frames to filter-descriptive possibility frames. Specifically, we define the possibility semantics version of the algorithm ALBA, and an adapted interpretation of the expanded modal language used in the algorithm. We prove the soundness of the algorithm with respect to both (the dual algebras of) full possibility frames and (the dual algebras of) filter-descriptive possibility frames. We make some comparisons among different semantic settings in the design of the algorithms, and fit possibility semantics into this broader picture. One notable feature of the adaptation of ALBA to possibility frames setting is that the so-called nominal variables, which are interpreted as complete join-irreducibles in the standard setting, are interpreted as regular open closures of "singletons" in the present setting.

Motivation & Objective

• To extend the Sahlqvist correspondence and canonicity theory from Kripke semantics to possibility semantics, completing the picture for inductive formulas.
• To adapt the ALBA algorithm for use in possibility semantics, where nominals are interpreted as regular open closures of singletons rather than atoms.
• To prove soundness of the adapted ALBA algorithm over dual algebras of both full and filter-descriptive possibility frames.
• To demonstrate that inductive formulas have first-order correspondents in both frame types, resolving a key gap in the theory.
• To position possibility semantics as the constructive counterpart of Kripke semantics, especially in the context of constructive canonical extensions.

Proposed method

• Adapt the ALBA algorithm to possibility semantics by redefining the interpretation of nominal variables as regular open closures of singletons in the expanded modal language.
• Define two dual algebraic structures for full possibility frames: the Boolean algebra of regular open subsets and the Boolean algebra of arbitrary subsets, linked by a canonical order-embedding.
• Prove soundness of the adapted ALBA algorithm with respect to the dual algebras of full possibility frames and filter-descriptive possibility frames.
• Use the bimodal perspective of possibility frames (as Kripke frames with additional constraints) to analyze connective combinations and their order-theoretic properties.
• Leverage the duality between possibility frames and general lattices to relate the results to constructive canonical extensions.
• Compare the algorithmic design across different semantic settings, including Kripke, general lattices, and possibility semantics, to highlight structural parallels and differences.

Experimental results

Research questions

• RQ1 Can the ALBA algorithm be successfully adapted to compute first-order correspondents in possibility semantics, where nominals are interpreted as regular open closures of singletons rather than atoms?
• RQ2 Do inductive formulas over possibility frames have first-order correspondents in both full and filter-descriptive frames, despite potential loss of correspondence in the transition?
• RQ3 How does the order-theoretic behavior of connective combinations (e.g., $\Box_{\sqsubseteq}\Diamond_{\sqsubseteq}$) in the bimodal perspective affect correspondence analysis in possibility semantics?
• RQ4 Is the persistence of validity from filter-descriptive to full possibility frames equivalent to constructive canonical extension, as suggested by the duality with general lattices?
• RQ5 Can possibility semantics serve as a relational framework for constructive canonicity, replacing the need for the Axiom of Choice in canonical extension constructions?

Key findings

• The adapted ALBA algorithm successfully computes first-order correspondents for inductive formulas in both full and filter-descriptive possibility frames, proving soundness over their dual algebras.
• Nominal variables in the algorithm are interpreted as regular open closures of singletons in possibility semantics, which is a key structural adaptation from standard Kripke semantics.
• The algorithm establishes that inductive formulas have first-order correspondents in both frame types, resolving a key theoretical gap in possibility semantics.
• The persistence of validity from filter-descriptive to full possibility frames corresponds exactly to constructive canonical extension, linking possibility semantics to constructive algebraic logic.
• The bimodal perspective of possibility frames reveals additional order-theoretic properties in connective combinations, such as meet-preservation in $\Box_{\sqsubseteq}\Diamond_{\sqsubseteq}$, which support more robust correspondence analysis.
• Possibility semantics is identified as the constructive counterpart of Kripke semantics, providing a relational environment for constructive canonicity without relying on the Axiom of Choice.