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Convex Team Logics

Aleksi Anttila, Søren Brinck Knudstorp

TL;DR

This work develops the theory of convex team logics by defining convex propositional and modal logics (notably CONDEP, CONINQ, and PLIM) and proving expressive completeness for the entire class of convex properties. It analyzes how standard connectives interact with convexity, replaces non-convex-preserving disjunctions with convex-friendly variants, and demonstrates modal counterparts invariant under bounded bisimulation. A parallel study of convex+union-closed properties yields a complete propositional logic (PLNE) and modal analogues, including connections to BSML. Finally, the authors generalize uniform definability to articulate precise notions of extension between logics, establish inner uniform extensions for the convex variants, and pose several directions for axiomatization and broader logical variants. Overall, the paper advances the understanding of convexity in team semantics and provides foundational tools for comparing and extending convex logics.

Abstract

We prove expressive completeness results for convex propositional and modal team logics, where a logic is convex if, for each formula, if it is true in two teams $t$ and $u$ and $t\subseteq s\subseteq u$, then it is also true in $s$. We introduce multiple propositional/modal logics which are expressively complete for the class of all convex propositional/modal team properties. We also answer an open question concerning the expressive power of classical propositional logic extended with the nonemptiness atom NE: we show that it is expressively complete for the class of all convex and union-closed properties. A modal analogue of this result additionally yields an expressive completeness theorem for Aloni's Bilateral State-based Modal Logic. In a specific sense, one of the novel propositional convex logics extends propositional dependence logic and another, propositional inquisitive logic. We generalize the notion of uniform definability, as considered in the team semantics literature, to formalize the notion of extension pertaining to the convex logics.

Convex Team Logics

TL;DR

This work develops the theory of convex team logics by defining convex propositional and modal logics (notably CONDEP, CONINQ, and PLIM) and proving expressive completeness for the entire class of convex properties. It analyzes how standard connectives interact with convexity, replaces non-convex-preserving disjunctions with convex-friendly variants, and demonstrates modal counterparts invariant under bounded bisimulation. A parallel study of convex+union-closed properties yields a complete propositional logic (PLNE) and modal analogues, including connections to BSML. Finally, the authors generalize uniform definability to articulate precise notions of extension between logics, establish inner uniform extensions for the convex variants, and pose several directions for axiomatization and broader logical variants. Overall, the paper advances the understanding of convexity in team semantics and provides foundational tools for comparing and extending convex logics.

Abstract

We prove expressive completeness results for convex propositional and modal team logics, where a logic is convex if, for each formula, if it is true in two teams and and , then it is also true in . We introduce multiple propositional/modal logics which are expressively complete for the class of all convex propositional/modal team properties. We also answer an open question concerning the expressive power of classical propositional logic extended with the nonemptiness atom NE: we show that it is expressively complete for the class of all convex and union-closed properties. A modal analogue of this result additionally yields an expressive completeness theorem for Aloni's Bilateral State-based Modal Logic. In a specific sense, one of the novel propositional convex logics extends propositional dependence logic and another, propositional inquisitive logic. We generalize the notion of uniform definability, as considered in the team semantics literature, to formalize the notion of extension pertaining to the convex logics.

Paper Structure

This paper contains 12 sections, 19 theorems, 35 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Convex Properties
  3. Propositional Properties
  4. Preliminaries: Logics and (Classical) Languages
  5. Preliminaries: Closure Properties
  6. Expressive Completeness
  7. Modal Properties
  8. Convex and Union-closed Properties
  9. Propositional Properties
  10. Modal Properties
  11. Uniform Definability and Uniform Extensions
  12. Conclusion

Key Result

proposition 1

Formulas of $\DEP$ and $\INQ$ are downward closed (and hence also convex). Formulas of $\CONDEP$, $\CONINQ$, and $\PLIM$ are convex.

Figures (2)

  • Figure 1: Examples of convex/non-convex subsets of a lattice. Note that in a powerset lattice, upward-directedness corresponds to union closure.
  • Figure 2: Examples of the semantics.

Theorems & Definitions (42)

  • definition 1: Syntax
  • definition 2: Semantics
  • definition 3: Closure properties
  • proposition 1
  • proof
  • proof
  • definition 4: Properties and Expressive Completeness
  • lemma 1
  • proof
  • lemma 2
  • ...and 32 more