Advancements in Lattice Languages and Modal Logics

The field of lattice languages and modal logics is experiencing significant growth, driven by innovations in algebraic approaches, description logics, and realizability theory. Researchers are exploring new frameworks, such as fuzzy lattice-based description logics and monadic combinatory algebras, to better represent and reason about complex computational systems. Notably, the development of paraconsistent relations and swap Kripke models is enabling the expression of programs and tests yielding vague or inconsistent outcomes. Furthermore, advancements in syntactic effectful realizability and linear logic are providing new insights into the cut-elimination of the modal mu-calculus. Overall, these developments are advancing our understanding of computation and its relationship with realizability models and programming languages. Noteworthy papers include:

  • 'From Partial to Monadic: Combinatory Algebra with Effects' which introduces monadic combinatory algebras to internalize a wide range of computational effects.
  • 'Syntactic Effectful Realizability in Higher-Order Logic' which presents EffHOL, a novel framework that expands syntactic realizability to uniformly support modern programming paradigms with side effects.

Sources

Positive Varieties of Lattice Languages

Fuzzy Lattice-based Description Logic

Nominal Equational Rewriting and Narrowing

Paraconsistent Relations as a Variant of Kleene Algebras

Swap Kripke models for deontic LFIs

From Partial to Monadic: Combinatory Algebra with Effects

Syntactic Effectful Realizability in Higher-Order Logic

On the cut-elimination of the modal $\mu$-calculus: Linear Logic to the rescue

Growing a Modular Framework for Modal Systems- HOLMS: a HOL Light Library

Notes on applicative matching logic

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