Algebraic tools for default modal systems

V. Cassano, R. Fervari, C. Areces, and P. Castro. Algebraic tools for default modal systems. Journal of Logic and Computation, 2022.

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Abstract

Default Logics are a family of non-monotonic formalisms having so-called defaults and extensions as their common foundation. Traditionally, default logics have been defined and dealt with via syntactic notions of consequence in propositional or first-order logic. Here, we build default logics on modal logics. First, we present these default logics syntactically. Then, we elaborate on an algebraic counterpart. More precisely, we extend the notion of a modal algebra to accommodate for defaults and extensions. Our algebraic view of default logics concludes with an algebraic completeness result and a way of comparing default logics borrowing ideas from the concept of bisimulation in modal logic. To our knowledge, this take on default logics approach is novel. Interestingly, it also lays the groundwork for studying default logics from a dynamic logic perspective.

BibTeX

@Article{exac051,
  author =       "V. Cassano and R. Fervari and C. Areces and P.
                 Castro",
  title =        "{Algebraic tools for default modal systems}",
  journal =      "Journal of Logic and Computation",
  year =         "2022",
  abstract =     "{Default Logics are a family of non-monotonic
                 formalisms having so-called defaults and extensions as
                 their common foundation. Traditionally, default logics
                 have been defined and dealt with via syntactic notions
                 of consequence in propositional or first-order logic.
                 Here, we build default logics on modal logics. First,
                 we present these default logics syntactically. Then, we
                 elaborate on an algebraic counterpart. More precisely,
                 we extend the notion of a modal algebra to accommodate
                 for defaults and extensions. Our algebraic view of
                 default logics concludes with an algebraic completeness
                 result and a way of comparing default logics borrowing
                 ideas from the concept of bisimulation in modal logic.
                 To our knowledge, this take on default logics approach
                 is novel. Interestingly, it also lays the groundwork
                 for studying default logics from a dynamic logic
                 perspective.}",
  ISSN =         "0955-792X",
}