Minimisation in Logical Form

Bezhanishvili, Nick; Bonsangue, Marcello; Hansen, Helle Hvid; Kozen, Dexter; Kupke, Clemens; Panangaden, Prakash; Silva, Alexandra

doi:10.48550/arxiv.2005.11551

Cited by 1 publication

(2 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As before, let F be the set endofunctor satisfying F X = X A ×2. One verifies that coalgebras for the composed endofunctor F N can be recognised as unpointed alternating automata [30]. The set 2 can be equipped with a H-algebra structure h which satisfies h(ϕ) = ϕ(id 2 ) and induces a canonical distributive law between H and F [18].…”

Section: Neighbourhood Monadmentioning

confidence: 93%

See 1 more Smart Citation

Generators and Bases for Monadic Closures

Zetzsche¹,

Silva²,

Sammartino³

2020

Preprint

Self Cite

View full text Add to dashboard Cite

One of the fundamental notions of linear algebra is the concept of a basis for a vector space. In the category theoretic formulation of universal algebra, vector spaces are the Eilenberg-Moore algebras over the free vector space monad on the category of sets. In this paper we show that the notion of a basis can be extended to algebras of arbitrary monads on arbitrary categories. On the one hand, we establish purely algebraic results, for instance about the existence and uniqueness of bases and the representation of algebra morphisms. On the other hand, we use the general notion of a basis in the context of coalgebraic systems and show that a basis for the underlying algebra of a bialgebra gives rise to an equivalent free bialgebra. As a result, we are, for instance, able to recover known constructions from automata theory, namely the socalled canonical residual finite state automaton. Finally, we instantiate the framework to a variety of example monads, including the powerset, downset, distribution, and neighbourhood monad.

show abstract

Section: Neighbourhood Monadmentioning

confidence: 93%

“…In more detail [30], the equivalence (i) assigns to a H-algebra (X, h) the complete atomic Boolean algebra on X with pointwise induced operations…”

Section: Neighbourhood Monadmentioning

confidence: 99%