Minimisation in Logical Form

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

*Bijbehorende auteur voor dit werk

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Stone-type dualities provide a powerful mathematical framework for studying properties of logical systems. They have recently been fruitfully explored in understanding minimisation of various types of automata. In Bezhanishvili et al. (2012), a dual equivalence between a category of coalgebras and a category of algebras was used to explain minimisation. The algebraic semantics is dual to a coalgebraic semantics in which logical equivalence coincides with trace equivalence. It follows that maximal quotients of coalgebras correspond to minimal subobjects of algebras. Examples include partially observable deterministic finite automata, linear weighted automata viewed as coalgebras over finite-dimensional vector spaces, and belief automata, which are coalgebras on compact Hausdorff spaces. In Bonchi et al. (2014), Brzozowski's double-reversal minimisation algorithm for deterministic finite automata was described categorically and its correctness explained via the duality between reachability and observability. This work includes generalisations of Brzozowski's algorithm to Moore and weighted automata over commutative semirings.
In this paper we propose a general categorical framework within which such minimisation algorithms can be understood. The goal is to provide a unifying perspective based on duality. Our framework consists of a stack of three interconnected adjunctions: a base dual adjunction that can be lifted to a dual adjunction between coalgebras and algebras and also to a dual adjunction between automata. The approach provides an abstract understanding of reachability and observability. We illustrate the general framework on range of concrete examples, including deterministic Kripke frames, weighted automata, topological automata (belief automata), and alternating automata.
Originele taal-2English
TitelSamson Abramsky on Logic and Structure in Computer Science and Beyond
RedacteurenAlessandra Palmigiano, Mehrnoosh Sadrzadeh
Plaats van productieCham
UitgeverijSpringer
Hoofdstuk3
Pagina's89-127
Aantal pagina's38
ISBN van elektronische versie978-3-031-24117-8
ISBN van geprinte versie978-3-031-24116-1
DOI's
StatusPublished - 2023

Publicatie series

NaamOutstanding Contributions to Logic
UitgeverijSpringer
Volume25
ISSN van geprinte versie2211-2758
ISSN van elektronische versie2211-2766

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