9 research outputs found
Minimisation in Logical Form
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
Coalgebra Encoding for Efficient Minimization
Recently, we have developed an efficient generic partition refinement algorithm, which computes behavioural equivalence on a state-based system given as an encoded coalgebra, and implemented it in the tool CoPaR. Here we extend this to a fully fledged minimization algorithm and tool by integrating two new aspects: (1) the computation of the transition structure on the minimized state set, and (2) the computation of the reachable part of the given system. In our generic coalgebraic setting these two aspects turn out to be surprisingly non-trivial requiring us to extend the previous theory. In particular, we identify a sufficient condition on encodings of coalgebras, and we show how to augment the existing interface, which encapsulates computations that are specific for the coalgebraic type functor, to make the above extensions possible. Both extensions have linear run time
Coalgebra Encoding for Efficient Minimization
Recently, we have developed an efficient generic partition refinement
algorithm, which computes behavioural equivalence on a state-based system given
as an encoded coalgebra, and implemented it in the tool CoPaR. Here we extend
this to a fully fledged minimization algorithm and tool by integrating two new
aspects: (1) the computation of the transition structure on the minimized state
set, and (2) the computation of the reachable part of the given system. In our
generic coalgebraic setting these two aspects turn out to be surprisingly
non-trivial requiring us to extend the previous theory. In particular, we
identify a sufficient condition on encodings of coalgebras, and we show how to
augment the existing interface, which encapsulates computations that are
specific for the coalgebraic type functor, to make the above extensions
possible. Both extensions have linear run time
A Coalgebraic Approach to Dualities for Neighborhood Frames
We develop a uniform coalgebraic approach to J\'onsson-Tarski and Thomason
type dualities for various classes of neighborhood frames and neighborhood
algebras. In the first part of the paper we construct an endofunctor on the
category of complete and atomic Boolean algebras that is dual to the double
powerset functor on . This allows us to show that Thomason
duality for neighborhood frames can be viewed as an algebra-coalgebra duality.
We generalize this approach to any class of algebras for an endofunctor
presented by one-step axioms in the language of infinitary modal logic. As a
consequence, we obtain a uniform approach to dualities for various classes of
neighborhood frames, including monotone neighborhood frames, pretopological
spaces, and topological spaces.
In the second part of the paper we develop a coalgebraic approach to
J\'{o}nsson-Tarski duality for neighborhood algebras and descriptive
neighborhood frames. We introduce an analogue of the Vietoris endofunctor on
the category of Stone spaces and show that descriptive neighborhood frames are
isomorphic to coalgebras for this endofunctor. This allows us to obtain a
coalgebraic proof of the duality between descriptive neighborhood frames and
neighborhood algebras. Using one-step axioms in the language of finitary modal
logic, we restrict this duality to other classes of neighborhood algebras
studied in the literature, including monotone modal algebras and contingency
algebras.
We conclude the paper by connecting the two types of dualities via canonical
extensions, and discuss when these extensions are functorial
Dualities in modal logic
Categorical dualities are an important tool in the study of (modal) logics. They offer conceptual understanding and enable the transfer of results between the different semantics of a logic. As such, they play a central role in the proofs of completeness theorems, Sahlqvist theorems and Goldblatt-Thomason theorems. A common way to obtain dualities is by extending existing ones. For example, Jonsson-Tarski duality is an extension of Stone duality. A convenient formalism to carry out such extensions is given by the dual categorical notions of algebras and coalgebras. Intuitively, these allow one to isolate the new part of a duality from the existing part. In this thesis we will derive both existing and new dualities via this route, and we show how to use the dualities to investigate logics. However, not all (modal logical) paradigms fit the (co)algebraic perspective. In particular, modal intuitionistic logics do not enjoy a coalgebraic treatment, and there is a general lack of duality results for them. To remedy this, we use a generalisation of both algebras and coalgebras called dialgebras. Guided by the research field of coalgebraic logic, we introduce the framework of dialgebraic logic. We show how a large class of modal intuitionistic logics can be modelled as dialgebraic logics and we prove dualities for them. We use the dialgebraic framework to prove general completeness, Hennessy-Milner, representation and Goldblatt-Thomason theorems, and instantiate this to a wide variety of modal intuitionistic logics. Additionally, we use the dialgebraic perspective to investigate modal extensions of the meet-implication fragment of intuitionistic logic. We instantiate general dialgebraic results, and describe how modal meet-implication logics relate to modal intuitionistic logics