324 research outputs found
On Well-Founded and Recursive Coalgebras
This paper studies fundamental questions concerning category-theoretic models
of induction and recursion. We are concerned with the relationship between
well-founded and recursive coalgebras for an endofunctor. For monomorphism
preserving endofunctors on complete and well-powered categories every coalgebra
has a well-founded part, and we provide a new, shorter proof that this is the
coreflection in the category of all well-founded coalgebras. We present a new
more general proof of Taylor's General Recursion Theorem that every
well-founded coalgebra is recursive, and we study under which hypothesis the
converse holds. In addition, we present a new equivalent characterization of
well-foundedness: a coalgebra is well-founded iff it admits a
coalgebra-to-algebra morphism to the initial algebra
Well-Pointed Coalgebras
For endofunctors of varieties preserving intersections, a new description of
the final coalgebra and the initial algebra is presented: the former consists
of all well-pointed coalgebras. These are the pointed coalgebras having no
proper subobject and no proper quotient. The initial algebra consists of all
well-pointed coalgebras that are well-founded in the sense of Osius and Taylor.
And initial algebras are precisely the final well-founded coalgebras. Finally,
the initial iterative algebra consists of all finite well-pointed coalgebras.
Numerous examples are discussed e.g. automata, graphs, and labeled transition
systems
Abstract GSOS Rules and a Modular Treatment of Recursive Definitions
Terminal coalgebras for a functor serve as semantic domains for state-based
systems of various types. For example, behaviors of CCS processes, streams,
infinite trees, formal languages and non-well-founded sets form terminal
coalgebras. We present a uniform account of the semantics of recursive
definitions in terminal coalgebras by combining two ideas: (1) abstract GSOS
rules l specify additional algebraic operations on a terminal coalgebra; (2)
terminal coalgebras are also initial completely iterative algebras (cias). We
also show that an abstract GSOS rule leads to new extended cia structures on
the terminal coalgebra. Then we formalize recursive function definitions
involving given operations specified by l as recursive program schemes for l,
and we prove that unique solutions exist in the extended cias. From our results
it follows that the solutions of recursive (function) definitions in terminal
coalgebras may be used in subsequent recursive definitions which still have
unique solutions. We call this principle modularity. We illustrate our results
by the five concrete terminal coalgebras mentioned above, e.\,g., a finite
stream circuit defines a unique stream function
An abstract view on syntax with sharing
The notion of term graph encodes a refinement of inductively generated syntax
in which regard is paid to the the sharing and discard of subterms. Inductively
generated syntax has an abstract expression in terms of initial algebras for
certain endofunctors on the category of sets, which permits one to go beyond
the set-based case, and speak of inductively generated syntax in other
settings. In this paper we give a similar abstract expression to the notion of
term graph. Aspects of the concrete theory are redeveloped in this setting, and
applications beyond the realm of sets discussed.Comment: 26 pages; v2: final journal versio
Representations of stream processors using nested fixed points
We define representations of continuous functions on infinite streams of discrete values, both in the case of discrete-valued functions, and in the case of stream-valued functions. We define also an operation on the representations of two continuous functions between streams that yields a representation of their composite. In the case of discrete-valued functions, the representatives are well-founded (finite-path) trees of a certain kind. The underlying idea can be traced back to Brouwer's justification of bar-induction, or to Kreisel and Troelstra's elimination of choice-sequences. In the case of stream-valued functions, the representatives are non-wellfounded trees pieced together in a coinductive fashion from well-founded trees. The definition requires an alternating fixpoint construction of some ubiquity
Generic Trace Semantics via Coinduction
Trace semantics has been defined for various kinds of state-based systems,
notably with different forms of branching such as non-determinism vs.
probability. In this paper we claim to identify one underlying mathematical
structure behind these "trace semantics," namely coinduction in a Kleisli
category. This claim is based on our technical result that, under a suitably
order-enriched setting, a final coalgebra in a Kleisli category is given by an
initial algebra in the category Sets. Formerly the theory of coalgebras has
been employed mostly in Sets where coinduction yields a finer process semantics
of bisimilarity. Therefore this paper extends the application field of
coalgebras, providing a new instance of the principle "process semantics via
coinduction."Comment: To appear in Logical Methods in Computer Science. 36 page
Inductive and Coinductive Components of Corecursive Functions in Coq
In Constructive Type Theory, recursive and corecursive definitions are
subject to syntactic restrictions which guarantee termination for recursive
functions and productivity for corecursive functions. However, many terminating
and productive functions do not pass the syntactic tests. Bove proposed in her
thesis an elegant reformulation of the method of accessibility predicates that
widens the range of terminative recursive functions formalisable in
Constructive Type Theory. In this paper, we pursue the same goal for productive
corecursive functions. Notably, our method of formalisation of coinductive
definitions of productive functions in Coq requires not only the use of ad-hoc
predicates, but also a systematic algorithm that separates the inductive and
coinductive parts of functions.Comment: Dans Coalgebraic Methods in Computer Science (2008
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