907 research outputs found
Guard Your Daggers and Traces: On The Equational Properties of Guarded (Co-)recursion
Motivated by the recent interest in models of guarded (co-)recursion we study
its equational properties. We formulate axioms for guarded fixpoint operators
generalizing the axioms of iteration theories of Bloom and Esik. Models of
these axioms include both standard (e.g., cpo-based) models of iteration
theories and models of guarded recursion such as complete metric spaces or the
topos of trees studied by Birkedal et al. We show that the standard result on
the satisfaction of all Conway axioms by a unique dagger operation generalizes
to the guarded setting. We also introduce the notion of guarded trace operator
on a category, and we prove that guarded trace and guarded fixpoint operators
are in one-to-one correspondence. Our results are intended as first steps
leading to the description of classifying theories for guarded recursion and
hence completeness results involving our axioms of guarded fixpoint operators
in future work.Comment: In Proceedings FICS 2013, arXiv:1308.589
Terminal semantics for codata types in intensional Martin-L\"of type theory
In this work, we study the notions of relative comonad and comodule over a
relative comonad, and use these notions to give a terminal coalgebra semantics
for the coinductive type families of streams and of infinite triangular
matrices, respectively, in intensional Martin-L\"of type theory. Our results
are mechanized in the proof assistant Coq.Comment: 14 pages, ancillary files contain formalized proof in the proof
assistant Coq; v2: 20 pages, title and abstract changed, give a terminal
semantics for streams as well as for matrices, Coq proof files updated
accordingl
Distributive Laws for Monotone Specifications
Turi and Plotkin introduced an elegant approach to structural operational
semantics based on universal coalgebra, parametric in the type of syntax and
the type of behaviour. Their framework includes abstract GSOS, a categorical
generalisation of the classical GSOS rule format, as well as its categorical
dual, coGSOS. Both formats are well behaved, in the sense that each
specification has a unique model on which behavioural equivalence is a
congruence. Unfortunately, the combination of the two formats does not feature
these desirable properties. We show that monotone specifications - that
disallow negative premises - do induce a canonical distributive law of a monad
over a comonad, and therefore a unique, compositional interpretation.Comment: In Proceedings EXPRESS/SOS 2017, arXiv:1709.0004
Bialgebraic Semantics for Logic Programming
Bialgebrae provide an abstract framework encompassing the semantics of
different kinds of computational models. In this paper we propose a bialgebraic
approach to the semantics of logic programming. Our methodology is to study
logic programs as reactive systems and exploit abstract techniques developed in
that setting. First we use saturation to model the operational semantics of
logic programs as coalgebrae on presheaves. Then, we make explicit the
underlying algebraic structure by using bialgebrae on presheaves. The resulting
semantics turns out to be compositional with respect to conjunction and term
substitution. Also, it encodes a parallel model of computation, whose soundness
is guaranteed by a built-in notion of synchronisation between different
threads
Unguarded Recursion on Coinductive Resumptions
We study a model of side-effecting processes obtained by starting from a
monad modelling base effects and adjoining free operations using a cofree
coalgebra construction; one thus arrives at what one may think of as types of
non-wellfounded side-effecting trees, generalizing the infinite resumption
monad. Correspondingly, the arising monad transformer has been termed the
coinductive generalized resumption transformer. Monads of this kind have
received some attention in the recent literature; in particular, it has been
shown that they admit guarded iteration. Here, we show that they also admit
unguarded iteration, i.e. form complete Elgot monads, provided that the
underlying base effect supports unguarded iteration. Moreover, we provide a
universal characterization of the coinductive resumption monad transformer in
terms of coproducts of complete Elgot monads.Comment: 47 pages, extended version of
http://www.sciencedirect.com/science/article/pii/S157106611500079
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