8 research outputs found
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
The Proof Technique of Unique Solutions of Contractions
International audienceWe review some recent work aimed at understanding proof techniques for behavioural equivalence on processes based on the concept of unique solution of equations. The schema of equations is refined to that of contraction, based on partial orders rather than equalities
GSOS for non-deterministic processes with quantitative aspects
Recently, some general frameworks have been proposed as unifying theories for
processes combining non-determinism with quantitative aspects (such as
probabilistic or stochastically timed executions), aiming to provide general
results and tools. This paper provides two contributions in this respect.
First, we present a general GSOS specification format (and a corresponding
notion of bisimulation) for non-deterministic processes with quantitative
aspects. These specifications define labelled transition systems according to
the ULTraS model, an extension of the usual LTSs where the transition relation
associates any source state and transition label with state reachability weight
functions (like, e.g., probability distributions). This format, hence called
Weight Function SOS (WFSOS), covers many known systems and their bisimulations
(e.g. PEPA, TIPP, PCSP) and GSOS formats (e.g. GSOS, Weighted GSOS,
Segala-GSOS, among others).
The second contribution is a characterization of these systems as coalgebras
of a class of functors, parametric on the weight structure. This result allows
us to prove soundness of the WFSOS specification format, and that
bisimilarities induced by these specifications are always congruences.Comment: In Proceedings QAPL 2014, arXiv:1406.156
Foundational Extensible Corecursion
This paper presents a formalized framework for defining corecursive functions
safely in a total setting, based on corecursion up-to and relational
parametricity. The end product is a general corecursor that allows corecursive
(and even recursive) calls under well-behaved operations, including
constructors. Corecursive functions that are well behaved can be registered as
such, thereby increasing the corecursor's expressiveness. The metatheory is
formalized in the Isabelle proof assistant and forms the core of a prototype
tool. The corecursor is derived from first principles, without requiring new
axioms or extensions of the logic
The Guarded Lambda-Calculus: Programming and Reasoning with Guarded Recursion for Coinductive Types
We present the guarded lambda-calculus, an extension of the simply typed
lambda-calculus with guarded recursive and coinductive types. The use of
guarded recursive types ensures the productivity of well-typed programs.
Guarded recursive types may be transformed into coinductive types by a
type-former inspired by modal logic and Atkey-McBride clock quantification,
allowing the typing of acausal functions. We give a call-by-name operational
semantics for the calculus, and define adequate denotational semantics in the
topos of trees. The adequacy proof entails that the evaluation of a program
always terminates. We introduce a program logic with L\"ob induction for
reasoning about the contextual equivalence of programs. We demonstrate the
expressiveness of the calculus by showing the definability of solutions to
Rutten's behavioural differential equations.Comment: Accepted to Logical Methods in Computer Science special issue on the
18th International Conference on Foundations of Software Science and
Computation Structures (FoSSaCS 2015
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
Abstract GSOS Rules and a Modular Treatment of Recursive Definitions
Terminal coalgebras for a functor serve as semantic domains for state-basedsystems of various types. For example, behaviors of CCS processes, streams,infinite trees, formal languages and non-well-founded sets form terminalcoalgebras. We present a uniform account of the semantics of recursivedefinitions in terminal coalgebras by combining two ideas: (1) abstract GSOSrules l specify additional algebraic operations on a terminal coalgebra; (2)terminal coalgebras are also initial completely iterative algebras (cias). Wealso show that an abstract GSOS rule leads to new extended cia structures onthe terminal coalgebra. Then we formalize recursive function definitionsinvolving given operations specified by l as recursive program schemes for l,and we prove that unique solutions exist in the extended cias. From our resultsit follows that the solutions of recursive (function) definitions in terminalcoalgebras may be used in subsequent recursive definitions which still haveunique solutions. We call this principle modularity. We illustrate our resultsby the five concrete terminal coalgebras mentioned above, e.\,g., a finitestream circuit defines a unique stream function
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