114 research outputs found
Coinduction up to in a fibrational setting
Bisimulation up-to enhances the coinductive proof method for bisimilarity,
providing efficient proof techniques for checking properties of different kinds
of systems. We prove the soundness of such techniques in a fibrational setting,
building on the seminal work of Hermida and Jacobs. This allows us to
systematically obtain up-to techniques not only for bisimilarity but for a
large class of coinductive predicates modelled as coalgebras. By tuning the
parameters of our framework, we obtain novel techniques for unary predicates
and nominal automata, a variant of the GSOS rule format for similarity, and a
new categorical treatment of weak bisimilarity
Indexed induction and coinduction, fibrationally.
This paper extends the fibrational approach to induction and coinduction pioneered by Hermida and Jacobs, and developed by the current authors, in two key directions. First, we present a sound coinduction rule for any data type arising as the final coalgebra of a functor, thus relaxing Hermida and Jacobsâ restriction to polynomial data types. For this we introduce the notion of a quotient category with equality (QCE), which both abstracts the standard notion of a fibration of relations constructed from a given fibration, and plays a role in the theory of coinduction dual to that of a comprehension category with unit (CCU) in the theory of induction. Second, we show that indexed inductive and coinductive types also admit sound induction and coinduction rules. Indexed data types often arise as initial algebras and final coalgebras of functors on slice categories, so our key technical results give sufficent conditions under which we can construct, from a CCU (QCE) U : E -> B, a fibration with base B/I that models indexing by I and is also a CCU (QCE)
Indexed Induction and Coinduction, Fibrationally
This paper extends the fibrational approach to induction and coinduction
pioneered by Hermida and Jacobs, and developed by the current authors, in two
key directions. First, we present a dual to the sound induction rule for
inductive types that we developed previously. That is, we present a sound
coinduction rule for any data type arising as the carrier of the final
coalgebra of a functor, thus relaxing Hermida and Jacobs' restriction to
polynomial functors. To achieve this we introduce the notion of a quotient
category with equality (QCE) that i) abstracts the standard notion of a
fibration of relations constructed from a given fibration; and ii) plays a role
in the theory of coinduction dual to that played by a comprehension category
with unit (CCU) in the theory of induction. Secondly, we show that inductive
and coinductive indexed types also admit sound induction and coinduction rules.
Indexed data types often arise as carriers of initial algebras and final
coalgebras of functors on slice categories, so we give sufficient conditions
under which we can construct, from a CCU (QCE) U:E \rightarrow B, a fibration
with base B/I that models indexing by I and is also a CCU (resp., QCE). We
finish the paper by considering the more general case of sound induction and
coinduction rules for indexed data types when the indexing is itself given by a
fibration
Indexed Induction And Coinduction, Fibrationally
This paper extends the fibrational approach to induction and coinductionpioneered by Hermida and Jacobs, and developed by the current authors, in two keydirections. First, we present a dual to the sound induction rule for inductive types thatwe developed previously. That is, we present a sound coinduction rule for any data typearising as the carrier of the final coalgebra of a functor, thus relaxing Hermida and Jacobsârestriction to polynomial functors. To achieve this we introduce the notion of a quotientcategory with equality (QCE) that i) abstracts the standard notion of a fibration of relationsconstructed from a given fibration; and ii) plays a role in the theory of coinduction dualto that played by a comprehension category with unit (CCU) in the theory of induction.Secondly, we show that inductive and coinductive indexed types also admit sound inductionand coinduction rules. Indexed data types often arise as carriers of initial algebras andfinal coalgebras of functors on slice categories, so we give sufficient conditions under which we can construct, from a CCU (QCE) U : E : B, a fibration with base B/I that models indexing by I and is also a CCU (resp., QCE). We finish the paper by considering themore general case of sound induction and coinduction rules for indexed data types whenthe indexing is itself given by a fibration
Fibrational induction rules for initial algebras
This paper provides an induction rule that can be used to prove properties of data structures whose types are inductive, i.e., are carriers of initial algebras of functors. Our results are semantic in nature and are inspired by Hermida and Jacobsâ elegant algebraic formulation of induction for polynomial data types. Our contribution is to derive, under slightly different assumptions, an induction rule that is generic over all inductive types, polynomial or not. Our induction rule is generic over the kinds of properties to be proved as well: like Hermida and Jacobs, we work in a general fibrational setting and so can accommodate very general notions of properties on inductive types rather than just those of particular syntactic forms. We establish the correctness of our generic induction rule by reducing induction to iteration. We show how our rule can be instantiated to give induction rules for the data types of rose trees, finite hereditary sets, and hyperfunctions. The former lies outside the scope of Hermida and Jacobsâ work because it is not polynomial; as far as we are aware, no induction rules have been known to exist for the latter two in a general fibrational framework. Our instantiation for hyperfunctions underscores the value of working in the general fibrational setting since this data type cannot be interpreted as a set
Up-To Techniques for Behavioural Metrics via Fibrations
Up-to techniques are a well-known method for enhancing coinductive proofs of behavioural equivalences. We introduce up-to techniques for behavioural metrics between systems modelled as coalgebras and we provide abstract results to prove their soundness in a compositional way.
In order to obtain a general framework, we need a systematic way to lift functors: we show that the Wasserstein lifting of a functor, introduced in a previous work, corresponds to a change of base in a fibrational sense. This observation enables us to reuse existing results about soundness of up-to techniques in a fibrational setting. We focus on the fibrations of predicates and relations valued in a quantale, for which pseudo-metric spaces are an example. To illustrate our approach we provide an example on distances between regular languages
Fibrational induction meets effects
This paper provides several induction rules that can be used to prove properties of effectful data types. Our results are semantic in nature and build upon Hermida and Jacobsâ fibrational formulation of induction for polynomial data types and its extension to all inductive data types by Ghani, Johann, and Fumex. An effectful data type ÎŒ(TF) is built from a functor F that describes data, and a monad T that computes effects. Our main contribution is to derive induction rules that are generic over all functors F and monads T such that ÎŒ(TF) exists. Along the way, we also derive a principle of definition by structural recursion for effectful data types that is similarly generic. Our induction rule is also generic over the kinds of properties to be proved: like the work on which we build, we work in a general fibrational setting and so can accommodate very general notions of properties, rather than just those of particular syntactic forms. We give examples exploiting the generality of our results, and show how our results specialize to those in the literature, particularly those of Filinski and StĂžvring
Expressive Logics for Coinductive Predicates
The classical Hennessy-Milner theorem says that two states of an image-finite transition system are bisimilar if and only if they satisfy the same formulas in a certain modal logic. In this paper we study this type of result in a general context, moving from transition systems to coalgebras and from bisimilarity to coinductive predicates. We formulate when a logic fully characterises a coinductive predicate on coalgebras, by providing suitable notions of adequacy and expressivity, and give sufficient conditions on the semantics. The approach is illustrated with logics characterising similarity, divergence and a behavioural metric on automata
Dependent Inductive and Coinductive Types are Fibrational Dialgebras
In this paper, I establish the categorical structure necessary to interpret
dependent inductive and coinductive types. It is well-known that dependent type
theories \`a la Martin-L\"of can be interpreted using fibrations. Modern
theorem provers, however, are based on more sophisticated type systems that
allow the definition of powerful inductive dependent types (known as inductive
families) and, somewhat limited, coinductive dependent types. I define a class
of functors on fibrations and show how data type definitions correspond to
initial and final dialgebras for these functors. This description is also a
proposal of how coinductive types should be treated in type theories, as they
appear here simply as dual of inductive types. Finally, I show how dependent
data types correspond to algebras and coalgebras, and give the correspondence
to dependent polynomial functors.Comment: In Proceedings FICS 2015, arXiv:1509.0282
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