Results 1  10
of
15
Memoryful Geometry of Interaction From Coalgebraic Components to Algebraic Effects
"... Girard’s Geometry of Interaction (GoI) is interaction based semantics of linear logic proofs and, via suitable translations, of functional programs in general. Its mathematical cleanness identifies essential structures in computation; moreover its use as a compilation technique from programs to s ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
(Show Context)
Girard’s Geometry of Interaction (GoI) is interaction based semantics of linear logic proofs and, via suitable translations, of functional programs in general. Its mathematical cleanness identifies essential structures in computation; moreover its use as a compilation technique from programs to state machines—“GoI implementation, ” so to speak—has been worked out by Mackie, Ghica and others. In this paper, we develop Abramsky’s idea of resumption based GoI systematically into a generic framework that accounts for computational effects (nondeterminism, probability, exception, global states, interactive I/O, etc.). The framework is categorical: Plotkin & Power’s algebraic operations provide an interface to computational effects; the framework is built on the categorical axiomatization of GoI by Abramsky, Haghverdi and Scott; and, by use of the coalgebraic formalization of component calculus, we describe explicit construction of state machines as interpretations of functional programs. The resulting interpretation is shown to be sound with respect to equations between algebraic operations, as well as to Moggi’s equations for the computational lambda calculus. We illustrate the construction by concrete examples.
Pointwise Extensions of GSOSDefined Operations
"... Distributive laws of syntax over behaviour (cf. [1, 3]) are, among other things, a wellstructured way of defining algebraic operations on final coalgebras. For a simple example, consider the set B ω of infinite streams of elements of B; this carries a final coalgebra w = 〈hd,tl〉: B ω → B × B ω for ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
Distributive laws of syntax over behaviour (cf. [1, 3]) are, among other things, a wellstructured way of defining algebraic operations on final coalgebras. For a simple example, consider the set B ω of infinite streams of elements of B; this carries a final coalgebra w = 〈hd,tl〉: B ω → B × B ω for the endofunctor F = B × − on Set. If B comes with a binary operation +, one can define an addition operation ⊕ on streams coinductively: hd(σ ⊕ τ) = hd(σ) + hd(τ) tl(σ ⊕ τ) = tl(σ) ⊕ tl(τ). It is easy to see that these equations define a distributive law, i.e., a natural transformation λ: ΣF ⇒ FΣ, where ΣX = X 2 is the signature endofunctor corresponding to a single binary operation. The operation ⊕: B ω × B ω → B ω is now defined as the unique morphism to the final coalgebra as in: ΣB ω B ω
Generic forward and backward simulations II: Probabilistic simulations
 International Conference on Concurrency Theory (CONCUR 2010), Lect. Notes Comp. Sci
, 2010
"... Abstract. Jonsson and Larsen’s notion of probabilistic simulation is studied from a coalgebraic perspective. The notion is compared with two generic coalgebraic definitions of simulation: Hughes and Jacobs ’ one, and the one introduced previously by the author. We show that the first almost coincid ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
(Show Context)
Abstract. Jonsson and Larsen’s notion of probabilistic simulation is studied from a coalgebraic perspective. The notion is compared with two generic coalgebraic definitions of simulation: Hughes and Jacobs ’ one, and the one introduced previously by the author. We show that the first almost coincides with the second, and that the second is a special case of the last. We investigate implications of this characterization; notably the JonssonLarsen simulation is shown to be sound, i.e. its existence implies trace inclusion. 1
Structural Operational Semantics and Modal Logic, Revisited
"... A previously introduced combination of the bialgebraic approach to structural operational semantics with coalgebraic modal logic is reexamined and improved in some aspects. Firstly, a more abstract, conceptual proof of the main compositionality theorem is given, based on an understanding of modal l ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
A previously introduced combination of the bialgebraic approach to structural operational semantics with coalgebraic modal logic is reexamined and improved in some aspects. Firstly, a more abstract, conceptual proof of the main compositionality theorem is given, based on an understanding of modal logic as a study of coalgebras in slice categories of adjunctions. Secondly, a more concrete understanding of the assumptions of the theorem is provided, where proving compositionality amounts to finding a syntactic distributive law between two collections of predicate liftings. Keywords: structural operational semantics, modal logic, coalgebra 1
Traces for Coalgebraic Components
 MATH. STRUCT. IN COMP. SCIENCE
, 2010
"... This paper contributes a feedback operator, in the form of a monoidal trace, to the theory of coalgebraic, statebased modelling of components. The feedback operator on components is shown to satisfy the trace axioms of Joyal, Street and Verity. We employ McCurdy’s tube diagrams, an extension of sta ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
This paper contributes a feedback operator, in the form of a monoidal trace, to the theory of coalgebraic, statebased modelling of components. The feedback operator on components is shown to satisfy the trace axioms of Joyal, Street and Verity. We employ McCurdy’s tube diagrams, an extension of standard string diagrams for monoidal categories, for representing and manipulating component diagrams. The microcosm principle then yields a canonical “inner” traced monoidal structure on the category of resumptions (elements of final coalgebras / components). This generalises an observation by Abramsky, Haghverdi and Scott.
Stream Differential Equations: Specification Formats and Solution Methods
, 2014
"... Streams, or infinite sequences, are infinite objects of a very simple type, yet they have a rich theory partly due to their ubiquity in mathematics and computer science. Stream differential equations are a coinductive method for specifying streams and stream operations, and their theory has been dev ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Streams, or infinite sequences, are infinite objects of a very simple type, yet they have a rich theory partly due to their ubiquity in mathematics and computer science. Stream differential equations are a coinductive method for specifying streams and stream operations, and their theory has been developed in many papers over the past two decades. In this paper we present a survey of the many results in this area. Our focus is on the classification of different formats of stream differential equations, their solution methods, and the classes of streams they can define. Moreover, we describe in detail the connection between the socalled syntactic solution method and abstract GSOS.
Towards Bialgebraic Semantics for CSP
, 2010
"... This paper extends bialgebraic semantics [1, 2] to take into account notions of behaviour that lead to process equivalences coarser than bisimulation. For that purpose, the requirement of finality for the characterisation of behaviours is relaxed to quasifinality, which informally consists in releg ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
This paper extends bialgebraic semantics [1, 2] to take into account notions of behaviour that lead to process equivalences coarser than bisimulation. For that purpose, the requirement of finality for the characterisation of behaviours is relaxed to quasifinality, which informally consists in relegating to the underlying category the conditions that finality must satisfy in the main category of coalgebras. This setting is then applied to the failure semantics of CSP.
Towards Effects in Mathematical Operational Semantics
"... In this paper, we study extensions of mathematical operational semantics with algebraic effects. Our starting point is an effectfree coalgebraic operational semantics, given by a natural transformation of syntax over behaviour. The operational semantics of the extended language arises by distributi ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
In this paper, we study extensions of mathematical operational semantics with algebraic effects. Our starting point is an effectfree coalgebraic operational semantics, given by a natural transformation of syntax over behaviour. The operational semantics of the extended language arises by distributing program syntax over effects, again inducing a coalgebraic operational semantics, but this time in the Kleisli category for the monad derived from the algebraic effects. The final coalgebra in this Kleisli category then serves as the denotational model. For it to exist, we ensure that the the Kleisli category is enriched over CPOs by considering the monad of possibly infinite terms, extended with a bottom element. Unlike the effectless setting, not all operational specifications give rise to adequate and compositional semantics. We give a proof of adequacy and compositionality provided the specifications can be described by evaluationincontext. We illustrate our techniques with a simple extension of (stateless) while programs with global store, i.e. variable lookup.
1.1 Structural Operational Semantics and Its Bialgebraic Modeling
"... Abstract. In the previous work by Jacobs, Sokolova and the author, synchronous parallel composition of coalgebras—yielding a coalgebra—and parallel composition of behaviors—yielding a behavior, where behaviors are identified with states of the final coalgebra—were observed to form an instance of th ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. In the previous work by Jacobs, Sokolova and the author, synchronous parallel composition of coalgebras—yielding a coalgebra—and parallel composition of behaviors—yielding a behavior, where behaviors are identified with states of the final coalgebra—were observed to form an instance of the microcosm principle. The microcosm principle, a term by Baez and Dolan, refers to the general phenomenon of nested algebraic structures such as a monoid in a monoidal category. Suitable organization of these two levels of parallel composition led to a general compositionality theorem: the behavior of the composed system relies only on the behaviors of its constituent parts. In the current paper this framework is extended so that it accommodates any process operator—not restricted to parallel composition—whose meaning is specified by means of GSOS rules. This generalizes Turi and Plotkin’s bialgebraic modeling of GSOS, by allowing a process operator to act as a connector between components as coalgebras.