10 research outputs found
A duality between exceptions and states
In this short note we study the semantics of two basic computational effects,
exceptions and states, from a new point of view. In the handling of exceptions
we dissociate the control from the elementary operation which recovers from the
exception. In this way it becomes apparent that there is a duality, in the
categorical sense, between exceptions and states
Decorated proofs for computational effects: Exceptions
We define a proof system for exceptions which is close to the syntax for
exceptions, in the sense that the exceptions do not appear explicitly in the
type of any expression. This proof system is sound with respect to the intended
denotational semantics of exceptions. With this inference system we prove
several properties of exceptions.Comment: 11 page
Bases as Coalgebras
The free algebra adjunction, between the category of algebras of a monad and
the underlying category, induces a comonad on the category of algebras. The
coalgebras of this comonad are the topic of study in this paper (following
earlier work). It is illustrated how such coalgebras-on-algebras can be
understood as bases, decomposing each element x into primitives elements from
which x can be reconstructed via the operations of the algebra. This holds in
particular for the free vector space monad, but also for other monads, like
powerset or distribution. For instance, continuous dcpos or stably continuous
frames, where each element is the join of the elements way below it, can be
described as such coalgebras. Further, it is shown how these
coalgebras-on-algebras give rise to a comonoid structure for copy and delete,
and thus to diagonalisation of endomaps like in linear algebra
Patterns for computational effects arising from a monad or a comonad
This paper presents equational-based logics for proving first order
properties of programming languages involving effects. We propose two dual
inference system patterns that can be instanciated with monads or comonads in
order to be used for proving properties of different effects. The first pattern
provides inference rules which can be interpreted in the Kleisli category of a
monad and the coKleisli category of the associated comonad. In a dual way, the
second pattern provides inference rules which can be interpreted in the
coKleisli category of a comonad and the Kleisli category of the associated
monad. The logics combine a 3-tier effect system for terms consisting of pure
terms and two other kinds of effects called 'constructors/observers' and
'modifiers', and a 2-tier system for 'up-to-effects' and 'strong' equations.
Each pattern provides generic rules for dealing with any monad (respectively
comonad), and it can be extended with specific rules for each effect. The paper
presents two use cases: a language with exceptions (using the standard monadic
semantics), and a language with state (using the less standard comonadic
semantics). Finally, we prove that the obtained inference system for states is
Hilbert-Post complete
Exploring the Boundaries of Monad Tensorability on Set
We study a composition operation on monads, equivalently presented as large
equational theories. Specifically, we discuss the existence of tensors, which
are combinations of theories that impose mutual commutation of the operations
from the component theories. As such, they extend the sum of two theories,
which is just their unrestrained combination. Tensors of theories arise in
several contexts; in particular, in the semantics of programming languages, the
monad transformer for global state is given by a tensor. We present two main
results: we show that the tensor of two monads need not in general exist by
presenting two counterexamples, one of them involving finite powerset (i.e. the
theory of join semilattices); this solves a somewhat long-standing open
problem, and contrasts with recent results that had ruled out previously
expected counterexamples. On the other hand, we show that tensors with bounded
powerset monads do exist from countable powerset upwards
Adjunctions for exceptions
An algebraic method is used to study the semantics of exceptions in computer
languages. The exceptions form a computational effect, in the sense that there
is an apparent mismatch between the syntax of exceptions and their intended
semantics. We solve this apparent contradiction by efining a logic for
exceptions with a proof system which is close to their syntax and where their
intended semantics can be seen as a model. This requires a robust framework for
logics and their morphisms, which is provided by categorical tools relying on
adjunctions, fractions and limit sketches.Comment: In this Version 2, minor improvements are made to Version
States and exceptions considered as dual effects
In this paper we consider the two major computational effects of states and
exceptions, from the point of view of diagrammatic logics. We get a surprising
result: there exists a symmetry between these two effects, based on the
well-known categorical duality between products and coproducts. More precisely,
the lookup and update operations for states are respectively dual to the throw
and catch operations for exceptions. This symmetry is deeply hidden in the
programming languages; in order to unveil it, we start from the monoidal
equational logic and we add progressively the logical features which are
necessary for dealing with either effect. This approach gives rise to a new
point of view on states and exceptions, which bypasses the problems due to the
non-algebraicity of handling exceptions
Bootstrapping Inductive and Coinductive Types in HasCASL
We discuss the treatment of initial datatypes and final process types in the
wide-spectrum language HasCASL. In particular, we present specifications that
illustrate how datatypes and process types arise as bootstrapped concepts using
HasCASL's type class mechanism, and we describe constructions of types of
finite and infinite trees that establish the conservativity of datatype and
process type declarations adhering to certain reasonable formats. The latter
amounts to modifying known constructions from HOL to avoid unique choice; in
categorical terminology, this means that we establish that quasitoposes with an
internal natural numbers object support initial algebras and final coalgebras
for a range of polynomial functors, thereby partially generalising
corresponding results from topos theory. Moreover, we present similar
constructions in categories of internal complete partial orders in
quasitoposes
Generic Exception Handling and the Java Monad
We develop an equational definition of exception monads that characterizes Moggi's exception monad transformer. This axiomatization is then used to define an extension of previously described monad-independent computational logics by abnormal termination. Instantiating this generic formalism with the Java monad used in the LOOP project yields in particular the known Hoare calculi with abnormal termination and JML's method specifications; this opens up the possibility of extending these formalisms by hitherto missing computational features such as I/O and nondeterminism