An abstract monadic semantics for value recursion

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 f...

extended abstract -We try to combine the 'syntactic composition of tree transducers' [FV98, KV01] on the one hand side and 'short cut fusion' [GLP93] on the other hand side.

2002

We model notions of computation using algebraic operations and equations. We show that these generate several of the monads of primary interest that have been used to model computational effects, with the striking omission of the continuations monad. We focus on semantics for global and local state, showing that taking operations and equations as primitive yields a mathematical relationship that reflects their computational relationship. This work is supported by EPSRC grant GR/M56333 and a British Council grant. may also prove useful when manipulating programs, e.g., for optimisation. We regard it as positive that there are no algebraic operations and equations in the sense of the theorem yielding the continuations monad, taking that as reflecting its differing computational nature.

2012

Abstract: In functional programming, monads are supposed to encapsulate computations, effectfully producing the final result, but keeping to themselves the means of acquiring it. For various reasons, we sometimes want to reveal the internals of a computation. To make that possible, in this paper we introduce monad transformers that add the ability to automatically accumulate observations about the course of execution as an effect.

Information and Computation, 2010

Iterative monads of Calvin Elgot were introduced to treat the semantics of recursive equations purely algebraically. They are Lawvere theories with the property that all ideal systems of recursive equations have unique solutions. We prove that the unique solutions in iterative monads satisfy all the equational properties of iteration monads of Stephen Bloom and Zoltán Ésik, whenever the base category is hyper-extensive and locally finitely presentable. This result is a step towards proving that functorial iteration monads form a monadic category over sets in context. This shows that functoriality is an equational property when considered w.r.t. sets in context.

ACM SIGPLAN Notices, 2013

Recent developments in the systematic construction of abstract interpreters hinted at the possibility of a broad unification of concepts in static analysis. We deliver that unification by showing context-sensitivity, polyvariance, flow-sensitivity, reachabilitypruning, heap-cloning and cardinality-bounding to be independent of any particular semantics. Monads become the unifying agent between these concepts and between semantics. For instance, by plugging the same "context-insensitivity monad" into a monadicallyparameterized semantics for Java or for the lambda calculus, it yields the expected context-insensitive analysis.

Lecture Notes in Computer Science, 2009

For ideal monads in Set (e. g. the finite list monad, the finite bag monad etc.) we have recently proved that every set generates a free iterative algebra. This gives rise to a new monad. We prove now that this monad is iterative in the sense of Calvin Elgot, in fact, this is the iterative reflection of the given ideal monad. This shows how to freely add unique solutions of recursive equations to a given algebraic theory. Examples: the monad of free commutative binary algebras has the monad of binary rational unordered trees as iterative reflection, and the finite list monad has the iterative reflection given by adding an absorbing element.

Electronic Notes in Theoretical Computer Science, 1999

We de ne a typed compiler intermediate language, MIL-lite, which incorporates computational types re ned with e ect information. We characterise MIL-lite observational congruence by using Howe's method to prove a ciu theorem for the language in terms of a termination predicate de ned directly on the term. We then de ne a logical predicate which captures an observable version of the intended meaning of each of our e ect annotations. Having proved the fundamental theorem for this predicate, we use it with the ciu theorem to validate a number of e ect-based transformations performed by the MLj compiler for Standard ML.

Electronic Notes in Theoretical Computer Science, 2011

In this paper, we study extensions of mathematical operational semantics with algebraic effects. Our starting point is an effect-free 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 evaluation-in-context. We illustrate our techniques with a simple extension of (stateless) while programs with global store, i.e. variable lookup.

Information and Computation, 1991

The λ-calculus is considered an useful mathematical tool in the study of programming languages, since programs can be identified with λ-terms. However, if one goes further and uses βη-conversion to prove equivalence of programs, then a gross simplification is introduced (programs are identified with total functions from values to values), that may jeopardise the applicability of theoretical results. In this paper we introduce calculi based on a categorical semantics for computations, that provide a correct basis for proving equivalence of programs, for a wide range of notions of computation. * Research partially supported by EEC Joint Collaboration Contract # ST2J-0374-C(EDB).