Modular SOS (MSOS) is a variant of conventional Structural Operational Semantics (SOS). Using MSOS, the transition rules for each construct of a programming language can be given incrementally, once and for all, and do not need reformulation when further constructs are added to the language. MSOS thus provides an exceptionally high degree of modularity in language descriptions, removing a shortcoming of the original SOS framework. After sketching the background and reviewing the main features of SOS, the paper explains the crucial differences between SOS and MSOS, and illustrates how MSOS descriptions are written. It also discusses standard notions of semantic equivalence based on MSOS. Appendix A shows how the illustrative MSOS rules given in the paper would be formulated in conventional SOS.

We give semantics for notions of computation, also called computational effects, by means of 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, demonstrating the latter to be of a different character, as is computationally true. 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.

Given a complete and cocomplete symmetric monoidal closed category V and a symmetric monoidal V-category C with cotensors and a strong V-monad T on C, we investigate axioms under which an ObC-indexed family of operations of the form α_x : (Tx)^ν → (Tx)^ω provides semantics for algebraic operations on the computational λ-calculus. We recall a definition for which we have elsewhere given adequacy results, and we show that an enrichment of it is equivalent to a range of other possible natural definitions of algebraic operation. In particular, we define the notion of generic effect and show that to give a generic effect is equivalent to giving an algebraic operation. We further show how the usual monadic semantics of the computational λ-calculus extends uniformly to incorporate generic effects. We outline examples and non-examples and we show that our definition also enriches one for call-by-name languages with e#ects.

We seek a unified account of modularity for computational effects. We begin by reformulating Moggi’s monadic paradigm for modelling computational effects using the notion of enriched Lawvere theory, together with its relationship with strong monads; this emphasises the importance of the operations that produce the effects. Effects qua theories are then combined by appropriate bifunctors on the category of theories. We give a theory for the sum of computational effects, which in particular yields Moggi’s exceptions monad transformer and an interactive input/output monad transformer. We further give a theory of the commutative combination of effects, their tensor, which yields Moggi’s side-effects monad transformer. Finally we give a theory of operation transformers, for redefining operations when adding new effects; we derive explicit forms for the operation transformers associated to the above monad transformers.

We begin to develop a unified account of modularity for computational effects. We use the notion of enriched Lawvere theory, together with its relationship with strong monads, to reformulate Moggi's paradigm for modelling computational effects; we emphasise the importance here of the operations that induce computational effects. Effects qua theories are then combined by appropriate bifunctors (on the category of theories). We give a theory of the commutative combination of effects, which in particular yields Moggi's side-effects monad transformer (an application is the combination of side-effects with nondeterminism). And we give a theory...

Given a complete and cocomplete symmetric monoidal closed category V and a symmetric monoidal V -category C with cotensors and a strong V -monad T on C, we investigate axioms under which an ObC- indexed family of operations of the form #x : (Tx) v -# (Tx) w provides semantics for algebraic operations, which may be used to extend the usual monadic semantics of the computational #-calculus uniformly. We recall a definition for which we have elsewhere given adequacy results, and we show that an enrichment of it is equivalent to a range of other possible natural definitions of algebraic operation. We outline examples and non-examples and we show that our definition also enriches one for call-by-name languages with e#ects. 1

Lawvere theories have been one of the two main category theoretic formulations of universal algebra, the other being monads. Monads have appeared extensively over the past fifteen years in the theoretical computer science literature, specifically in connection with computational effects, but Lawvere theories have not. So we define the notion of (countable) Lawvere theory and give a precise statement of its relationship with the notion of monad on the category Set. We illustrate with examples arising from the study of computational effects, explaining how the notion of Lawvere theory keeps one closer to computational practice. We then describe constructions that one can make with Lawvere theories, notably sum, tensor, and distributive tensor, reflecting the ways in which the various computational effects are usually combined, thus giving denotational semantics for the combinations.