Syntax and Semantics for Operations with Scopes

with Maciej Piróg, Tom Schrijvers and Nicolas Wu. LICS 2018.PDF


Motivated by the problem of separating syntax from semantics in programming with algebraic effects and handlers, we propose a categorical model of abstract syntax with so-called scoped operations. As a building block of a term, a scoped operation is not merely a node in a tree, as it can also encompass a whole part of the term (a scope). Some examples from the area of programming are given by the operation catch for handling exceptions, in which the part in the scope is the code that may raise an exception, or the operation once, which selects a single solution from a nondeterministic computation. A distinctive feature of such operations is their behaviour under program composition, that is, syntactic substitution. Our model is based on what Ghani et al. call the monad of explicit substitutions, defined using the initial-algebra semantics in the category of endofunctors. We also introduce a new kind of multi-sorted algebras, called scoped algebras, which serve as interpretations of syntax with scopes. In generality, scoped algebras are given in the style of the presheaf formalisation of syntax with binders of Fiore et al. As the main technical result, we show that our monad indeed arises from free objects in the category of scoped algebras. Importantly, we show that our results are immediately applicable. In particular, we show a Haskell implementation together with practical, real-life examples. It is well-known that monads are monoids in the category of endofunctors, and in fact so are applicative functors. Unfortunately, monoids do not have enough structure to account for computational effects with non-determinism operators. This article recovers a unified view of computational effects with non- determinism by extending monoids to near-semirings with both additive and multiplicative structure. This enables us to generically define free constructions as well as a novel double Cayley representation that optimises both left-nested sums and left-nested products.


  title = {Syntax and Semantics for Operations with Scopes},
  author = {Maciej PirĂ³g and Tom Schrijvers and Nicolas Wu and Mauro Jaskelioff},
  year = {2018},
  booktitle = {Thirty-Third Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)},