Syntax and Semantics for Operations with Scopes

Maciej Piróg, Tom Schrijvers, Nicolas Wu, Mauro Jaskelioff
2018 Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '18  
General rights This document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: Abstract 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
more » ... 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. 1 Introduction The Big Picture: Formalised Abstract Syntax This paper is about formal models of abstract syntax and their connection with semantics, mainly in the context of programming languages. The simplest model of abstract syntax is given by the set of terms over a signature, parametrised by the set of variables. Terms can be generalised to free monads generated by an endofunctor, which allows one to capture more intricate kinds of signatures, and move to different categories, possibly more amenable for modelling and reasoning about programming languages. However, to model syntax of programming calculi, one needs more advanced
doi:10.1145/3209108.3209166 dblp:conf/lics/PirogSWJ18 fatcat:ugfepo5cijdk5fent4av4srnra