# An experiment with denotational semantics

## TL;DR

This paper presents a systematic algebraic approach to designing programming languages from their denotational semantics, demonstrated through a layered virtual language with complex type systems and SQL integration.

## Contribution

It introduces a novel algebraic framework for language design from denotations to syntax, including a set-theoretic model and a detailed language example with rich types and SQL features.

## Key findings

- Successful construction of a layered language model with denotational semantics
- Integration of SQL-like types and constraints into the language model
- Demonstration of the approach through a comprehensive language example

## Abstract

The paper is devoted to showing how to systematically design a programming language in 'reverse order', i.e. from denotations to syntax. This construction is developed in an algebraic framework consisting of three many-sorted algebras: of denotations, of an abstract syntax and of a concrete syntax. These algebras are constructed in such a way that there is a unique homomorphism from concrete syntax to denotations, which constitutes the denotational semantics of the language. Besides its algebraic framework, the model is set-theoretic, i.e. the denotational domains are just sets, rather than Scott's reflexive domains. The method is illustrated by a layer-by-layer development of a virtual language Lingua: an applicative layer, an imperative layer (with recursive procedures) and an SQL layer where Lingua is regarded as an API (Application Programming Interface) for an SQL engine. The latter is given a denotational semantics as well. The langue is equipped with a strong typing mechanism which covers basic types (numbers, Booleans, etc.), lists, arrays, record and their arbitrary combinations plus SQL-like types: rows, tables and databases. The model of types includes SQL integrity constraints.

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Source: https://tomesphere.com/paper/1905.01469