Initial Algebra Semantics for Cyclic Sharing Tree Structures

TL;DR

This paper introduces a new algebraic framework for representing cyclic sharing tree structures, enabling structural induction and recursion, and demonstrating its implementation in Haskell and Agda.

Contribution

It proposes a simple term syntax for cyclic sharing structures with initial algebra semantics, extending inductive principles to graph-like data.

Findings

Syntax is directly usable in Haskell and Agda.

Supports structural induction and recursion for cyclic structures.

Framework generalizes traditional tree representations to graphs with sharing and cycles.

Abstract

Terms are a concise representation of tree structures. Since they can be naturally defined by an inductive type, they offer data structures in functional programming and mechanised reasoning with useful principles such as structural induction and structural recursion. However, for graphs or "tree-like" structures - trees involving cycles and sharing - it remains unclear what kind of inductive structures exists and how we can faithfully assign a term representation of them. In this paper we propose a simple term syntax for cyclic sharing structures that admits structural induction and recursion principles. We show that the obtained syntax is directly usable in the functional language Haskell and the proof assistant Agda, as well as ordinary data structures such as lists and trees. To achieve this goal, we use a categorical approach to initial algebra semantics in a presheaf category. That approach follows the line of Fiore, Plotkin and Turi's models of abstract syntax with variable binding.