Modular Tree Automata

Abstract

Abstract. Tree automata are traditionally used to study properties of tree languages and tree transformations. In this paper, we consider tree automata as the basis for modular and extensible recursion schemes. We show, using well-known techniques, how to derive from standard tree automata highly modular recursion schemes. Functions that are defined in terms of these recursion schemes can be combined, reused and transformed in many ways. This flexibility facilitates the specification of complex transformations in a concise manner, which is illustrated with a number of examples.

Composing and decomposing data types

“…put :: (e :≺: p) ⇒ p → e → p get :: (e :≺: p) ⇒ p → e These functions satisfy the expected equations: put p (get p) = p get (put p e) = e put (put p e) e ′ = put p e ′ This setup is especially useful for implementing automata in a modular fashion (Bahr 2012) as it allows us to easily combine state spaces of different automata using binary products.…”

Generalising Tree Traversals to DAGs

“…Using closed type families [13], the type class ∈ can be defined such that it works on arbitrarily nested product types, but disallows ambiguous instances such as Int ∈ (Int, (Bool , Int)) for which multiple projections exist. But there are also simpler implementations of ∈ that only use type classes [5]. We can thus represent the semantic function for a synthesised attribute of type s as follows:…”

“…tree automata that characterise tree transformations, and our representation of them in Haskell is based on Hasuo et al [21]. Our representation of AGs in Haskell is directly taken from Bahr's modular tree automata [5], which are in turn derived from representations of tree automata based on the work of Hasuo et al [21].…”

“…This paper presents a method for running tree traversals on directed acyclic graphs (DAGs), taking full account of the sharing structure. The traversals are expressed as attribute grammars (AGs) using Bahr's representation of tree automata in Haskell [5]. The underlying DAG structure is completely transparent to the AGs, which means that the same AG can be run on both trees and DAGs.…”