Parametric Compositional Data Types

In previous work we have illustrated the benefits that compositional data types (CDTs) offer for implementing languages and in general for dealing with abstract syntax trees (ASTs). Based on Swierstra's data types \'a la carte, CDTs are implemented as a Haskell library that enables the definition of recursive data types and functions on them in a modular and extendable fashion. Although CDTs provide a powerful tool for analysing and manipulating ASTs, they lack a convenient representation of variable binders. In this paper we remedy this deficiency by combining the framework of CDTs with Chlipala's parametric higher-order abstract syntax (PHOAS). We show how a generalisation from functors to difunctors enables us to capture PHOAS while still maintaining the features of the original implementation of CDTs, in particular its modularity. Unlike previous approaches, we avoid so-called exotic terms without resorting to abstract types: this is crucial when we want to perform transformations on CDTs that inspect the recursively computed CDTs, e.g. constant folding.Comment: In Proceedings MSFP 2012, arXiv:1202.240

A Finite Semantics of Simply-Typed Lambda Terms for Infinite Runs of<br> Automata

Model checking properties are often described by means of finite automata. Any particular such automaton divides the set of infinite trees into finitely many classes, according to which state has an infinite run. Building the full type hierarchy upon this interpretation of the base type gives a finite semantics for simply-typed lambda-trees. A calculus based on this semantics is proven sound and complete. In particular, for regular infinite lambda-trees it is decidable whether a given automaton has a run or not. As regular lambda-trees are precisely recursion schemes, this decidability result holds for arbitrary recursion schemes of arbitrary level, without any syntactical restriction.Comment: 23 page

The Sketch of a Polymorphic Symphony

In previous work, we have introduced functional strategies, that is, first-class generic functions that can traverse into terms of any type while mixing uniform and type-specific behaviour. In the present paper, we give a detailed description of one particular Haskell-based model of functional strategies. This model is characterised as follows. Firstly, we employ first-class polymorphism as a form of second-order polymorphism as for the mere types of functional strategies. Secondly, we use an encoding scheme of run-time type case for mixing uniform and type-specific behaviour. Thirdly, we base all traversal on a fundamental combinator for folding over constructor applications. Using this model, we capture common strategic traversal schemes in a highly parameterised style. We study two original forms of parameterisation. Firstly, we design parameters for the specific control-flow, data-flow and traversal characteristics of more concrete traversal schemes. Secondly, we use overloading to postpone commitment to a specific type scheme of traversal. The resulting portfolio of traversal schemes can be regarded as a challenging benchmark for setups for typed generic programming. The way we develop the model and the suite of traversal schemes, it becomes clear that parameterised + typed strategic programming is best viewed as a potent combination of certain bits of parametric, intensional, polytypic, and ad-hoc polymorphism

Cost Automata, Safe Schemes, and Downward Closures

Higher-order recursion schemes are an expressive formalism used to define languages of possibly infinite ranked trees. They extend regular and context-free grammars, and are equivalent to simply typed ?Y-calculus and collapsible pushdown automata. In this work we prove, under a syntactical constraint called safety, decidability of the model-checking problem for recursion schemes against properties defined by alternating B-automata, an extension of alternating parity automata for infinite trees with a boundedness acceptance condition. We then exploit this result to show how to compute downward closures of languages of finite trees recognized by safe recursion schemes

Unified Analysis of Collapsible and Ordered Pushdown Automata via Term Rewriting

We model collapsible and ordered pushdown systems with term rewriting, by encoding higher-order stacks and multiple stacks into trees. We show a uniform inverse preservation of recognizability result for the resulting class of term rewriting systems, which is obtained by extending the classic saturation-based approach. This result subsumes and unifies similar analyses on collapsible and ordered pushdown systems. Despite the rich literature on inverse preservation of recognizability for term rewrite systems, our result does not seem to follow from any previous study.Comment: in Proc. of FRE

The Expressive Power of One Variable Used Once: The Chomsky Hierarchy and First-Order Monadic Constructor Rewriting

We study the implicit computational complexity of constructor term rewriting systems where every function and constructor symbol is unary or nullary. Surprisingly, adding simple and natural constraints to rule formation yields classes of systems that accept exactly the four classes of languages in the Chomsky hierarchy