This paper gives a static semantics for Haskell 98, a non-strict purely functional programming language. The semantics formally specifies nearly all the details of the Haskell 98 type system, including the resolution of overloading, kind inference (including defaulting) and polymorphic recursion, the only major omission being a proper treatment of ambiguous overloading and its resolution.

. Precise mode information is important for compiler optimisations and in program development tools. Within the framework of abstract compilation, the precision of a mode analysis depends, in part, on the expressiveness of the abstract domain and its associated abstraction function. This paper considers abstract domains for polymorphically typed logic programs and shows how specialised domains may be constructed for each type in the program. These domains capture the degree of instantiation to a high level of precision. By providing a generic definition of abstract unication, the abstraction of a program using these domains is formalised. The domain construction procedure is fully implemented using the Godel language and tested on a number of example programs to demonstrate the viability of the approach. Note: Some proofs have been omitted for space reasons. They can be found in the full version of this paper [17]. 1 Introduction 1.1 Background Typed logic programming languages su...

x 1 Introduction 1 1.1 Dependent Types : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.2 Dependent Type Inference and Reconstruction : : : : : : : : : : : : : : : : 8 2 Primitive Recursive Functionals with Dependent Types 17 2.1 A Dependent Type System for T : : : : : : : : : : : : : : : : : : : : : : : 17 2.1.1 Terms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 17 2.1.2 Types : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 19 2.1.3 Typing Rules : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 24 2.1.4 Strong Normalization of T Terms : : : : : : : : : : : : : : : : : : : 28 2.2 Dependent Typing Examples : : : : : : : : : : : : : : : : : : : : : : : : : : 29 2.3 A Term Model Semantics for T : : : : : : : : : : : : : : : : : : : : : : : : 34 3 Principal Types and Dependent Type Reconstruction 58 3.1 Type Subsumption and Unification : : : : : : : : : : : : : : : : : : : : : : : 58 3.2 Matching : : : : : :...

) Martin Muller Joachim Niehren y German Research Center for Artificial Intelligence (DFKI) Stuhlsatzenhausweg 3, 66123 Saarbrucken, Germany fmmueller,niehreng@dfki.uni-sb.de March 15, 1995 Abstract We introduce constraints necessary for type checking a higher-order concurrent constraint language, and solve them with an incremental algorithm. Our constraint system extends rational unification by constraints x`y saying that "x has at least the structure of y", modelled by a weak instance relation between trees. This notion of instance has been carefully chosen to be weaker than the usual one which renders semi-unification undecidable. Semi-unification has more than once served to link unification problems arising from type inference and those considered in computational linguistics. Just as polymorphic recursion corresponds to subsumption through the semi-unification problem, our type constraint problem corresponds to weak subsumption of feature graphs in linguistics. The decida...

. Higher-order programming languages, such as ML, permit a flexible programming style by using compile-time type inference together with the concept of type polymorphism, which allows to specify the types of generic functions. In ML, however, recursive functions must always be given a unique (monomorphic) type inside their definition. Giving polymorphic types to recursive functions is known as the problem of polymorphic recursion which has been shown equivalent to the problem of semi-unification, known as undecidable. We show that the absence of a decidable specification to give polymorphic types for recursive definitions lies on the non-adequacy of representing type polymorphism by using type variables as primitive elements. We introduce the notion of syntactic type polymorphism to relate polymorphic types with syntactic information. We formulate a decidable calculus which gives polymorphic types to recursive functions in ML. We present an inference algorithm which we prove the term...