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EelcoVisser. Multi-level specifications. In ArieVanDeursen, JanHeering, and PaulKlint, editors,
**Abstract**

This chapter introduces a modular, applicative, multi-level equational specification formalism that supports algebraic specification with user-definable type constructors, polymorphic functions and higher-order functions. Specifications consist of one or more levels numbered 0 to n. Level 0 defines the object level terms. Level 1 defines the types used in the signature of level 0. In general, the terms used as types in level n are defined in level n+1. This setup makes the algebra of types and the algebra of types of types, etc., user-definable. The applicative term structure makes functions first-class citizens and facilitates higher-order functions. The use of variables in terms used as types provides polymorphism (including higher-order polymorphism, i.e., abstraction over type constructors). Functions and variables can be overloaded. Specifications can be divided into modules. Modules can be imported at several levels by means of a specification lifting operation. Equations define the semantics of terms over a signature. The formalism also allows equations over types, by means of which many type systems can be described. The typechecker presented in this chapter does not take into account type equations.

The specification, in ASFandSDF, of the syntax, type system and semantics of the formalism is presented in three stages: (1) untyped equational specifications (2) applicative one-level specifications (3) modular multi-level specifications. The definition of a typechecker for stages (2) and (3) is divided into four parts: (a) well-formedness judgements verifying type correctness of fully annotated terms and specifications, (b) non well-formedness rules giving descriptive error messages for the cases not covered under (a), (c) a type assignment function annotating the terms in a plain specification with types, and (d) a typechecking function which checks well-formedness after applying type assignment. These functions are defined uniformly for all levels of a specification.

Aside of defining a new specification formalism, this chapter illustrates the use of ASFandSDF for the design and prototyping of sophisticated specification formalisms.

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