Initial Algebra Specifications Research Articles

Dual Number Meadows

The class of dual number meadows is introduced. By definition this class is a quasivariety. Dual number meadows contain a non-zero element the square of which is zero. These structures are non-involutive and coregular. Some properties of the equational theory of dual number meadows are discussed and an initial algebra specification is given for the minimal dual number meadow of characteristic zero which contains the dual rational numbers. Several open problems are stated.

On the Power of Higher-Order Algebraic Specification Methods

Soundness and adequacy theorems are presented for the expressive power of higher-order initial algebra specifications with respect to the arithmetical and analytical hierarchies. These results demonstrate that higher-order initial algebra semantics substantially extends the power of both first-order initial and first-order final algebra semantics. It thus provides a unifying framework for all three different approaches to the semantics of algebraic specifications.

Equational specifications, complete term rewriting systems, and computable and semicomputable algebras

We classify the computable and semicomputable algebras in terms of finite equational initial algebra specifications and their properties as term term rewriting systems, such as completeness. Further results on properties of these specifications, such as on their size and orthogonality, are provided which show that our main results are the best possible.

Homomorphism Preserving Algebraic Specifications Require Hidden Sorts

Although every computable data type has an initial algebra specification with hidden functions, it may happen that some of the homomorphic images of the data type are not models of the specification. The latter are reducts of algebras that would be models of the specification if all its functions were visible, whereas the homomorphic images of the data type are independent of the specification and need not be compatible with the hidden functions used in it. A hidden function specification that does not exclude any of the homomorphic images of its initial model from its model class will be called homomorphism preserving . It turns out that, unlike unrestricted initial homomorphism preserving. It turns out that, unlike unrestricted initial algebra specification, homomorphism preserving initial algebra specification of computable data types requires both hidden sorts and hidden functions.

A recursive second order initial algebra specification of primitive recursion

Theoretical results on the scope and limits of first order algebraic specifications can be used to show that certain natural algebras have no recursively enumerable equational specification under first order initial algebra semantics. A well known example is the algebraP? of primitive recursive functions over the natural numbers. In this paper we show thatP? has a recursive equational specification under second order initial algebra semantics. It follows that higher order initial algebra specifications are strictly more powerful than first order initial algebra specifications.

Which data types have ω-complete initial algebra specifications?

An algebraic specification is called ω-complete or inductively complete if all (open as well as closed) equations valid in its initial model are equationally derivable from it, i.e., if the equational theory of the initial model is identical to the equational theory of the specification. As the latter is recursively enumerable, the initial model of an ω-complete algebraic specification is a data type with a recursively enumerable equational theory. We show that if hidden sorts and functions are allowed in the specification, the converse is also true: every data type with a recursively enumerable equational theory has an ω-complete initial algebra specification with hidden sorts and functions. We also show that in the case of finite data types the hidden sorts can be dispensed with.

Inductively defined functions in functional programming languages

This paper proposes a notation for defining functions or procedures in such a way that their termination is guaranteed by the scope rules. It uses an extension of case expressions. Suggested uses include programming languages and logical languages; an application is also given to the problem of proving inequations from initial algebra specifications.

Partial evaluation and ω-completeness of algebraic specifications

Suppose P( x, y) is a program with two arguments, whose first argument has a known value c, but whose second argument is not yet known. Partial evaluation of P( c, y) results (or rather: should result) in a specialized residual program P c ( y) in which ‘as much as possible’ has been computed on the basis of c. In the literature on partial evaluation this is often more or less loosely expressed by saying that partial evaluation amounts to ‘making maximal use of incomplete information’. In this paper a precise meaning is given to this notion in the context of equational logic, initial algebra specification, and term rewriting systems. If maximal propagation of incomplete information is to be achieved within this context, as a first step it is necessary to add equations to the algebraic specification in question until it is ω-complete (if ever). The basic properties of ω-complete specifications are discussed and some examples of ω-complete specifications as well as of specifications that do not have a finite ω-complete enrichment are given.

Multi-level specifications based on traces

A methodology for the formal specification of data base applications is presented. Central to the methodology is the idea of proceeding through successive levels of "traces", which enables an initial algebra specification to be obtained from one based on preconditions and effects. Besides being executable these trace levels provide a way to go from a natural language specification to an algebraic one.

A simple transfer lemma for algebraic specifications

(If we say that V and ckp are equivalent w.r.t. expressiveness, in future we shall always assume #(D v) = #$(D w), because it has no meaning otherwise.) ln this paper we shall prove the following theorem: Theorem. Let c)3 and W be two algebraic structures with equal expressiveness. Then it holds that: 99 has an initial algebra specification (in the sense of the ADJ Group) iff W has. When proving this, which we shall do in section 2, we shall make explicit how I specification must be ‘translated’. In section 3 we shall illustrate this theorem