Abstract

The Milner-Damas typing algorithm W is one of the classic algorithms in computer science. In this paper we describe a formalized soundness and completeness proof for this algorithm. Our formalization is based on names for both term and type variables, and is carried out in Isabelle/HOL using the Nominal Datatype Package. It turns out that in our formalization we have to deal with a number of issues that are often overlooked in informal presentations of W.

Introduction

Milner's polymorphic type system for ML is probably the most influential programming language type system. The second author learned about it from a paper by Clément et al. He was immediately taken by their view that type inference can be viewed as Prolog execution, in particular because the Isabelle system, which he had started to work on, was based on a similar paradigm as the Typol language developed by Kahn and his coworkers. Milner himself had provided the explicit type inference algorithm W and proved its soundness. Completeness was later shown by Damas and Milner. Neither soundness nor completeness of W are trivial because of the presence of the Let-construct (which is not expanded during type inference).

Term rewriting is a branch of theoretical computer science which combines elements of logic, universal algebra, automated theorem proving and functional programming. Its foundation is equational logic. What distinguishes term rewriting from equational logic is that equations are used as directed replacement rules, i.e. the left-hand side can be replaced by the right-hand side, but not vice versa. This constitutes a Turing-complete computational model which is very close to functional programming. It has applications in algebra (e.g. Boolean algebra, group theory and ring theory), recursion theory (what is and is not computable with certain sets of rewrite rules), software engineering (reasoning about equationally defined data types such as numbers, lists, sets etc.), and programming languages (especially functional and logic programming). In general, term rewriting applies in any context where efficient methods for reasoning with equations are required.

To date, most of the term rewriting literature has been published in specialist conference proceedings (especially Rewriting Techniques and Applications and Automated Deduction in Springer's LNCS series) and journals (e.g. Journal of Symbolic Computation and Journal of Automated Reasoning). In addition, several overview articles provide introductions into the field, and references to the relevant literature [141, 74, 204]. This is the first English book devoted to the theory and applications of term rewriting. It is ambitious in that it tries to serve two masters:

• The researcher, who needs a unified theory that covers, in detail and in a single volume, material that has previously only been collected in overview articles, and whose technical details are spread over the literature.

• The teacher or student, who needs a readable textbook in an area where there is hardly any literature for the non-specialist.

We have seen in the previous chapter that in some cases completion generates a convergent term rewriting system for a given equational theory, and that such a system can then be used to decide the word problem for this equational theory. A very similar approach has independently been developed in the area of computer algebra, where Gröbner bases are used to decide the ideal congruence and the ideal membership problem in polynomial rings. The close connection to rewriting is given by the fact that Gröbner bases define convergent reduction relations on polynomials, and that the ideal congruence problem can be seen as a word problem. In addition, Buchberger's algorithm, which is very similar to the basic completion procedure presented above, can be used to compute Gröbner bases. In contrast to the situation for term rewriting, however, termination of the reduction relation can always be guaranteed, and Buchberger's algorithm always terminates with success. The purpose of this chapter is, on the one hand, to provide another example for the usefulness of the rewriting and completion approach introduced above. On the other hand, the basic definitions and results from the area of Gröbner bases are presented using the notations and results for abstract reduction systems introduced in Chapter 2.

The ideal membership problem

Let us first introduce the basic algorithmic problems that can be solved with the help of Gröbner bases.

This chapter studies the problem of determining whether a TRS is confluent. After a brief look at the (undecidable) decision problem, the rest of the chapter divides neatly into two parts:

The first part deals with terminating systems, for which confluence turns out to be decidable. This is a key result in our search for decidable equational theories: if E constitutes a terminating TRS, we can decide if it is also confluent, in which case we know by Theorem 4.1.1 that ≈E is decidable.

The second part deals with those systems not covered by the first part, namely (potentially) nonterminating ones. The emphasis here is not on deciding ≈E by rewriting, which requires termination, but on the computational content of a TRS. Viewing a TRS as a program, confluence simply means that the program is deterministic. We show that for the class of so-called orthogonal systems, where no two rules interfere with each other, confluence holds irrespective of termination. This result has immediate consequences for the theory and design of functional programming languages.

The decision problem

Just as for termination and most other interesting properties (of term rewriting systems or otherwise), confluence is in general undecidable:

Theorem 6.1.1The following problem is undecidable:

Instance: A finite TRS R.

Question: Is R confluent?

Proof Given a set of identities E such that Var(l) = Var(r) and neither l nor r is a variable for all l ≈ r ∈ E, we can reduce the ground word problem for E to the confluence problem of a related TRS as follows.