This is a collection of some more or less chaotic remarks on the ML type system, definitely not sufficient to fill a research paper of reasonable quality, but perhaps interesting enough to be written down as a note. At the beginning the idea was to investigate the complexity of type reconstruction and typability in bounded order fragments of ML. Unexpectedly the problem turned out to be hard, and finally I obtained only partial results. I do not feel like spending more time on this topic, so the text is not polished, the proofs --- if included at all --- are only sketched and of rather poor mathematical quality. I believe however, that some remarks, especially those of "philosophical" nature, shed some light on the ML type system and may be of some value to the reader interested especially in the interaction between theory and practice of ML type reconstruction. 1 Introduction The ML type system was developed by Robin Milner in the late seventies [26, 3], but was influenced by much ol...

this documentation. The fault for remaining errors remains with the author. The implementation described here was created with the support of DFG project `Semiunifikation' Le 788/1-2.

. We describe an extension of the type inference of Standard ML that covers polymorphic recursion. For any term t of SML, a type scheme ø and a system L of inequations between (simple) types is computed, such that the types of t are the instances of ø by substitutions S that satisfy L. The inequation constraints L are computed bottom-up in a modification of Milner's algorithm W . The correctness proof is complicated by the fact that unknowns for polytypes are needed -- in contrast to type inference for SML. 1 Introduction Functional programming languages like ML[19], Miranda[23], or Haskell[?], have made statically typed polymorphic languages popular. Their success depends to a large extend on the following properties of the underlying type system of Damas/Milner[2]: - typability of an untyped term is decidable, - for typable terms, a schema representing the set of its types can be inferred automatically, - the declaration of polymorphic values by the user is supported, - well-typed...

Simultaneous rigid E-unification has been introduced in the area of theorem proving with equality. It is used in extension procedures, like the tableau method or the connection method. Many articles in this area tacitly assume the existence of an algorithm for simultaneous rigid E-unification. There were several faulty proofs of the decidability of this problem. In this article we prove several results about the simultaneous rigid E-unification. Two results are reductions of known problems to simultaneous rigid E-unification. Both these problems are very hard. The word equation solving (unification under associativity) is reduced to the monadic case of simultaneous rigid E-unification. The variable-bounded semi-unification problem is reduced to the general simultaneous rigid E-unification. The word equation problem used in the first reduction is known to be decidable, but the decidability result is extremely non-trivial. As for the variablebounded semi-unification, its decidability is ...