The Damas-Milner Calculus is the typed -calculus underlying the type system for ML and several other strongly typed polymorphic functional languages such as Miranda 1 and Haskell. Mycroft has extended its problematic monomorphic typing rule for recursive definitions with a polymorphic typing rule. He proved the resulting type system, which we call the Milner-Mycroft Calculus, sound with respect to Milner's semantics, and showed that it preserves the principal typing property of the Damas-Milner Calculus. The extension is of practical significance in typed logic programming languages and, more generally, in any language with (mutually) recursive definitions. In this paper we show that the type inference problem for the Milner-Mycroft Calculus is log-space equivalent to semi-unification, the problem of solving subsumption inequations between first-order terms. This result has been proved independently by Kfoury, Tiuryn, and Urzyczyn. In connection with the recently establish...

In the last ten years declaration-free programming languages with a polymorphic typing discipline (ML, B) have been developed to approximate the flexibility and conciseness of dynamically typed languages (LISP, SETL) while retaining the safety and execution efficiency of conventional statically typed languages (Algol68, Pascal). These polymorphic languages can be type checked at compile time, yet allow functions whose arguments range over a variety of types. We investigate several polymorphic type systems, the most powerful of which, termed Milner-Mycroft Calculus, extends the so-called let-polymorphism found in, e.g., ML with a polymorphic typing rule for recursive definitions. We show that semi-unification, the problem of solving inequalities over firstorder terms, characterizes type checking in the Milner-Mycroft Calculus to polynomial time, even in the restricted case where nested definitions are disallowed. This permits us to extend some infeasibility results for related combinato...

Various restrictions on the terms allowed for substitution give rise to different cases of semi-unification. Semi-unification on finite and regular terms has already been considered in the literature. We introduce a general case of semi-unification where substitutions are allowed on non-regular terms, and we prove the equivalence of this general case to a well-known undecidable data base dependency problem , thus establishing the undecidability of general semi-unification. We present a unified way of looking at the various problems of semi-unification. We give some properties that are common to all the cases of semi-unification. We also the principality property and the solution set for those problems. We prove that semi-unification on general terms has the principality property. Finally, we present a recursive inseparability result between semi-unification on regular terms and semi-unification on general terms. Partly supported by NSF grant CCR-9113196. Address: Department of Compu...

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 ...