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Type Inference with Polymorphic Recursion
 Transactions on Programming Languages and Systems
, 1991
"... The DamasMilner Calculus is the typed Acalculus underlying the type system for ML and several other strongly typed polymorphic functional languages such as Mirandal and Haskell. Mycroft has extended its problematic monomorphic typing rule for recursive definitions with a polymorphic typing rule. H ..."
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Cited by 135 (0 self)
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The DamasMilner Calculus is the typed Acalculus underlying the type system for ML and several other strongly typed polymorphic functional languages such as Mirandal 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 MilnerMycroft Calculus, sound with respect to Milner’s semantics, and showed that it preserves the principal typing property of the DamasMilner 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 MilnerMycroft Calculus is logspace equivalent to semiunification, the problem of solving subsumption inequations between firstorder terms. This result has been proved independently by Kfoury et al. In connection with the recently established undecidability of semiunification this implies that typability in the MilnerMycroft Calculus is undecidable. We present some reasons why type inference with polymorphic recursion appears to be practical despite its undecidability. This also sheds some light on the observed practicality of ML
Efficient Type Inference for HigherOrder BindingTime Analysis
 In Functional Programming and Computer Architecture
, 1991
"... Bindingtime analysis determines when variables and expressions in a program can be bound to their values, distinguishing between early (compiletime) and late (runtime) binding. Bindingtime information can be used by compilers to produce more efficient target programs by partially evaluating prog ..."
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Cited by 91 (4 self)
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Bindingtime analysis determines when variables and expressions in a program can be bound to their values, distinguishing between early (compiletime) and late (runtime) binding. Bindingtime information can be used by compilers to produce more efficient target programs by partially evaluating programs at compiletime. Bindingtime analysis has been formulated in abstract interpretation contexts and more recently in a typetheoretic setting. In a typetheoretic setting bindingtime analysis is a type inference problem: the problem of inferring a completion of a λterm e with bindingtime annotations such that e satisfies the typing rules. Nielson and Nielson and Schmidt have shown that every simply typed λterm has a unique completion ê that minimizes late binding in TML, a monomorphic type system with explicit bindingtime annotations, and they present exponential time algorithms for computing such minimal completions. 1 Gomard proves the same results for a variant of his twolevel λcalculus without a socalled “lifting ” rule. He presents another algorithm for inferring completions in this somewhat restricted type system and states that it can be implemented in time O(n 3). He conjectures that the completions computed are minimal.
Coinductive Axiomatization of Recursive Type Equality and Subtyping
, 1998
"... e present new sound and complete axiomatizations of type equality and subtype inequality for a firstorder type language with regular recursive types. The rules are motivated by coinductive characterizations of type containment and type equality via simulation and bisimulation, respectively. The mai ..."
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Cited by 64 (2 self)
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e present new sound and complete axiomatizations of type equality and subtype inequality for a firstorder type language with regular recursive types. The rules are motivated by coinductive characterizations of type containment and type equality via simulation and bisimulation, respectively. The main novelty of the axiomatization is the fixpoint rule (or coinduction principle), which has the form A; P ` P A ` P (Fix) where P is either a type equality = 0 or type containment 0 and the proof of the premise must be contractive in a formal sense. In particular, a proof of A; P ` P using the assumption axiom is not contractive. The fixpoint rule embodies a finitary coinduction principle and thus allows us to capture a coinductive relation in the fundamentally inductive framework of inference systems. The new axiomatizations are more concise than previous axiomatizations, particularly so for type containment since no separate axiomatization of type equality is required, as in A...
Type inference and semiunification
 In Proceedings of the ACM Conference on LISP and Functional Programming (LFP ) (Snowbird
, 1988
"... In the last ten years declarationfree 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 type ..."
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Cited by 25 (6 self)
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In the last ten years declarationfree 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 MilnerMycroft Calculus, extends the socalled letpolymorphism found in, e.g., ML with a polymorphic typing rule for recursive definitions. We show that semiunification, the problem of solving inequalities over firstorder terms, characterizes type checking in the MilnerMycroft Calculus to polynomial time, even in the restricted case where nested definitions are disallowed. This permits us to extend some infeasibility results for related combinatorial problems to type inference and to correct several claims and statements in the literature. We prove the existence of unique most general solutions of term inequalities, called most general semiunifiers, and present an algorithm for computing them that terminates for all known inputs due to a novel “extended occurs check”. We conjecture this algorithm to be