Reductions, intersection types, and explicit substitutions

Abstract

We present a new system of intersection types for a composition-free calculus of explicit substitutions with a rule for garbage collection, and show that it characterizes those terms which are strongly normalizing. This system extends previous work on the natural generalization of the classical intersection types system, which characterized head normalization and weak normalization, but was not complete for strong normalization. An important role is played by the notion of available variable in a term, which is a generalization of the classical notion of free variable.

Higher-order semantic labelling for inductive datatype systems

“…Proving termination of the simply-typed λX-calculus is not an obvious task. Actually, the direct proofs using the reducibility method are rather involved and long [15,23]. Moreover, the λX-calculus as an IDTS does not follow the General Schema [1] directly since a precedence between the application and the explicit substitution operators cannot be determined from the rules.…”

Intersection Type System and Lambda Calculus With Director Strings

“…Several intersection types for explicit substitution were studied. Dougherty and Lescanne [18] studied the relationship between intersection types and reduction (left reduction and head reduction) of λx.Lengrand [13] characterized strongly normalizing terms of λx gc with intersection types.Ventura et al [19] presented an intersection type system for λ db and showed the subject reduction property. Ventura et al [20] introduced intersection type systems for , , -calculus and proved the subject reduction property for them.…”

Intersection types for explicit substitutions

“…As discussed in [20], one can see an explicit substitution calculus as an improvement on both the system of combinators and lc, since it is a system whose mechanics are first-order and as simple as those of combinatory logic, yet which retains the same intensional character as lc. Observe that lc can be viewed as a subsystem of explicit substitution systems, defined by the strategy of "eagerly" applying the substitution induced by contracting a β-redex.…”

“…Previous work [19,20] explored some reduction properties of this system using intersection types. The natural generalizations of the classical type systems were able to characterize the sets of normalizing and head-normalizing terms by means of typability.…”

“…The natural generalizations of the classical type systems were able to characterize the sets of normalizing and head-normalizing terms by means of typability. But it was shown in [19] that the naive generalization of the classical system did not characterize the strongly normalizing terms. Typable terms were strongly normalizing but the converse fails.…”