On strong normalization and type inference in the intersection type discipline

Abstract

We introduce a new unification procedure for the type inference problem in the intersection type discipline. It is well known that type inference in this case should succeed exactly for the strongly normalizing expressions. We give a proof for the strong normalization result in the intersection type discipline, which we obtain by putting together some well-known results and proof techniques. Our proof uses a variant of Klop's extended λ-calculus, for which it is shown that strong normalization is equivalent to weak normalization. This is proved here by means of a finiteness of developments theorem, obtained following de Vrijer's combinatory technique. The main property of this extended calculus is uniformity, i.e. weak and strong normalizabilities coincide. The strong normalizability result is therefore a consequence of the fact, first established by Coppo and Dezani (for the λ-calculus) that typability implies weak normalizability. We then show that the unification process which is the basis for type inference exactly corresponds to reduction in the extended λ-calculus. Finally we show that our notion of unification allows us to compute a principal typing for any typable λ-expression.

Model Checking Higher-Order Programs

“…Second, our type system is decidable, while the intersection-type systems studied in the literature are usually undecidable (as the typability coincides with strong normalization). Third (and most importantly), the intersection-type systems studied in Coppo et al [1980], Ronchi Della Rocca and Venneri [1984], Kfoury and Wells [2004], and Boudol [2008] guarantee that typable terms (possibly with certain additional conditions) have the strong normalization property, while our type system does not. That is why our algorithm is hybrid: type information is extracted after a finite number of reduction steps, and then another algorithm is used for deciding whether the recursion scheme is typable using extracted type information.…”

“…Since our model-checking algorithm is a type inference algorithm for the intersection-type system presented in Section 4, there may be some connection between our algorithm and type inference algorithms for intersection types [Boudol 2008;Coppo et al 1980;Kfoury and Wells 2004;Ronchi Della Rocca and Venneri 1984]. In particular, earlier algorithms for intersection type inference [Coppo et al 1980;Ronchi Della Rocca and Venneri 1984] first find a normal form, and then obtain a principal typing for the normal form; this is somewhat similar to Steps 1 and 2 of our hybrid algorithm, which first reduces a given recursion scheme, and then extracts type information.…”

Model-checking higher-order functions

“…In general, any instance of the flow analysis problem ("Given a program and its subterms v and e, does the value of v flow to the value of e?") can be encoded into a resource usage verification problem (by replacing each value with a pair consisting of the value and a resource to keep track of its use 8 ), and then to a model checking problem of a recursion scheme.…”

“…Since our model checking algorithm is a type inference algorithm for the intersection type system presented in Section 2.3, there may be some connection between our algorithm and type inference algorithms for intersection types [8,9,14,27]. In particular, earlier algorithms for intersection type inference [9,27] first finds a normal form, and then obtains a principal typing for the normal form; this seems somewhat similar to Steps 1 and 2 of our algorithm, which first reduce a given recursion scheme, and then extract type information.…”

“…Secondly, our type system is decidable, while the intersection type systems studied in the literature are usually undecidable (as the typability coincides with strong normalization). Thirdly (and most importantly), the intersection type systems studied in [8,9,14,27] guarantee that typable terms (possibly with certain additionial conditions) have the strong normalization property, while the intersection type system for recursion schemes does not. That is why our algorithm is hybrid: type information is extracted after a finite number of reduction steps, and then another algorithm is used for deciding whether the recursion scheme is typable using extracted type information.…”