Self-Representation in Girard's System U

In 1991, Pfenning and Lee studied whether System F could support a typed self-interpreter. They concluded that typed selfrepresentation for System F “seems to be impossible”, but were able to represent System F in Fω. Further, they found that the representation of Fω requires kind polymorphism, which is outside Fω. In 2009, Rendel, Ostermann and Hofer conjectured that the representation of kind-polymorphic terms would require another, higher form of polymorphism. Is this a case of infinite regress? We show that it is not and present a typed self-representation for Girard’s System U, the first for a λ-calculus with decidable type checking. System U extends System Fω with kind polymorphic terms and types. We show that kind polymorphic types (i.e. types that depend on kinds) are sufficient to “tie the knot” – they enable representations of kind polymorphic terms without introducing another form of polymorphism. Our self-representation supports operations that iterate ...

• Type: Journal
• Year: 2015
• Where: POPL
• Authors: Matt Brown, Jens Palsberg
• Keywords: POPL 2015, Programming Languages