The Expressiveness of Simple and Second-Order Type Structures

Abstract

Typed lambda (?`-) calculi provtde convement mathematical settings in which to investigate the effects of type structure on the function definmon mechamsm m programming languages. Lambda expressaons mtm~c programs that do not use while loops or carcular function definitions. Two typed ?`-calculi are investigated, the sunply typed ?`-calculus, whose types are similar to Pascal types, and the second-order typed ?,-calculus, which has a type abstractaon mechamsm simdar to that of modern data abstraction languages such as ALPHARD. Two related questions are considered for each calculus:(1) What functaons are definable m the calculus? and (2) How difficult is the proof that all expressions in the calculus are normahzable 0.e., that all computaUons termmate)* The simply typed calculus only defines elementary functtons. Normahzation for this calculus ~s provable using commonplace forms of reasoning formalazable m Peano arithmetic The second-order calculus defines a huge hierarchy of funcuons going far beyond Ackermann's function These funcuons are so rapidly increasing that Peano arithmetic cannot prove that they are total In fact, normalizataon for the second-order calculus cannot be proved even m second-order Peano artthmetic, nor m Peano anthmettc augmented by all true statements Also d~scussed are the lmphcataons of the present ?`-calculus results for the programming languages PASCAL, ALPHARD, RUSSELL, and MODEL.

Inductively defined types

“…This corollary generalizes one direction of results obtained by Girard [14], and Fortune, Leivant & O'Donnell [12], and Leivant [17,18] which may be summarized as "The number-theoretic functions representable in F n are exactly the functions provably recursive in n th -order arithmetic." The induction principle will look more familiar after we curry at the argument V to eliminate the dependent pair and also apply uniform parameterization over the argument A.…”

“…Closely related is work by Girard [13,14], Fortune, Leivant & O'Donnell [12], and Leivant [17,18] who are concerned with the relationship between higher-order logic and polymorphic A-calculi.…”

Functional Programming and Lambda Calculus

“…The resulting system is a variant (the "Curry version" in the terminology of [8]) of the polymorphic second-order lambda calculus independently due to [37] On the Church numerals, many computable functions can be represented by terms of type N at-»N at. The next result is due to [36]; see also [34].…”

Three approaches to type structure

“…functions may be the key to a novel programming style. They have studied an austere subset of the language we are considering in which the fixpoint operator fix is excluded, and have shown that this restricted language has extraordinary properties [32,33]. On the one hand, all expressions have normal forms, i.e.…”