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The benefits of strong typing to disciplined programming, to cou:pile-time error detection and to program verification are well known. S!mng typing is especially natural for functional (applicative) mcnc... This gives rise to a hi,crarcby on polymorphic types, a notion that is equally natural for the type abstraction and type quantification disciplines. WC show that our algolithm V (of $3) cm be naturally cxtcndcd to the restriction of Lhc conjunctive discipline to types of rank 2 in the hierarchy, and we outline an argument showing that already for a low rank, scen)i1181Y ~, tYPc ill fcrcncc js not cffcctivcly decidable.

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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.

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