Abstract
An analysis of recursive procedures in ALGOL 68 with finite modes shows, that a denotational semantics of this language can be described on the level of program schemes using a typed λ-calculus with fixed-point operators. In the first part of this paper, we derive classical schematological theorems for the resulting class of level-n schemes. In part two, we investigate the language families obtained by call-by-value and call-by-name interpretation of level- n schemes over the algebra of formal languages. It is proved, that differentiating according to the functional level of recursion leads to two infinite hierarchies of recursive languages, the IO- and OI-hierarchies, which can be characterized as canonical extensions of the regular, context-free, and IO- and OI-macro languages, respectively. Sufficient conditions are derived to establish strictness of IO-like hierarchies. Finally we derive, that recursion on higher types induces an infinite hierarchy of control structures by proving that level- n schemes are strictly less powerful than level- n+1 schemes.
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