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"... upper bound for reduction sequences in the typed λ–calculus. Helmut Schwichtenberg Dedicated to Kurt Schütte on the occasion of his 80th birthday It is well known that the full reduction tree for any term of the typed λ–calculus is finite. However, it is not obvious how a reasonable estimate for its ..."

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upper bound for reduction sequences in the typed λ–calculus. Helmut Schwichtenberg Dedicated to Kurt Schütte on the occasion of his 80th birthday It is well known that the full reduction tree for any term of the typed λ–calculus is finite. However, it is not obvious how a reasonable estimate for its height might be obtained. Here we note that the head reduction tree has the property that the number of its nodes with conversions bounds the length of any reduction sequence*. The height of that tree, and hence also the number of its nodes, can be estimated using a technique due to Howard [3], which in turn is based on work of Sanchis [4] and Diller [1]. This gives the desired upper bound. The method of Gandy [2] can also be used to obtain a bound for the length of arbitrary reduction sequences; this is carried out in [5]. However, the bound derived here, apart from being more intelligible, is also better. Let r, s, t denote terms of the typed λ–calculus. The level lev(r) of r is defined to

### Nordic Journal of Computing Programming Languages Capturing Complexity Classes

"... Abstract. We investigate an imperative and a functional programming language. The computational power of fragments of these languages induce two hierarchies of com-plexity classes. Our first main theorem says that these hierarchies match, level by level, a complexity-theoretic alternating space-time ..."

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Abstract. We investigate an imperative and a functional programming language. The computational power of fragments of these languages induce two hierarchies of com-plexity classes. Our first main theorem says that these hierarchies match, level by level, a complexity-theoretic alternating space-time hierarchy known from the literature. Our second main theorems says that a slightly different complexity-theoretic hierarchy (the Goerdt-Seidl hierarchy) also can be captured by hierarchies induced by fragments of the programming languages. Well known complexity classes like logspace, linspace, p, pspace, etc., occur in the hierarchies.