VEC is a higher-order functional language of nested arrays, which includes a general folding operation. Static computation of the shape, of its programs is used to support a compositional cost calculus based:on a cost monad. This, in turn, is based on a cost algebra, whose operations may be customized to handle different cost regimes, especially for parallel programming. We present examples based on sequential costing and~on the PRAM model of parallel computation. The latter has been implemented in Haskell, and applied to some linear algebra examples. I n t r o d u c t i o nSecond-order combinators such as map, fold and zip provide programmers with a concise, abstract language for writing skeletons for implicitly parallel programs, as in [Ski94], but there is a hitch. These combinators are defined for list programs (see [BW88]), but efficient implementations (which is the point of parallelism, after all) are based on arrays. This disparity becomes acute when working with nested arrays, which are rarely supported in parallel practice. NESL [BCH+94] is a notable exception, but still does not support folding over them. Our approach is to redevelop the theory of arrays to support the combinators while retaining constant-time access. We illustrate the efficacy of this approach by implementing static estimates of shapes and parallel work, as compared to, say, the dynamically obtained estimates for NESL [BG96].Efficient array access is based on direct access to the storage of array entries, typically in contiguous blocks of memory. List storage, however, is by allocating cons cells during evaluation, which introduces significant access costs. Our approach to arrays uses the syntax of lists, so that combinators can be introduced in the functional style, but constrained so that a compiler can still determine the length, or more generally the shape, of any array expression. This process of shape analysis can be thought of as compile-time/run-time separation in a two-level operational semantics INN92]. Success requires that the program be shapely, that is, the shape of the result is determined by those of its inputs, as is typical in, say, linear algebra. Shapeliness, and successful shape analysis, require a new approach, which we have illustrated in a small functional language, VEC.1. This operation is shapely: the result is an integer, a datum whose shape is trivial and therefore unaffected by the condition.

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz Časopis pro pěstování matematiky, rol. 91 (1966), Praha REFERÁTY SUMMER SESSION ON ORDERED SETS AT CIKHAJ From August 23 to 31,1965, the mathematical departments of Brno and Bratislava universities in cooperation with the Czechoslovak Academy of Sciences, organized at Cikhaj a summer session on ordered sets and abstract algebra. 26 mathematicians took part in this summer session, and 11 lectures were given. The contents of all these lectures are presented here; on behalf of J. Jakubik, who was absent the lecture was read by K. Molnarova. DIE DEDEKINDSCHEN SCHNITTE IM DIREKTEN PRODUKT VON HALBGEORDNETEN MENGEN J. JAKUBIK, Kosice Es sei G 4= 0 eine halbgeordnete Menge. Für A c G bezeichnen wir mit L(A) bzw. U(A) die Menge aller unteren Schranken bzw. aller oberen Schranken von A. Ferner sei D(G) bzw. E(G) das System aller Mengen L(U(A)) 9 wobei A eine beliebige Teil menge von G bzw. eine beliebige nichtleere nach oben begrenzte Teilmenge von G ist. Jedes der Systeme D(G) 9 E(G) ist durch die mengentheoretische Inklusion teilweise geordnet. Es ist bekannt, dass D(G) ein vollständiger Verband ist; E(G) ist im all gemeinen kein Verband. Wir bezeichnen mit I7G a das direkte Produkt der halbgeord neten Mengen G a. Es gilt der Satz: (1) Aus dem Isomorphismus G ~ ÜG a folgt die Existenz eines Isomorphismus E(G) ~ II E(G a). Eine analoge Behauptung für D anstatt E gilt nicht. Ferner sei G eine gerichtete vollständig abgeschlossene Gruppe. Es ist bekannt, dass E(G) im solchen Fall eine vollständige verbandsgeordnete Gruppe ist. Es gilt ein analoger Satz zu (1): Es sei G eine gerichtete vollständig abgeschlossene Gruppe* die mit dem direkten Produkt der halbgeordneten Gruppen G a isomorph ist. Dann ist E(G) mit dem direkten Produkt der verbandsgeordneten Gruppen E(G a) iso morph.