Postma, Stef W. The \(\Sigma{}\lambda\)-calculus and derived program forms. (English) Zbl 0741.03005 Quaest. Math. 14, No. 2, 137-159 (1991). Summary: The \(\Sigma\lambda\)-calculus is an extension of an applied \(\lambda\)- calculus designed to abstract parallel and non-deterministic function application evaluation. The abstraction is extended to include Landin’s closure, and constant terms and a special form of the conditional allows the regular logics to be developed along the lines of the Hilbert- Ackermann approach. The regular logics are used to characterize derived conditionals in extensions to Dijkstra’s weakest preconditions. It is shown that various styles of programming — top-down, bottom-up, recursive and iterative — are expressible in the \(\Sigma\lambda\)- calculus. Lists and streams are found to be useful in expressing time- dependent behaviour of computational systems. An approximate Church- Rosser Theorem is given. MSC: 03B40 Combinatory logic and lambda calculus 68Q10 Modes of computation (nondeterministic, parallel, interactive, probabilistic, etc.) 03B70 Logic in computer science 68N01 General topics in the theory of software 03B50 Many-valued logic Keywords:\(\lambda\)-calculus; Landin’s closure; constant terms; Hilbert-Ackermann approach; Dijkstra’s weakest preconditions; time-dependent behaviour of computational systems; approximate Church-Rosser Theorem PDFBibTeX XMLCite \textit{S. W. Postma}, Quaest. Math. 14, No. 2, 137--159 (1991; Zbl 0741.03005) Full Text: DOI References: [1] Beth E W, The Foundations of Mathematics (1968) [2] Burge W H, Techniques (1975) [3] Curry H B, Combinatory Logic 1 (1958) · Zbl 0158.24703 [4] DOI: 10.1145/360933.360975 · Zbl 0308.68017 [5] Henderson P, Functional Programming–Application and Implementation (1980) [6] Hindley J R, Introduction to Combinators and {\(\lambda\)}-Calculus (1986) [7] Kleene S C, Introduction to Metamathematics (1952) [8] Landin P J, Computer J. 6 pp 308– (1964) [9] Loveland D W, A Logical Basis (1978) [10] Manna Z, Mathematical Theory of Computation (1972) [11] McCarthy, J. 1963.A basis for a mathematical theory of computation33–70. Braffort, P; Hirschberg, D: Computer Programming and Formal Systems, North-Holland, Amsterdam [12] Postma S W, On The Definition and Implementation of The Program Language Quadlisp (1984) [13] Postma S W, Uni Natal, Computer Science Report, PMB-TR/86–01, PMB (1986) [14] Postma S W, Basic Definitions of Octolisp in {\(\Sigma\)}{\(\lambda\)}-Calculus [15] Postma S W, Octolisp Syntax & Language [16] Postma S W, Octolisp Training Manual (1989) [17] SLICED–machine environments [18] Postma S W, Momaticae 6 pp 109– (1988) [19] Stoy J E, Denotational Semantics: The Scott-Strachey Approach to Programming Language Theory (1977) [20] Wilder R L, Introduction to the Foundations of Mathematics (1967) [21] Yasuhara A, Recursive Function Theory and Logic (1971) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.