# Algebra of programming by Richard Bird, Oege de Moor

By Richard Bird, Oege de Moor

Describes an algebraic method of programming that enables the calculation of courses. Introduces the basics of algebra for programming. offers paradigms and techniques of software building that shape the center of set of rules layout. Discusses capabilities and different types; purposes; relatives and allegories; datatypes; recursive courses, optimization matters, thinning algorithms, dynamic programming and grasping algorithms. acceptable for all programmers.

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A type U consisting of codes for small types is introduced, together with a decoding function T , which maps codes to the types they denote. The deﬁnition is both inductive and recursive; the type U is deﬁned inductively, and the decoding function T is deﬁned recursively on the way the elements of U are generated. The deﬁnition needs to be simultaneous, since the introduction rules for U refer to T . We illustrate this by means of a concrete example: say we want to deﬁne a data type representing a universe containing a name for the natural numbers, closed under Σ-types.

In this paper, we take the first steps in marrying these two research areas and in using rule formats for algebraic properties (specifically, for commutativity) to enhance the The first three authors have been partially supported by the project ‘Meta-theory of Algebraic Process Theories’ (nr. 100014021) of the Icelandic Research Fund. Eugen-Ioan Goriac is also funded by the project ‘Extending and Axiomatizing Structural Operational Semantics: Theory and Tools’ (nr. 1102940061) of the Icelandic Research Fund.

In the remainder of this paper, following [10], we shall tacitly assume that each TSS in the GSOS format contains these operators with the rules given above. The import of this assumption is that, as is well known, within each TSS in the GSOS format it is possible to express each finite synchronization tree over L. Following [12], the TSS containing the operators 0, a. (a ∈ L) and + , with the above-given rules, is denoted by BCCSP. The transition relation associated with a TSS in the GSOS format is the one defined by structural induction over closed terms using the rules.