Formal calculation

Abstract: Publisher Summary This chapter describes the use of the Genie system in numerical calculation. The set of codes designed for interpreting formal expressions on the Rice University computer is termed the Genie system, or simply Genie. Genie is concerned in general with the definition of objects belonging to certain computable domains, and the execution of particular operations between or upon these objects. Formal calculation proceeds by naming objects of interest, and assigning values to such objects either by means of equations or, in the case of linguistic objects, by exhibiting specific examples of the objects. One of the objectives of Genie is to extend the basic forms of definition that can readily be understood, and simply and efficiently encoded. The chapter also discusses the static properties of algorithm descriptions with illustrative examples. At the highest organizational level, Genie is concerned with the interaction of two definition sets—one initially in the machine and one written by the coder. The result of the interaction is a new definition set in the machine and some printed output information of interest to the coder.

Abstract: In programming language and software engineering, the main mathematical tool is de facto some form of predicate logic. Yet, as elsewhere in applied mathematics, it is used mostly far below its potential, due to its traditional formulation as just a topic in logic instead of a calculus for everyday practical use.The proposed alternative combines a language of utmost simplicity (four constructs only) that is devoid of the defects of common mathematical conventions, with a set of convenient calculation rules that is sufficiently comprehensive to make it practical for everyday use in most (if not all) domains of interest.Its main elements are a functional predicate calculus and concrete generic functionals. The first supports formal calculation with quantifiers with the same fluency as with derivatives and integrals in classical applied mathematics and engineering. The second achieves the same for calculating with functionals, including smooth transition between pointwise and point-free expression.The extensive collection of examples pertains mainly to software specification, language semantics and its mathematical basis, program calculation etc., but occasionally shows wider applicability throughout applied mathematics and engineering. Often it illustrates how formal reasoning guided by the shape of the expressions is an instrument for discovery and expanding intuition, or highlights design opportunities in declarative and (functional) programming languages.

Abstract: The objects of programming semantics, namely, programs and languages, are inherently formal, but the derivation of semantic theories is all too often informal, deprived of the benefits of formal calculation “guided by the shape of the formulas.” Therefore, the main goal of this article is to provide for the study of semantics an approach with the same convenience and power of discovery that calculus has given for many years to applied mathematics, physics, and engineering. The approach uses functional predicate calculus and concrete generic functionals; in fact, a small part suffices. Application to a semantic theory proceeds by describing program behavior in the simplest possible way, namely by program equations, and discovering the axioms of the theory as theorems by calculation. This is shown in outline for a few theories, and in detail for axiomatic semantics, fulfilling a second goal of this article. Indeed, a chafing problem with classical axiomatic semantics is that some axioms are unintuitive at first, and that justifications via denotational semantics are too elaborate to be satisfactory. Derivation provides more transparency. Calculation of formulas for ante- and postconditions is shown in general, and for the major language constructs in particular. A basic problem reported in the literature, whereby relations are inadequate for handling nondeterminacy and termination, is solved here through appropriately defined program equations. Several variants and an example in mathematical analysis are also presented. One conclusion is that formal calculation with quantifiers is one of the most important elements for unifying continuous and discrete mathematics in general, and traditional engineering with computing science, in particular.

Abstract: In [3] Breuer and Kronholm gave in effect two proofs for an explicit formula for the generating function for partitions where the difference between largest and smallest part is bounded by a given integer t. Their first proof is geometric, involving counting lattice points within a polyhedral region; their second proof constructs an explicit bijection. In this paper we give another proof, a formal calculation involving elementary q-series manipulation, involving no results deeper than the q-binomial theorem. The results of [3] imply a theorem of Andrews, Beck and Robbins [2] on partitions where the difference between largest and smallest part is a fixed integer t. They use formal q-series methods which go beyond ours, for instance Heine’s transformation for basic hypergeometric series.