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-Ecclesiastes 12:12 Programs are invariably subjected to many rorms or transrormation. After an initial version of a program has been designed and developed, it undergoes debugging and certification. In addition, most long-lived pro­ grams have a liCe-cycle that includes modifications to meet amended specifications and extensions for expanded capabilities. Such evolution­ ary aspects of programming are the topic of this monograph. We present rormal methods for manipulating programs and illustrate their applica­ tion with numerous examples. Such methods could be incorporated in semi-automated programming environments, where they would serve to ease the burden on the programmer. We begin by describing a method whereby a given program that achieves one goal can be modified to achieve a different goal or a pro­ gram that computes wrong results can be debugged to achieve the 2 Preface intended results. The abstraction of a set of cognate programs to obtain a program schema, and the instantiation of abstract schemata to solve concrete problems, are approached from the same perspective. In addition, we describe synthesis rules for generating code from specifications and annotation rules for making assertions about code. The synthesis rules may be used when a program is first being developed, or when, in the course of modifying a program, the need arises to rewrite a program segment. Annotation rules may be used for the purpose of determining what an incorrect program really does before attempting to debug it or how a correct program works before attempting to modify it.