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Given prime number p, we let exp.n denote the p-exponent of n. It is defined by the Galois connection: for all natural numbers m and n, exp.n = pm\n . (Read 'exp.n is the greatest m such that pm divides n'.) With these preliminaries we can now pose the following questions. (a) Show that exp.p = 1. (b) Derive an operator e such that exp.gcd(m,n) = exp. we exp.n . (c) Derive an operator such that exp.(raxn) = exp. m exp.n . (You will need to use existential quantification see Section 11. 3.2 to complete this final part of the exercise. Specifically, for prime number p,
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Summary
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The maximum and minimum operators can be defined by an equivalence relation connecting the operators to conjunction and disjunction in the prepositional calculus. This facilitates proving that maximum and niinimum share many properties in common with conjunction and disjunction (such as associativity, idempotence and absorption). In particular, the tedious case analyses, unavoidable when maximum and minimum are defined by a conditional statement, are avoided entirely.
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Bibliographic Remarks
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Much of this chapter is drawn from Feijen and Bijlsma (1990).
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The Assignment
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Statement
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This chapter introduces the formal specification and construction of programs. A notation for formally specifying programs is introduced and applied to assignment statements. Simple illustrations of how assignments are constructed to meet their specifications are given.
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9.1 Hoare Triples
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When writing computer programs, it is very good practice to comment them thoroughly in order to explain what is going on. It helps the programmer to avoid errors by enforcing greater clarity, and it helps others who need to modify the program at a later date (including the one who wrote the program in the first place!). It is a good discipline, for example, to comment every variable declaration with a statement about the variable's function in the program. This has the additional benefit of disciplining the programmer to use distinct variables for distinct functions, rather than overloading a variable with several different functions. Comments can also be almost useless. The comment Increment i by 1 immediately preceding the C/Java statement is completely useless to the experienced programmer who can be expected to know that 'i ++' means 'increment i by one' in C/Java idiom.
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9: The Assignment Statement
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Useless comments are ones that simply repeat in natural language (with all its complexities, nuances and ambiguities) what is stated simply and precisely in the program statements. They are operational in the sense that they repeat in words the instructions given to the computer in symbols. Good comments, on the other hand, should have added value. They should supplement the program text with explanations of the program's function and why the code that is used achieves that function. In this text, comments will be indicated by enclosing them in curly brackets '!' and '}'. The comments we write state formal properties of the program variables at a particular point in the execution of the program. For example, the text of a program may look like where the dots represent some arbitrary program statements. The intended meaning is that, when execution of the program has reached the point in the program text where the comment appears, the value of the variable i is guaranteed to be zero1. Such comments are called assertions, conditions or properties. They are boolean-valued functions of the values of the program variables; by adding them to the program, the claim is being made that they are true at that point in the program's execution. Typical examples of such assertions (albeit written in English rather than as mathematical expressions) can be seen by referring back to the search program shown in Figure 4.1. When a program statement is bracketed by two comments, as for example in we reason about the correctness of the program statement (here 'i := i-1') on the basis that the first comment (here '0 < V) acts as an assumption. That is, we understand the comments as claiming that if 0 < i before the statement i := i-1 is executed, then 0 ^ i after the assignment has been executed. (This is, of course, a valid claim in this particular instance provided that i has type integer.) An expression of the form where P and Q are properties of the program variables and 5 is a program statement (some portion of the program text), is called a Hoare triple. The property P is called the precondition, and the property Q is called the postcondition of the statement 5. We read such a triple as the claim that, if the program variables satisfy property P before execution of statement 5, execution of S is guaranteed to terminate and, afterwards, the program variables will satisfy property Q. A Hoare triple thus denotes a boolean value; if the value is true, we say the triple is valid, and, if it is false, we say the triple is invalid.
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1 Important warning to C/Java programmers: the equality symbol '=' really means equality here and not assignment.
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