10: Sequential Composition and Conditional Statements in .NET

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10: Sequential Composition and Conditional Statements
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and then extracting the first k entries. The amount of fuel that a car consumes on a journey of k kilometres is computed by determining the number of litres consumed per kilometre and then multiplying this amount by fc. Counting the proportion of times that a particular letter occurs in the words in a text say the proportion of words in this text that contain the letter 'e' is computed by counting the total number of words in the text and, of these, the number that contain the given letter, and then finally dividing the latter by the former. This problem-solving strategy is captured in programming terms by the sequential composition of statements, here denoted by an infix semicolon. Hoare triples are used to provide the mathematical basis for the strategy. Suppose a programming problem is specified by giving a precondition P and postcondition Q. We are required to construct a program statement S to satisfy
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{P}5{Q}
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We can decompose this problem by inventing a suitable intermediate assertion R and then constructing program statements 51 and S2 such that
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{P}S1 { R }
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and { R } S2 { Q } .
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Examples of intermediate assertions are 'the deck of cards is sorted' (in the case of finding the best k entries in a deck of cards), 'I pk is the number of litres consumed per kilometre' (in the case of computing fuel consumption over a given number of kilometres), or 'w is the number of words in the text and I is the number of such words containing the given letter' (in the case of counting the proportion of times that a particular letter occurs in a sequence of words). The sequential composition SI;52 of statements 51 and 52 is executed by first executing statement 51 and then statement 52. If 51 has been constructed to satisfy {P}51 { R } , (10.1)
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it means that, if execution of 51 is begun in a state satisfying P, termination is guaranteed in a state satisfying R. Also, if 52 has been constructed to satisfy
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{ R } 52 { Q } , (10.2)
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it means that, if execution of 52 is begun in a state satisfying R, termination is guaranteed in a state satisfying Q. Clearly then, if execution of 51; 52 is begun in a state satisfying P, termination is guaranteed in a state satisfying Q. That is, given (10.1) and (10.2), it is the case that
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{ P } 51;52 { Q } . (10.3)
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10.2 The skip Statement
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This is the rule of sequential composition: for all statements 51 and S2 and all assertions P, Q and R, [Sequence] { P } SI ;S2 { Q } <= { P } SI { R } A { R } S2 { Q } , (10.4)
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We have introduced the rule of sequential composition in terms of inventing a suitable intermediate assertion R. This is just one way the rule may be used. Another way is that we invent a program statement 51 that we think may be of use in solving the given programming problem. From the specification of 51, we calculate a postcondition R that is guaranteed to hold after executing 51 beginning in a state satisfying P. Finally, we solve the problem of constructing a program statement 52 that will guarantee the postcondition Q if execution of it is begun in a state satisfying precondition JR. Symmetrically, we may invent a program statement 52 that we think may be of use in solving the given programming problem. From the specification of 52, we calculate a precondition -R that will guarantee Q after executing 52. Finally, we solve the problem of constructing a program statement 51 that will guarantee the postcondition R if execution of it is begun in a state satisfying precondition P. These three different strategies (first inventing R, first inventing 51, or first inventing 52) are interrelated but it can be helpful for the more difficult problems to keep them separate in one's mind. It is difficult to give realistic but simple examples at this stage. An example combining the rule of sequential composition with the construction of conditional statements is discussed in Section 10.6.
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