II. Notes on Data Structuring * - Cornell University

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112 c.A.R. HOARE On constructing a value of a discriminated union type, it is necessary to name the alternative type from which the value originated: patience card (red (spade, Jack)). This will automatically cause the value "red" to be assigned to the tag field of the result. A particular car may be denoted by car (Ford, "RUR157D", local (me, date (1, Sept, 1968))). In order to access and operate on the information encoded as a discriminated union, it is necessary to convert it back to its original type. This may be accomplished by theconvention of using the label of this type as if it were a selector, e.g." cardl, wild carl. foreign figl. tri is of type (joker 1, joker 2) is of type (origin' country) is of type T If the constituent type is a Cartesian product, its selectors may be validly applied to the resulting value, using the convention that the operator associates to the left. cardl, normal, r carl. local, owner figl. circ. diameter If the programmer attempts to convert a discriminated union value back to a type from which it did not originate, this is a serious programming error, which could lead to meaningless results. This error can be detected only by a runtime check, which tests the tag field whenever such a conversion is explicitly or implicitly invoked. Such a check is timeconsuming and when it fails, highly inconvenient. We therefore seek a notational technique which will guarantee that this error can never occur in a running program; and the guarantee is given by merely inspecting the text, without any knowledge of the runtime values being processed. Such a guarantee could be given by an automatic compiler, if available. The proposed notational technique is a mixture between the with construction for Cartesian products and the case construction for discrimination. Suppose that a value sv of union type is to be processed in one of several
NOTES ON DATA STRUCTURING 113 ways in accordance with which of the alternative types it came from. Then one may write with sv do {al"sl, a2;$2~ a n "a n }; where S~ is the statement to be selected for execution whenever the value of the tag field of sv is a~. Within S~ it is guaranteed safe to assume that the value came from the corresponding alternative type, provided that the value of sv remains unchanged. Consequently it is safe to use the component selectors which are defined for that alternative type by themselves to refer to the components of sv, just as in the case of a simple with statement described previously for a Cartesian product. If it is desired to regard a union type as ordered, the most natural ordering is that defined by taking all values corresponding to earlier alternatives in the list before any of the values of the later alternatives. Exercise Write a function that will compute the area of a figure as defined above. 5.2. REPRESENTATION In representing a value from a discriminated union it is necessary first to represent the tag as an integer between zero and n - 1, where n is the number of alternative types. The tag is followed directly by the representation of the value of the original type. As with the Cartesian product, there is a choice of the degree of packing used in a representation. In the unpacked representation the tag occupies a complete word, and the space occupied by each value of a union type is one word more than that occupied by values from the largest alternative type. In a packed representation, this overhead can be reduced to a few bits. In the minimal representation, each value is obtained by adding its minimal representation in the original type to the sum of the cardinalities of all preceding types in the union. Thus a value originating from the first type, for example (diamond, four), has exactly the same value as it has in the original type, namely 16. But joker 1, with value zero in the original enumeration type, has added to it the cardinality of the card type. The choice between unpacked, packed and tight packed representations is based on the same considerations as for Cartesian products; however the runtime speed penalty for the minimal representation is a great deal less,
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end; end gensupersets; NOTES ON DAT
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NOTES ON DATA STRUCTURING 165 An al
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NOTES ON DATA STRUCTURING 167 The a
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Abbreviations' NOTES ON DATA STRUCT
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12.5 POWERSETS NOTES ON DATA STRUCT
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(5) initial (x~T(d)) = x (6) x (y z
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II. Notes on Data Structuring * - Cornell University

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