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1. Identify the proper scale for each of the following measures: LOC McCabe s cyclomatic number Average depth of nesting Maximum depth of nesting Absolute Interval Ordinal Ordinal
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2. Show that the temperature scales of Celsius and Fahrenheit are an interval scale using the Celsius temperatures of 20, 30, and 40 degrees. The Fahrenheit equivalents are 68, 86, and 104 degrees. The di erence between the lowest and middle is 10 Celsius and 18 Fahrenheit. The di erence between the bottom and the top is twice that, 20 Celsius and 36 Fahrenheit. 3. Show that McCabe s cyclomatic number satis es the representational theory of measurement. For the empirical system, consider the set of all control ow graphs. The relation is that one CFG is less than or equal to the second CFG if the second CFG can be built out of the rst by adding nodes and arcs. The numerical system (Answer Set) can be the integers. The relation on the integers is the standard less than or equal. The mapping is the formula e n 2. There are only two operations, adding nodes and adding arcs. Adding an arc means increasing the e value. Adding a nodes means adding a node on an arc. This means that both e and n increase by 1, so the value stays the same. Thus, for any two CFGs x and y, if x is less than y, then y can be created from x by adding arcs and nodes. Thus, the value of the mapping must either increase or stay the same. Therefore, the less stringent representation condition is satis ed. (The more stringent representation condition cannot be satis ed, since the order on the CFGs is a partial order). 4. Show that McCabe s cyclomatic number is an interval scale measure. Since McCabe s cyclomatic number is one more than the number of decisions, every interval in the cyclomatic number is caused by that number of additional decisions. So the di erence between 2 and 3 is the same as the di erence between 10 and 11. It is not a ratio scale measure because there is not a clear zero. In fact, the cyclomatic number cannot be zero. 5. Calculate McCabe s cyclomatic number on the following source code. Draw a control ow graph. Label the regions with Roman numerals. read x,y,z; type = scalene ; if (x == y or x == z or y == z) type = isosceles ; if (x == y and x == z) type = equilateral ; if (x >= y+z or y >= x+z or z >= x+y) type = not a triangle ; if (x <= 0 or y <= 0 or| z <= 0) type = bad inputs ; print type;
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See Fig. 5-6 for the control ow graph.
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a II c III I e IV g V i bad inputs not a triangle equilateral isosceles
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Fig. 5-6.
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Control ow graph.
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The number of regions is 5, so the cyclomatic number is 5. It can also be counted with decisions. There is a decision about which way to exit nodes a, c, e, and g. Thus, there are 4 decisions. The cyclomatic number is the number of decisions plus 1, so it is 5. It can also be counted using the formula e n 2. Here, e 12, and n 9, so e n 2 5. 6. Calculate McCabe s cyclomatic number from the control ow graph shown in Fig. 5-5. Note that the graph is not planar (that is, drawn without crossing lines). Removing arcs until the graph is planar and then counting an additional region for each removed arc can calculate the cyclomatic number. In this graph (see Fig. 5-7), the arc e-g was removed, the regions were counted, and then region VIII was added for arc e-g. The count of decisions is seven (two decisions each on nodes a and c, one decision each on b, e, and f). The e n 2 16 10 2 8.
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