An Analysis of the Design and Definitions of Halstead's Metrics

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where S is the Stroud number 1 , defined as thenumber of elementary discriminationsperformed by the human brain per second. The Svalue for software scientists is set to 18 [17].The unit of measurement of T is the second.All the above ten equations are based on theresults of n 1 , n 2 , N 1 , N 2 , n * 1 and n * 2 , whichthemselves are based on a counting strategy toclassify the program tokens as operators or operands.Unfortunately, there is a problem in distinguishingbetween operators and operands. This problem occursbecause Halstead has provided an example 2 withspecific illustrations of operators and operands, butwithout generic definitions applicable to any programcontext. That is, Halstead has not explicitly describedthe generic measurable concepts of operators andoperands. He has asserted only that – in the examplehe provides – their description is intuitively obviousand requires no further explanation. In practice, formeasurement purposes, intuition is insufficient toobtain accurate, repeatable and reproduciblemeasurement results.Therefore, it is important that the countingstrategy be clearly defined and consistent, since allHalstead’s software science depends on counts ofoperators and operands [18]. However, there is nogeneral agreement among researchers on the mostmeaningful way to classify and count these tokens[19]. Hence, individual researchers (and practitionersas well) must state their own interpretation or,alternatively, use one of the available countingstrategies proposed by other researchers, such as in[20-23]. Furthermore, Li et al. have proposed rulesfor identifying operators and operands in the objectorientedprogramming (OOP) languages [24].Of course, it is to be expected that differentcounting strategies will produce different values of n 1 ,n 2 , N 1 and N 2, and, consequently, different values forthe above ten equations.4. Analysis of the Design and Definitionsof Halstead’s Metrics4.1. Defining the ContextThe objective of Halstead’s metrics is to measurethe following characteristics of a program: length,vocabulary, volume, level, difficulty and intelligence1 In 1967, a psychologist, John M. Stroud, suggestedthat the human mind is capable of making a limitednumber of mental discrimination per second (StroudNumber), in the range of 5 to 20.2 Halstead’s book [3], pp. 6-8.content. In addition, they are used to measuring whatis referred to as “other characteristics” of thedeveloper: programming effort and requiredprogramming time. All these measures are based onlyon the number of operators and the number ofoperands the given program or algorithm contains.The last two attributes, which refer to adeveloper’s attributes (programming effort andrequired programming time), seem to be identical,since ‘effort to write a program’ is similar to‘required programming time’.4.2. Designing the measurement4.2.1. The empirical and numerical worlds andtheir mappingThe entities that can be used to apply Halstead’smetrics are the source code itself or the algorithm ofthat source code. However, applying Halstead’smetrics to these two entities will produce differentvalues for the same base measures. For example, inJava language, the number of operators in the sourcecode is different from the number of operators in theequivalent algorithm for that source code, since – asan example – in Java source code, each statementmust be end with a semicolon (;), which is anoperator.Halstead’s metrics are based on two attributes: thenumber of operators and the number of operands. Asmentioned in section 3, there is no agreement on howto distinguish between operators and operands.Therefore, different counting strategies will producedifferent numbers of operators and operands for thesame program or algorithm. The two attributes can beeasily mapped to a mathematical structure bycounting the number of operators and operands in theprogram source code or the equivalent algorithm.Furthermore, Kiricenko and Ormandjieva [25]investigated the validation of the representationcondition for Halstead’s program length metric.4.2.2. The measurement methodTo obtain a value for each of Halstead’s metrics,ten equations have to be computed (see section 3). Itis to be noted that all of these equations (equations 1to 10) correspond to a ‘derived measure’, as definedby the international vocabulary of basic and generalterms in metrology (VIM) and the ISO 15939.Equation (3) is of a ratio scale type, whileequation (5) is of an ordinal scale type, as noted byFenton and Pfleeger [16]. By contrast, Zuse [26]maintains that equation (1) is of the ratio scale type4
and equations (2), (3), (6) and (9) are of an ordinalscale type. Moreover, it can be observed that equation(4) is also of the ratio scale type. However, it is notclear to which scale type equations (7), (8) and (10)belong.These conclusions on the scale types of Halstead’smetrics need to be revisited when the units ofmeasurement in Halstead’s equations are taken intoconsideration.For instance, in equation (1), the program length(N) is calculated by the addition of the total numberof occurrences of operators and the total number ofoccurrences of operands. However, since their unitsare different, operators and operands cannot bedirectly added together unless the concept common tothem (and its related unit) is taken into considerationin the addition of these numbers, that is, ‘occurrencesof tokens’: then, the right-hand side of equation (1)gives ‘occurrences of tokens’ as a measurement uniton the ratio scale:occurrences occurrences occurrencesof tokens of operators of operandsN= N+ N.12From equation (2), the program vocabulary (n)can be constructed by adding the number of distinctoperators and the number of distinct operands:distinct distinct distincttokens operators operandsn = n1+ n 2 .The measurement unit here is ‘distinct tokens’. Thismeasurement unit must then also be assigned to theleft-hand side of this equation, labeled ‘vocabulary’,and associating it to the related concepts.It can be noted that, while the concept of ‘length’is associated with a number, the concept of‘vocabulary’ is not. Indeed, the program vocabulary(n) reflects a different view of program size [15], andit is a measure of ‘the repertoire of elements that aprogrammer must deal with to implement theprogram’ [27]. Most probably, an expression such as‘size of a vocabulary’ would have been moreappropriate.From equation (3), program volume (V) has beeninterpreted with two different units of measurement;‘the number of bits required to code the program’[17] and ‘the number of mental comparisons neededto write the program’ [14] on the left-hand side of theequation:Vbitsormentalcomparisonsoccurrences distinctof tokenstokens= N * log2n .Thus, there is no relationship between themeasurement unit on the left-hand side and those onthe right-hand side of this equation. Furthermore, onthe right-hand side, the true meaning of themultiplication of ‘occurrences of tokens’ and ‘distincttokens’ is not clear. Such a multiplication wouldnormally produce a number without a measurementunit, see Fig. 2.In general, in engineering applications we do nottake the logarithm of a dimensioned number, onlyof dimensionless quantities. For instance, incalculating decibels, we take the logarithm of aratio of two quantities. A ratio of quantities withthe same dimensions is itself dimensionless. Wecan writelog(a/b) = log(a) - log(b)making it appear that we are taking the logs ofdimensioned quantities (a) and (b), but thedimensions come out in the wash: by the time wehave finished (subtracting one log from theother), we have effectively taken the log of adimensionless quantity, (a/b).We can regard units as factors in an expression,for instance:8 meters = 8 * [1 meter]800 cm = 800 * [1 cm]= 800 * 0.01 * [1 meter]In these terms, we have:(8m)*log 2 (8m) = 8*[1m]*log 2 (8*[1m])= 8*[1m]*(log 2 (8)+log 2 [1m])= (8*log 2 (8)+8*log 2 [1m])*[1m]That inconvenient 8*log 2 [1m] is an additive termthat depends on the units being used. If it is partof a valid engineering calculation, this term willbe canceled out somewhere in the process. It maybe, for instance, that when we take the log of 8meters, we are actually taking the log of a ratio of8 meters to a one-meter standard length.Fig. 2: Explanation of the measurement unitproduced by log 1 2 .Equation (4) gives the definition of the programpotential volume (V * ), which is a prediction of theprogram volume:1 Contact by e-mail with Mr. Richard Peterson, TheMath Forum (Ask Dr. Math) at Drexel University,http://www.mathforum.org/dr.math/.5
Page 3: n 1 : Number of distinct operators.

An Analysis of the Design and Definitions of Halstead's Metrics

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