This unit gives an in-depth overview of Paths of various flow graphs, their interpretations and application.

Transcription

1 PATHS, PATH PRODUCTS AND REGULAR EXPRESSIONS This unit gives an in-depth verview f Paths f varius flw graphs, their interpretatins and applicatin. At the end f this unit, the student will be able t: Interpret the cntrl flw graph and identify the path prducts, path sums and path expressins. Identify hw the mathematical laws (distributive, assciative, cmmutative etc) hld fr the paths. Apply reductin prcedure algrithm t a cntrl flwgraph and simplify it int a single path expressin. Find the all pssible paths (Max. Path Cunt) f a given flw graph. Find the minimum paths required t cver a given flw graph. Calculate the prbability f paths and understand the need fr finding the prbabilities. Differentiate between Structured and Un-structured flw graphs. Calculate the mean prcessing time f a rutine f a given flw graph. Understand hw cmplimentary peratins such as PUSH / POP r GET / RETURN are interpreted in a flw graph. Identify the limitatins f the abve appraches. Understand the prblems due t flw-anmalies and identify whether anmalies exist in the given path expressin. PATH PRODUCTS AND PATH EXPRESSION: TOP MOTIVATION: Flw graphs are being an abstract representatin f prgrams. Any questin abut a prgram can be cast int an equivalent questin abut an apprpriate flw graph. Mst sftware develpment, testing and debugging tls use flw graphs analysis techniques. PATH PRODUCTS: Nrmally flw graphs used t dente nly cntrl flw cnnectivity. The simplest weight we can give t a link is a name. Using link names as weights, we then cnvert the graphical flw graph int an equivalent algebraic like expressins which dentes the set f all pssible paths frm entry t exit fr the flw graph. Every link f a graph can be given a name.

2 The link name will be dented by lwer case italic letters. In tracing a path r path segment thrugh a flw graph, yu traverse a successin f link names. The name f the path r path segment that crrespnds t thse links is expressed naturally by cncatenating thse link names. Fr example, if yu traverse links a, b, c and d alng sme path, the name fr that path segment is abcd. This path name is als called a path prduct. Figure 5.1 shws sme examples: Figure 5.1: Examples f paths. PATH EXPRESSION: Cnsider a pair f ndes in a graph and the set f paths between thse nde. Dente that set f paths by Upper case letter such as X,Y. Frm Figure 5.1c, the members f the path set can be listed as fllws: ac, abc, abbc, abbbc, abbbbc...

3 Alternatively, the same set f paths can be dented by : ac+abc+abbc+abbbc+abbbbc+... The + sign is understd t mean "r" between the tw ndes f interest, paths ac, r abc, r abbc, and s n can be taken. Any expressin that cnsists f path names and "OR"s and which dentes a set f paths between tw ndes is called a "Path Expressin.". PATH PRODUCTS: The name f a path that cnsists f tw successive path segments is cnveniently expressed by the cncatenatin r Path Prduct f the segment names. Fr example, if X and Y are defined as X=abcde,Y=fghij,then the path crrespnding t X fllwed by Y is dented by XY=abcdefghij Similarly, If X and Y represent sets f paths r path expressins, their prduct represents the set f paths that can be btained by fllwing every element f X by any element f Y in all pssible ways. Fr example, X = abc + def + ghi Y = uvw + z YX=fghijabcde ax=aabcde Xa=abcdea XaX=abcdeaabcde Then, XY = abcuvw + defuvw + ghiuvw + abcz + defz + ghiz If a link r segment name is repeated, that fact is dented by an expnent. The expnent's value dentes the number f repetitins: a1 = a; a2 = aa; a3 = aaa; an = aaaa... n times. Similarly, if X = abcde then X1 = abcde X2 = abcdeabcde = (abcde)2 X3 = abcdeabcdeabcde = (abcde)2abcde = abcde(abcde)2 = (abcde)3

4 The path prduct is nt cmmutative (that is XY!=YX). The path prduct is Assciative. RULE 1: A(BC)=(AB)C=ABC where A,B,C are path names, set f path names r path expressins. The zerth pwer f a link name, path prduct, r path expressin is als needed fr cmpleteness. It is dented by the numeral "1" and dentes the "path" whse length is zer - that is, the path that desn't have any links. a0 = 1 X0 = 1 PATH SUMS: The "+" sign was used t dente the fact that path names were part f the same set f paths. The "PATH SUM" dentes paths in parallel between ndes. Links a and b in Figure 5.1a are parallel paths and are dented by a + b. Similarly, links c and d are parallel paths between the next tw ndes and are dented by c + d. The set f all paths between ndes 1 and 2 can be thught f as a set f parallel paths and dented by eacf+eadf+ebcf+ebdf. If X and Y are sets f paths that lie between the same pair f ndes, then X+Y dentes the UNION f thse set f paths. Fr example, in Figure 5.2: Figure 5.2: Examples f path sums. The first set f parallel paths is dented by X + Y + d and the secnd set by U + V + W + h + i + j. The set f all paths in this flwgraph is f(x + Y + d)g(u + V + W + h + i + j)k The path is a set unin peratin, it is clearly Cmmutative and Assciative. RULE 2: X+Y=Y+X RULE 3: (X+Y)+Z=X+(Y+Z)=X+Y+Z

5 DISTRIBUTIVE LAWS: The prduct and sum peratins are distributive, and the rdinary rules f multiplicatin apply; that is RULE 4: A(B+C)=AB+AC and (B+C)D=BD+CD Applying these rules t the belw Figure 5.1a yields e(a+b)(c+d)f=e(ac+ad+bc+bd)f = eacf+eadf+ebcf+ebdf ABSORPTION RULE: If X and Y dente the same set f paths, then the unin f these sets is unchanged; cnsequently, RULE 5: X+X=X (Absrptin Rule) If a set cnsists f paths names and a member f that set is added t it, the "new" name, which is already in that set f names, cntributes nthing and can be ignred. Fr example, if X=a+aa+abc+abcd+def then X+a = X+aa = X+abc = X+abcd = X+def = X It fllws that any arbitrary sum f identical path expressins reduces t the same path expressin. LOOPS: Lps can be understd as an infinite set f parallel paths. Say that the lp cnsists f a single link b. then the set f all paths thrugh that lp pint is b0+b1+b2+b3+b4+b5+... Figure 5.3: Examples f path lps. This ptentially infinite sum is dented by b* fr an individual link and by X* when X is a path expressin.

6 Figure 5.4: Anther example f path lps. The path expressin fr the abve figure is dented by the ntatin: ab*c=ac+abc+abbc+abbbc+... Evidently, aa*=a*a=a+ and XX*=X*X=X+ It is mre cnvenient t dente the fact that a lp cannt be taken mre than a certain, say n, number f times. A bar is used under the expnent t dente the fact as fllws: Xn = X0+X1+X2+X3+X4+X5+...+Xn RULES 6-16: The fllwing rules can be derived frm the previus rules: RULE 6: Xn + Xm = Xn if n>m RULE 6: Xn + Xm = Xm if m>n RULE 7: XnXm = Xn+m RULE 8: XnX* = X*Xn = X* RULE 9: XnX+ = X+Xn = X+ RULE 10: X*X+ = X+X* = X+ RULE 11: = 1 RULE 12: 1X = X1 = X Fllwing r preceding a set f paths by a path f zer length des nt change the set. RULE 13: 1n = 1n = 1* = 1+ = 1 N matter hw ften yu traverse a path f zer length,it is a path f zer length. RULE 14: 1++1 = 1*=1 The null set f paths is dented by the numeral 0. it beys the fllwing rules: RULE 15: X+0=0+X=X RULE 16: 0X=X0=0 If yu blck the paths f a graph fr r aft by a graph that has n paths, there wnt be any paths. TOP REDUCTION PROCEDURE:

7 TOP REDUCTION PROCEDURE ALGORITHM: This sectin presents a reductin prcedure fr cnverting a flwgraph whse links are labeled with names int a path expressin that dentes the set f all entry/exit paths in that flwgraph. The prcedure is a nde-by-nde remval algrithm. The steps in Reductin Algrithm are as fllws: 1. Cmbine all serial links by multiplying their path expressins. 2. Cmbine all parallel links by adding their path expressins. 3. Remve all self-lps (frm any nde t itself) by replacing them with a link f the frm X*, where X is the path expressin f the link in that lp. STEPS 4-8 ARE IN THE ALGORIHTM'S LOOP: 4. Select any nde fr remval ther than the initial r final nde. Replace it with a set f equivalent links whse path expressins crrespnd t all the ways yu can frm a prduct f the set f inlinks with the set f utlinks f that nde. 5. Cmbine any remaining serial links by multiplying their path expressins. 6. Cmbine all parallel links by adding their path expressins. 7. Remve all self-lps as in step Des the graph cnsist f a single link between the entry nde and the exit nde? If yes, then the path expressin fr that link is a path expressin fr the riginal flwgraph; therwise, return t step 4. A flwgraph can have many equivalent path expressins between a given pair f ndes; that is, there are many different ways t generate the set f all paths between tw ndes withut affecting the cntent f that set. The appearance f the path expressin depends, in general, n the rder in which ndes are remved. CROSS-TERM STEP (STEP 4): The crss - term step is the fundamental step f the reductin algrithm. It remves a nde, thereby reducing the number f ndes by ne. Successive applicatins f this step eventually get yu dwn t ne entry and ne exit nde. The fllwing diagram shws the situatin at an arbitrary nde that has been selected fr remval: Frm the abve diagram, ne can infer: (a + b)(c + d + e) = ac + ad + + ae + bc + bd + be LOOP REMOVAL OPERATIONS: There are tw ways f lking at the lp-remval peratin:

8 In the first way, we remve the self-lp and then multiply all utging links by Z*. In the secnd way, we split the nde int tw equivalent ndes, call them A and A' and put in a link between them whse path expressin is Z*. Then we remve nde A' using steps 4 and 5 t yield utging links whse path expressins are Z*X and Z*Y. A REDUCTION PROCEDURE - EXAMPLE: Let us see by applying this algrithm t the fllwing graph where we remve several ndes in rder; that is Figure 5.5: Example Flwgraph fr demnstrating reductin prcedure. Remve nde 10 by applying step 4 and cmbine by step 5 t yield

9 Remve nde 9 by applying step4 and 5 t yield Remve nde 7 by steps 4 and 5, as fllws: Remve nde 8 by steps 4 and 5, t btain: PARALLEL TERM (STEP 6): Remval f nde 8 abve led t a pair f parallel links between ndes 4 and 5. cmbine them t create a path expressin fr an equivalent link whse path expressin is c+gkh; that is

10 LOOP TERM (STEP 7): Remving nde 4 leads t a lp term. The graph has nw been replaced with the fllwing equivalent simpler graph: Cntinue the prcess by applying the lp-remval step as fllws: Remving nde 5 prduces:

11 Remve the lp at nde 6 t yield: Remve nde 3 t yield: Remving the lp and then nde 6 result in the fllwing expressin: a(bgjf)*b(c+gkh)d((ilhd)*imf(bjgf)*b(c+gkh)d)*(ilhd)*e Yu can practice by applying the algrithm n the fllwing flwgraphs and generate their respective path expressins:

12 Figure 5.6: Sme graphs and their path expressins. TOP APPLICATIONS: TOP APPLICATIONS: The purpse f the nde remval algrithm is t present ne very generalized cnceptthe path expressin and way f getting it. Every applicatin fllws this cmmn pattern: 1. Cnvert the prgram r graph int a path expressin. 2. Identify a prperty f interest and derive an apprpriate set f "arithmetic" rules that characterizes the prperty. 3. Replace the link names by the link weights fr the prperty f interest. The path expressin has nw been cnverted t an expressin in sme algebra, such as

13 rdinary algebra, regular expressins, r blean algebra. This algebraic expressin summarizes the prperty f interest ver the set f all paths. 4. Simplify r evaluate the resulting "algebraic" expressin t answer the questin yu asked. HOW MANY PATHS IN A FLOWGRAPH? The questin is nt simple. Here are sme ways yu culd ask it: 1. What is the maximum number f different paths pssible? 2. What is the fewest number f paths pssible? 3. Hw many different paths are there really? 4. What is the average number f paths? Determining the actual number f different paths is an inherently difficult prblem because there culd be unachievable paths resulting frm crrelated and dependent predicates. If we knw bth f these numbers (maximum and minimum number f pssible paths) we have a gd idea f hw cmplete ur testing is. Asking fr "the average number f paths" is meaningless. MAXIMUM PATH COUNT ARITHMETIC: Label each link with a link weight that crrespnds t the number f paths that link represents. Als mark each lp with the maximum number f times that lp can be taken. If the answer is infinite, yu might as well stp the analysis because it is clear that the maximum number f paths will be infinite. There are three cases f interest: parallel links, serial links, and lps. This arithmetic is an rdinary algebra. The weight is the number f paths in each set. EXAMPLE: The fllwing is a reasnably well-structured prgram. Each link represents a single link and cnsequently is given a weight f "1" t

14 start. Lets say the uter lp will be taken exactly fur times and inner Lp Can be taken zer r three times Its path expressin, with a little wrk, is: Path expressin: a(b+c)d{e(fi)*fgj(m+l)k}*e(fi)*fgh A: The flw graph shuld be anntated by replacing the link name with the maximum f paths thrugh that link (1) and als nte the number f times fr lping. B: Cmbine the first pair f parallel lps utside the lp and als the pair in the uter lp. C: Multiply the things ut and remve ndes t clear the clutter. Fr the Inner Lp: D:Calculate the ttal weight f inner lp, which can execute a min. f 0 times and max. f 3 times. S, it inner lp can be evaluated as fllws: = = =4 E: Multiply the link weights inside the lp: 1 X 4 = 4 F: Evaluate the lp by multiplying the link wieghts: 2 X 4 = 8. G: Simpifying the lp further results in the ttal maximum number f paths in the flwgraph: 4 2 X 8 X 2 = 32,768.

15 Alternatively, yu culd have substituted a "1" fr each link in the path expressin and then simplified, as fllws: a(b+c)d{e(fi)*fgj(m+l)k}*e(fi)*fgh = 1(1 + 1)1(1(1 x 1) 1 x 1 x 1(1 + 1)1) 1(1 x 1) 1 x 1 x = 2(1 1 x (2)) 1 4 = 2(4 x 2) x 4 4 = 2 x 8 x 4 = 32,768 This is the same result we gt graphically. Actually, the uter lp shuld be taken exactly fur times. That desn't mean it will be taken zer r fur times. Cnsequently, there is a superfluus "4" n the utlink in the last step. Therefre the maximum number f different paths is 8192 rather than 32,768. STRUCTURED FLOWGRAPH: Structured cde can be defined in several different ways that d nt invlve ad-hc rules such as nt using GOTOs. A structured flwgraph is ne that can be reduced t a single link by successive applicatin f the transfrmatins f Figure 5.7.

16 Figure 5.7: Structured Flwgraph Transfrmatins. The nde-by-nde reductin prcedure can als be used as a test fr structured cde. Flw graphs that DO NOT cntain ne r mre f the graphs shwn belw (Figure 5.8) as subgraphs are structured. 0. Jumping int lps 1. Jumping ut f lps 2. Branching int decisins 3. Branching ut f decisins

17 Figure 5.8: Un-structured sub-graphs. LOWER PATH COUNT ARITHMETIC: A lwer bund n the number f paths in a rutine can be apprximated fr structured flw graphs. The arithmetic is as fllws:

18 The values f the weights are the number f members in a set f paths. EXAMPLE: Applying the arithmetic t the earlier example gives us the identical steps unitl step 3 (C) as belw: Frm Step 4, the it wuld be different frm the previus example:

19 If yu bserve the riginal graph, it takes at least tw paths t cver and that it can be dne in tw paths. If yu have fewer paths in yur test plan than this minimum yu prbably haven't cvered. It's anther check. CALCULATING THE PROBABILITY: Path selectin shuld be biased tward the lw - rather than the high-prbability paths. This raises an interesting questin: What is the prbability f being at a certain pint in a rutine? This questin can be answered under suitable assumptins, primarily that all prbabilities invlved are independent, which is t say that all decisins are independent and uncrrelated. We use the same algrithm as befre : nde-by-nde remval f uninteresting ndes. Weights, Ntatins and Arithmetic: Prbabilities can cme int the act nly at decisins (including decisins assciated with lps). Anntate each utlink with a weight equal t the prbability f ging in that directin. Evidently, the sum f the utlink prbabilities must equal 1 Fr a simple lp, if the lp will be taken a mean f N times, the lping prbability is N/(N + 1) and the prbability f nt lping is 1/(N + 1). A link that is nt part f a decisin nde has a prbability f 1. The arithmetic rules are thse f rdinary arithmetic.

20 In this table, in case f a lp, PA is the prbability f the link leaving the lp and PL is the prbability f lping. The rules are thse f rdinary prbability thery. 1. If yu can d smething either frm clumn A with a prbability f PA r frm clumn B with a prbability PB, then the prbability that yu d either is PA + PB. 2. Fr the series case, if yu must d bth things, and their prbabilities are independent (as assumed), then the prbability that yu d bth is the prduct f their prbabilities. Fr example, a lp nde has a lping prbability f PL and a prbability f nt lping f PA, which is bviusly equal t I - PL. Fllwing the abve rule, all we've dne is replace the utging prbability with 1 - s why the cmplicated rule? After a few steps in which yu've remved ndes, cmbined parallel terms, remved lps and the like, yu might find smething like this:

21 because PL + PA + PB + PC = 1, 1 - PL = PA + PB + PC, and which is what we've pstulated fr any decisin. In ther wrds, divisin by 1 - PL renrmalizes the utlink prbabilities s that their sum equals unity after the lp is remved. EXAMPLE: Here is a cmplicated bit f lgic. We want t knw the prbability assciated with cases A, B, and C. Let us d this in three parts, starting with case A. Nte that the sum f the prbabilities at each decisin nde is equal t 1. Start by thrwing away anything that isn't n the way t case A, and then apply the reductin prcedure. T avid clutter, we usually leave ut prbabilities equal t 1. CASE A:

22 Case B is simpler:

23 Case C is similar and shuld yield a prbability f = 0.717: This checks. It's a gd idea when ding this srt f thing t calculate all the prbabilities and t verify that the sum f the rutine's exit prbabilities des equal 1. If it desn't, then yu've made calculatin errr r, mre likely, yu've left ut sme branching prbability.

24 Hw abut path prbabilities? That's easy. Just trace the path f interest and multiply the prbabilities as yu g. Alternatively, write dwn the path name and d the indicated arithmetic peratin. Say that a path cnsisted f links a, b, c, d, e, and the assciated prbabilities were.2,.5, 1.,.01, and I respectively. Path abcbcbcdeabddea wuld have a -10 prbability f 5 x 10. Lng paths are usually imprbable. MEAN PROCESSING TIME OF A ROUTINE: Given the executin time f all statements r instructins fr every link in a flwgraph and the prbability fr each directin fr all decisins are t find the mean prcessing time fr the rutine as a whle. The mdel has tw weights assciated with every link: the prcessing time fr that link, dented by T, and the prbability f that link P. The arithmetic rules fr calculating the mean time: EXAMPLE: 0. Start with the riginal flw graph anntated with prbabilities and prcessing time. 1. Cmbine the parallel links f the uter lp. The result is just the mean f the prcessing times fr the links because there aren't any ther links leaving the first nde. Als cmbine the pair f links at the beginning f the flwgraph..

25 2. Cmbine as many serial links as yu can. 3. Use the crss-term step t eliminate a nde and t create the inner self - lp. 4. Finally, yu can get the mean prcessing time, by using the arithmetic rules as fllws:

26 PUSH/POP, GET/RETURN: This mdel can be used t answer several different questins that can turn up in debugging. It can als help decide which test cases t design. The questin is: Given a pair f cmplementary peratins such as PUSH (the stack) and POP (the stack), cnsidering the set f all pssible paths thrugh the rutine, what is the net effect f the rutine? PUSH r POP? Hw many times? Under what cnditins? Here are sme ther examples f cmplementary peratins t which this mdel applies: GET/RETURN a resurce blck. OPEN/CLOSE a file. START/STOP a device r prcess. EXAMPLE 1 (PUSH / POP): Here is the Push/Pp Arithmetic:

27 The numeral 1 is used t indicate that nthing f interest (neither PUSH nr POP) ccurs n a given link. "H" dentes PUSH and "P" dentes POP. The peratins are cmmutative, assciative, and distributive. Cnsider the fllwing flwgraph: P(P + 1)1{P(HH)n1HP1(P + H)1}n2P(HH)n1HPH Simplifying by using the arithmetic tables, =(P2 + P){P(HH)n1(P + H)}n1(HH)n1 =(P2 + P){H2n1(P2 + 1)}n2H2n1 Belw Table 5.9 shws several cmbinatins f values fr the tw lping terms - M1 is the number f times the inner lp will be taken and M2 the number f times the uter lp will be taken.

28 Figure 5.9: Result f the PUSH / POP Graph Analysis. These expressins state that the stack will be ppped nly if the inner lp is nt taken. The stack will be left alne nly if the inner lp is iterated nce, but it may als be pushed. Fr all ther values f the inner lp, the stack will nly be pushed. EXAMPLE 2 (GET / RETURN):

29 Exactly the same arithmetic tables used fr previus example are used fr GET / RETURN a buffer blck r resurce, r, in fact, fr any pair f cmplementary peratins in which the ttal number f peratins in either directin is cumulative. The arithmetic tables fr GET/RETURN are: "G" dentes GET and "R" dentes RETURN. Cnsider the fllwing flwgraph: G(G + R)G(GR)*GGR*R 3 = G(G + R)G R*R 3 = (G + R)G R* 4 2 = (G + G )R* This expressin specifies the cnditins under which the resurces will be balanced n leaving the rutine. If the upper branch is taken at the first decisin, the secnd lp must be taken fur times. If the lwer branch is taken at the first decisin, the secnd lp must be taken twice. Fr any ther values, the rutine will nt balance. Therefre, the first lp des nt have t be instrumented t verify this behavir because its impact shuld be nil. LIMITATIONS AND SOLUTIONS: The main limitatin t these applicatins is the prblem f unachievable paths. The nde-by-nde reductin prcedure, and mst graph-thery-based algrithms wrk well when all paths are pssible, but may prvide misleading results when sme paths are unachievable. The apprach t handling unachievable paths (fr any applicatin) is t partitin the graph int subgraphs s that all paths in each f the subgraphs are achievable. The resulting subgraphs may verlap, because ne path may be cmmn t several different subgraphs.

30 Each predicate's truth-functinal value ptentially splits the graph int tw subgraphs. Fr n n predicates, there culd be as many as 2 subgraphs. TOP REGULAR EXPRESSIONS AND FLOW ANOMALY DETECTION: TOP THE PROBLEM: The generic flw-anmaly detectin prblem (nte: nt just data-flw anmalies, but any flw anmaly) is that f lking fr a specific sequence f ptins cnsidering all pssible paths thrugh a rutine. Let the peratins be SET and RESET, dented by s and r respectively, and we want t knw if there is a SET fllwed immediately a SET r a RESET fllwed immediately by a RESET (an ss r an rr sequence). Sme mre applicatin examples: 1. A file can be pened (), clsed (c), read (r), r written (w). If the file is read r written t after it's been clsed, the sequence is nnsensical. Therefre, cr and cw are anmalus. Similarly, if the file is read befre it's been written, just after pening, we may have a bug. Therefre, r is als anmalus. Furthermre, and cc, thugh nt actual bugs, are a waste f time and therefre shuld als be examined. 2. A tape transprt can d a rewind (d), fast-frward (f), read (r), write (w), stp (p), and skip (k). There are rules cncerning the use f the transprt; fr example, yu cannt g frm rewind t fast-frward withut an intervening stp r frm rewind r fast-frward t read r write withut an intervening stp. The fllwing sequences are anmalus: df, dr, dw, fd, and fr. Des the flwgraph lead t anmalus sequences n any path? If s, what sequences and under what circumstances? 3. The data-flw anmalies discussed in Unit 4 requires us t detect the dd, dk, kk, and ku sequences. Are there paths with anmalus data flws? THE METHOD: Anntate each link in the graph with the apprpriate peratr r the null peratr 1. Simplify things t the extent pssible, using the fact that a + a = a and 12 = 1. Yu nw have a regular expressin that dentes all the pssible sequences f peratrs in that graph. Yu can nw examine that regular expressin fr the sequences f interest. EXAMPLE: Let A, B, C, be nnempty sets f character sequences whse smallest string is at least ne character lng. Let T be a tw-character string f characters. Then if T is a n 2 substring f (i.e., if T appears within) AB C, then T will appear in AB C. (HUANG's Therem) As an example, let A = pp B = srr C = rp T = ss n 2 The therem states that ss will appear in pp(srr) rp if it appears in pp(srr) rp.

31 Hwever, let A = p + pp + ps B = psr + ps(r + ps) C = rp 4 T=P 4 n Is it bvius that there is a p sequence in AB C? The therem states that we have nly t lk at 2 (p + pp + ps)[psr + ps(r + ps)] rp 4 Multiplying ut the expressin and simplifying shws that there is n p sequence. Incidentally, the abve bservatin is an infrmal prf f the wisdm f lping twice discussed in Unit 2. Because data-flw anmalies are represented by tw-character sequences, it fllws the abve therem that lping twice is what yu need t d t find such anmalies. LIMITATIONS: Huang's therem can be easily generalized t cver sequences f greater length than tw characters. Beynd three characters, thugh, things get cmplex and this methd has prbably reached its utilitarian limit fr manual applicatin. There are sme nice therems fr finding sequences that ccur at the beginnings and ends f strings but n nice algrithms fr finding strings buried in an expressin. Static flw analysis methds can't determine whether a path is r is nt achievable. Unless the flw analysis includes symblic executin r similar techniques, the impact f unachievable paths will nt be included in the analysis. The flw-anmaly applicatin, fr example, desn't tell us that there will be a flw anmaly - it tells us that if the path is achievable, then there will be a flw anmaly. Such analytical prblems g away, f curse, if yu take the truble t design rutines fr which all paths are achievable. SUMMARY: A flw graph anntated with link names fr every link can be cnverted int a path expressin that represents the set f all paths in that flwgraph. A nde-by-nde reductin prcedure is used. By substituting link weights fr all links, and using the apprpriate arithmetic rules, the path expressin is cnverted int an algebraic expressin that can be used t determine the minimum and maximum number f pssible paths in a flwgraph, the prbability that a given nde will be reached, the mean prcessing time f a rutine, and ther mdels. With different, suitable arithmetic rules, and by using cmplementary peratrs as weights fr the links, the path expressin can be cnverted int an expressin that dentes, ver the set f all pssible paths, what the net effect f the rutine is. With links anntated with the apprpriate weights, the path expressin is cnverted int a regular expressin that dentes the set f all peratr sequences ver the set f all paths in a rutine. Rules fr determining whether a given sequence f peratins are pssible are given. In ther wrds, we have a generalized flw-anmaly detectin methd that'll wrk fr data-flw anmalies r any ther flw anmaly.

32 All flw analysis methds lse accuracy and utility if there are unachievable paths. Expand the accuracy and utility f yur analytical tls by designs fr which all paths are achievable. Such designs are always pssible.

33 LOGIC BASED TESTING: This unit gives an indepth verview f lgic based testing and its implementatin. At the end f this unit, the student will be able t: Understand the cncept f Lgic based testing. Learn abut Decisin Tables and their applicatin Understand the use f decisin tables in test-case design and knw their limitatins. Understand and interpret KV Charts and knw their limitatins. Learn hw t transfrm specificatins int sentences and map them int KV charts. Understand the imprtance f dnt-care cnditins. OVERVIEW OF LOGIC BASED TESTING : TOP INTRODUCTION: The functinal requirements f many prgrams can be specified by decisin tables, which prvide a useful basis fr prgram and test design. Cnsistency and cmpleteness can be analyzed by using blean algebra, which can als be used as a basis fr test design. Blean algebra is trivialized by using Karnaugh-Veitch charts. "Lgic" is ne f the mst ften used wrds in prgrammers' vcabularies but ne f their least used techniques. Blean algebra is t lgic as arithmetic is t mathematics. Withut it, the tester r prgrammer is cut ff frm many test and design techniques and tls that incrprate thse techniques. Lgic has been, fr several decades, the primary tl f hardware lgic designers. Many test methds develped fr hardware lgic can be adapted t sftware lgic testing. Because hardware testing autmatin is 10 t 15 years ahead f sftware testing autmatin, hardware testing methds and its assciated thery is a fertile grund fr sftware testing methds. As prgramming and test techniques have imprved, the bugs have shifted clser t the prcess frnt end, t requirements and their specificatins. These bugs range frm 8% t 30% f the ttal and because they're first-in and last-ut, they're the cstliest f all. The truble with specificatins is that they're hard t express.

34 Blean algebra (als knwn as the sentential calculus) is the mst basic f all lgic systems. Higher-rder lgic systems are needed and used fr frmal specificatins. Much f lgical analysis can be and is embedded in tls. But these tls incrprate methds t simplify, transfrm, and check specificatins, and the methds are t a large extent based n blean algebra. KNOWLEDGE BASED SYSTEM: The knwledge-based system (als expert system, r "artificial intelligence" system) has becme the prgramming cnstruct f chice fr many applicatins that were nce cnsidered very difficult. Knwledge-based systems incrprate knwledge frm a knwledge dmain such as medicine, law, r civil engineering int a database. The data can then be queried and interacted with t prvide slutins t prblems in that dmain. One implementatin f knwledge-based systems is t incrprate the expert's knwledge int a set f rules. The user can then prvide data and ask questins based n that data. The user's data is prcessed thrugh the rule base t yield cnclusins (tentative r definite) and requests fr mre data. The prcessing is dne by a prgram called the inference engine. Understanding knwledge-based systems and their validatin prblems requires an understanding f frmal lgic. Decisin tables are extensively used in business data prcessing; Decisin-table preprcessrs as extensins t COBOL are in cmmn use; blean algebra is embedded in the implementatin f these prcessrs. Althugh prgrammed tls are nice t have, mst f the benefits f blean algebra can be reaped by whlly manual means if yu have the right cnceptual tl: the Karnaugh-Veitch diagram is that cnceptual tl. TOP DECISION TABLES: TOP Figure 6.1 is a limited - entry decisin table. It cnsists f fur areas called the cnditin stub, the cnditin entry, the actin stub, and the actin entry. Each clumn f the table is a rule that specifies the cnditins under which the actins named in the actin stub will take place. The cnditin stub is a list f names f cnditins.

35 Figure 6.1 : Examples f Decisin Table. A mre general decisin table can be as belw: Figure 6.2 : Anther Examples f Decisin Table. A rule specifies whether a cnditin shuld r shuld nt be met fr the rule t be satisfied. "YES" means that the cnditin must be met, "NO" means that the cnditin must nt be met, and "I" means that the cnditin plays n part in the rule, r it is immaterial t that rule.

36 The actin stub names the actins the rutine will take r initiate if the rule is satisfied. If the actin entry is "YES", the actin will take place; if "NO", the actin will nt take place. The table in Figure 6.1 can be translated as fllws: Actin 1 will take place if cnditins 1 and 2 are met and if cnditins 3 and 4 are nt met (rule 1) r if cnditins 1, 3, and 4 are met (rule 2). "Cnditin" is anther wrd fr predicate. Decisin-table uses "cnditin" and "satisfied" r "met". Let us use "predicate" and TRUE / FALSE. Nw the abve translatins becme: 1. Actin 1 will be taken if predicates 1 and 2 are true and if predicates 3 and 4 are false (rule 1), r if predicates 1, 3, and 4 are true (rule 2). 2. Actin 2 will be taken if the predicates are all false, (rule 3). 3. Actin 3 will take place if predicate 1 is false and predicate 4 is true (rule 4). In additin t the stated rules, we als need a Default Rule that specifies the default actin t be taken when all ther rules fail. The default rules fr Table in Figure 6.1 is shwn in Figure 6.3 Figure 6.3 : The default rules f Table in Figure 6.1 DECISION-TABLE PROCESSORS: Decisin tables can be autmatically translated int cde and, as such, are a higher-rder language If the rule is satisfied, the crrespnding actin takes place Otherwise, rule 2 is tried. This prcess cntinues until either a satisfied rule results in an actin r n rule is satisfied and the default actin is taken

37 Decisin tables have becme a useful tl in the prgrammers kit, in business data prcessing. DECISION-TABLES AS BASIS FOR TEST CASE DESIGN: 0. The specificatin is given as a decisin table r can be easily cnverted int ne. 1. The rder in which the predicates are evaluated des nt affect interpretatin f the rules r the resulting actin - i.e., an arbitrary permutatin f the predicate rder will nt, r shuld nt, affect which actin takes place. 2. The rder in which the rules are evaluated des nt affect the resulting actin - i.e., an arbitrary permutatin f rules will nt, r shuld nt, affect which actin takes place. 3. Once a rule is satisfied and an actin selected, n ther rule need be examined. 4. If several actins can result frm satisfying a rule, the rder in which the actins are executed desn't matter DECISION-TABLES AND STRUCTURE: Decisin tables can als be used t examine a prgram's structure. Figure 6.4 shws a prgram segment that cnsists f a decisin tree. These decisins, in varius cmbinatins, can lead t actins 1, 2, r 3. Figure 6.4 : A Sample Prgram If the decisin appears n a path, put in a YES r NO as apprpriate. If the decisin des nt appear n the path, put in an I, Rule 1 des nt cntain decisin C, therefre its entries are: YES, YES, I, YES. The crrespnding decisin table is shwn in Table 6.1

38 RULE 1 RULE 2 RULE 3 RULE 4 RULE 5 RULE 6 CONDITION A CONDITION B CONDITION C CONDITION D YES YES I YES YES NO I I YES YES I NO NO I YES I NO I NO YES NO I NO NO ACTION 1 ACTION 2 ACTION 3 YES NO NO YES NO NO NO YES NO NO YES NO NO YES NO NO NO YES Table 6.1 : Decisin Table crrespnding t Figure 6.4 As an example, expanding the immaterial cases results as belw: Similalrly, If we expand the immaterial cases fr the abve Table 6.1, it results in Table 6.2 as belw: CONDITION A CONDITION B CONDITION C CONDITION D R1 RULE 2 R3 RULE 4 R5 R6 YY YY YN YY YYYY NNNN NNYY YNNY YY YY YN NN NNNN YYNN YYYY NYYN NN NY NN YY NN YN NN NN Table 6.2 : Expansin f Table 6.1 Sixteen cases are represented in Table 6.1, and n case appears twice. Cnsequently, the flwgraph appears t be cmplete and cnsistent. As a first check, befre yu lk fr all sixteen cmbinatins, cunt the number f Y's and N's in each rw. They shuld be equal. We can find the bug that way. ANOTHER EXAMPLE - A TROUBLE SOME PROGRAM: Cnsider the fllwing specificatin whse putative flwgraph is shwn in Figure 6.5:

39 1. If cnditin A is met, d prcess A1 n matter what ther actins are taken r what ther cnditins are met. 2. If cnditin B is met, d prcess A2 n matter what ther actins are taken r what ther cnditins are met. 3. If cnditin C is met, d prcess A3 n matter what ther actins are taken r what ther cnditins are met. 4. If nne f the cnditins is met, then d prcesses A1, A2, and A3. 5. When mre than ne prcess is dne, prcess A1 must be dne first, then A2, and then A3. The nly permissible cases are: (A1), (A2), (A3), (A1,A3), (A2,A3) and (A1,A2,A3). Figure 6.5 shws a sample prgram with a bug. Figure 6.5 : A Trublesme Prgram The prgrammer tried t frce all three prcesses t be executed fr the cases but frgt that the B and C predicates wuld be dne again, thereby bypassing prcesses A2 and A3. Table 6.3 shws the cnversin f this flwgraph int a decisin table after expansin.

40 Table 6.3 : Decisin Table fr Figure 6.5 TOP PATH EXPRESSIONS: TOP GENERAL: Lgic-based testing is structural testing when it's applied t structure (e.g., cntrl flwgraph f an implementatin); it's functinal testing when it's applied t a specificatin. In lgic-based testing we fcus n the truth values f cntrl flw predicates. A predicate is implemented as a prcess whse utcme is a truthfunctinal value. Fr ur purpse, lgic-based testing is restricted t binary predicates. We start by generating path expressins by path tracing as in Unit V, but this time, ur purpse is t cnvert the path expressins int blean algebra, using the predicates' truth values (e.g., A and ) as weights. BOOLEAN ALGEBRA: STEPS: 1. Label each decisin with an uppercase letter that represents the truth value f the predicate. The YES r TRUE branch is labeled with a letter (say A) and the NO r FALSE branch with the same letter verscred (say ). 2. The truth value f a path is the prduct f the individual labels. Cncatenatin r prducts mean "AND". Fr example, the straightthrugh path f Figure 6.5, which ges via ndes 3, 6, 7, 8, 10, 11, 12, and 2, has a truth value f ABC. The path via ndes 3, 6, 7, 9 and 2 has a value f. 3. If tw r mre paths merge at a nde, the fact is expressed by use f a plus sign (+) which means "OR".

41 Figure 6.5 : A Trublesme Prgram Using this cnventin, the truth-functinal values fr several f the ndes can be expressed in terms f segments frm previus ndes. Use the nde name t identify the pint. There are nly tw numbers in blean algebra: zer (0) and ne (1). One means "always true" and zer means "always false". RULES OF BOOLEAN ALGEBRA: Blean algebra has three peratrs: X (AND), + (OR) and (NOT) X : meaning AND. Als called multiplicatin. A statement such as AB (A X B) means "A and B are bth true". This symbl is usually left ut as in rdinary algebra. + : meaning OR. "A + B" means "either A is true r B is true r bth". meaning NOT. Als negatin r cmplementatin. This is read as either "nt A" r "A bar". The entire expressin under the bar is negated.

42 The fllwing are the laws f blean algebra: In all f the abve, a letter can represent a single sentence r an entire blean algebra expressin. Individual letters in a blean algebra expressin are called Literals (e.g. A,B) The prduct f several literals is called a prduct term (e.g., ABC, DE). An arbitrary blean expressin that has been multiplied ut s that it cnsists f the sum f prducts (e.g., ABC + DEF + GH) is said t be in sum-f-prducts frm. The result f simplificatins (using the rules abve) is again in the sum f prduct frm and each prduct term in such a simplified versin is called a prime implicant. Fr example, ABC + AB + DEF reduces by rule 20 t AB + DEF; that is, AB and DEF are prime implicants. The path expressins f Figure 6.5 can nw be simplified by applying the rules. The fllwing are the laws f blean algebra:

43 Similarly, The deviatin frm the specificatin is nw clear. The functins shuld have been: Lps cmplicate things because we may have t slve a blean equatin t determine what predicate-value cmbinatins lead t where. TOP

44 KV CHARTS: INTRODUCTION: If yu had t deal with expressins in fur, five, r six variables, yu culd get bgged dwn in the algebra and make as many errrs in designing test cases as there are bugs in the rutine yu're testing. Karnaugh-Veitch chart reduces blean algebraic manipulatins t graphical trivia. Beynd six variables these diagrams get cumbersme and may nt be effective. SINGLE VARIABLE: Figure 6.6 shws all the blean functins f a single variable and their equivalent representatin as a KV chart. Figure 6.6 : KV Charts fr Functins f a Single Variable. The charts shw all pssible truth values that the variable A can have.

45 A "1" means the variable s value is "1" r TRUE. A "0" means that the variable's value is 0 r FALSE. The entry in the bx (0 r 1) specifies whether the functin that the chart represents is true r false fr that value f the variable. We usually d nt explicitly put in 0 entries but specify nly the cnditins under which the functin is true. TWO VARIABLES: Figure 6.7 shws eight f the sixteen pssible functins f tw variables. Figure 6.7 : KV Charts fr Functins f Tw Variables.

46 Each bx crrespnds t the cmbinatin f values f the variables fr the rw and clumn f that bx. A pair may be adjacent either hrizntally r vertically but nt diagnally. Any variable that changes in either the hrizntal r vertical directin des nt appear in the expressin. In the fifth chart, the B variable changes frm 0 t 1 ging dwn the clumn, and because the A variable's value fr the clumn is 1, the chart is equivalent t a simple A. Figure 6.8 shws the remaining eight functins f tw variables.

47 Figure 6.8 : Mre Functins f Tw Variables. The first chart has tw 1's in it, but because they are nt adjacent, each must be taken separately. They are written using a plus sign. It is clear nw why there are sixteen functins f tw variables. Each bx in the KV chart crrespnds t a cmbinatin f the variables' values. That cmbinatin might r might nt be in the functin (i.e., the bx crrespnding t that cmbinatin might have a 1 r 0 entry). Since n variables lead t 2n cmbinatins f 0 and 1 fr the variables, and each such cmbinatin (bx) can be filled r nt filled, leading t 2 2n ways f ding this. Cnsequently fr ne variable there are 221 = 4 functins, 16 functins f 2 variables, 256 functins f 3 variables, 16,384 functins f 4 variables, and s n. Given tw charts ver the same variables, arranged the same way, their prduct is the term by term prduct, their sum is the term by term sum, and the negatin f a chart is gtten by reversing all the 0 and 1 entries in the chart. OR THREE VARIABLES: KV charts fr three variables are shwn belw. As befre, each bx represents an elementary term f three variables with a bar appearing r nt appearing accrding t whether the rw-clumn heading fr that bx is 0 r 1.

48 A three-variable chart can have grupings f 1, 2, 4, and 8 bxes. A few examples will illustrate the principles:

49 Figure 6.8: KV Charts fr Functins f Three Variables. Yu'll ntice that there are several ways t circle the bxes int maximumsized cvering grups.