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REDUCTION PROCEDURE:

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• REDUCTION PROCEDURE ALGORITHM:
  • This section presents a reduction procedure for converting a flowgraph whose links are labeled with names into a path expression that denotes the set of all entry/exit paths in that flowgraph. The procedure is a node-by-node removal algorithm.
  • The steps in Reduction Algorithm are as follows:
    1. Combine all serial links by multiplying their path expressions.
    2. Combine all parallel links by adding their path expressions.
    3. Remove all self-loops (from any node to itself) by replacing them with a link of the form X*, where X is the path expression of the link in that loop.
       
       STEPS 4 - 8 ARE IN THE ALGORIHTM'S LOOP:
    4. Select any node for removal other than the initial or final node. Replace it with a set of equivalent links whose path expressions correspond to all the ways you can form a product of the set of inlinks with the set of outlinks of that node.
    5. Combine any remaining serial links by multiplying their path expressions.
    6. Combine all parallel links by adding their path expressions.
    7. Remove all self-loops as in step 3.
    8. Does the graph consist of a single link between the entry node and the exit node? If yes, then the path expression for that link is a path expression for the original flowgraph; otherwise, return to step 4.
  • A flowgraph can have many equivalent path expressions between a given pair of nodes; that is, there are many different ways to generate the set of all paths between two nodes without affecting the content of that set.
  • The appearance of the path expression depends, in general, on the order in which nodes are removed.
• CROSS-TERM STEP (STEP 4):
  • The cross - term step is the fundamental step of the reduction algorithm.
  • It removes a node, thereby reducing the number of nodes by one.
  • Successive applications of this step eventually get you down to one entry and one exit node. The following diagram shows the situation at an arbitrary node that has been selected for removal:
    
    
    
    
  • From the above diagram, one can infer:

              (a + b)(c + d + e) = ac + ad + + ae + bc + bd + be
              

• LOOP REMOVAL OPERATIONS:
  • There are two ways of looking at the loop-removal operation:
    
    
    
    
  • In the first way, we remove the self-loop and then multiply all outgoing links by Z*.
  • In the second way, we split the node into two equivalent nodes, call them A and A' and put in a link between them whose path expression is Z*. Then we remove node A' using steps 4 and 5 to yield outgoing links whose path expressions are Z*X and Z*Y.
• A REDUCTION PROCEDURE - EXAMPLE:
  • Let us see by applying this algorithm to the following graph where we remove several nodes in order; that is
    
    
    
    

    Figure 5.5: Example Flowgraph for demonstrating reduction procedure.

  • Remove node 10 by applying step 4 and combine by step 5 to yield
    
    
    
    
  • Remove node 9 by applying step4 and 5 to yield
    
    
    
    
  • Remove node 7 by steps 4 and 5, as follows:
    
    
    
    
  • Remove node 8 by steps 4 and 5, to obtain:
    
    
    
    
  • PARALLEL TERM (STEP 6):
    Removal of node 8 above led to a pair of parallel links between nodes 4 and 5. combine them to create a path expression for an equivalent link whose path expression is c+gkh; that is
    
    
    
    
  • LOOP TERM (STEP 7):
    Removing node 4 leads to a loop term. The graph has now been replaced with the following equivalent simpler graph:
    
    
    
    
  • Continue the process by applying the loop-removal step as follows:
    
    
    
    
  • Removing node 5 produces:
    
    
    
    
  • Remove the loop at node 6 to yield:
    
    
    
    
  • Remove node 3 to yield:
    
    
    
    
  • Removing the loop and then node 6 result in the following expression:

             a(bgjf)*b(c+gkh)d((ilhd)*imf(bjgf)*b(c+gkh)d)*(ilhd)*e
             

  • You can practice by applying the algorithm on the following flowgraphs and generate their respective path expressions:
    
    
    
    

    Figure 5.6: Some graphs and their path expressions.

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APPLICATIONS:

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• APPLICATIONS:
  • The purpose of the node removal algorithm is to present one very generalized concept- the path expression and way of getting it.
  • Every application follows this common pattern:
    1. Convert the program or graph into a path expression.
    2. Identify a property of interest and derive an appropriate set of "arithmetic" rules that characterizes the property.
    3. Replace the link names by the link weights for the property of interest. The path expression has now been converted to an expression in some algebra, such as ordinary algebra, regular expressions, or boolean algebra. This algebraic expression summarizes the property of interest over the set of all paths.
    4. Simplify or evaluate the resulting "algebraic" expression to answer the question you asked.
• HOW MANY PATHS IN A FLOWGRAPH ?
  • The question is not simple. Here are some ways you could ask it:
    1. What is the maximum number of different paths possible?
    2. What is the fewest number of paths possible?
    3. How many different paths are there really?
    4. What is the average number of paths?
  • Determining the actual number of different paths is an inherently difficult problem because there could be unachievable paths resulting from correlated and dependent predicates.
  • If we know both of these numbers (maximum and minimum number of possible paths) we have a good idea of how complete our testing is.
  • Asking for "the average number of paths" is meaningless.
• MAXIMUM PATH COUNT ARITHMETIC:
  • Label each link with a link weight that corresponds to the number of paths that link represents.
  • Also mark each loop with the maximum number of times that loop can be taken. If the answer is infinite, you might as well stop the analysis because it is clear that the maximum number of paths will be infinite.
  • There are three cases of interest: parallel links, serial links, and loops.
    
    
    
  • This arithmetic is an ordinary algebra. The weight is the number of paths in each set.
  • EXAMPLE:
    • The following is a reasonably well-structured program.
      
      
      Each link represents a single link and consequently is given a weight of "1" to start. Lets say the outer loop will be taken exactly four times and inner Loop Can be taken zero or three times Its path expression, with a little work, is:

      Path expression: a(b+c)d{e(fi)*fgj(m+l)k}*e(fi)*fgh

    • A: The flow graph should be annotated by replacing the link name with the maximum of paths through that link (1) and also note the number of times for looping.
    • B: Combine the first pair of parallel loops outside the loop and also the pair in the outer loop.
    • C: Multiply the things out and remove nodes to clear the clutter.
      
      
      
    • For the Inner Loop:
      D:Calculate the total weight of inner loop, which can execute a min. of 0 times and max. of 3 times. So, it inner loop can be evaluated as follows:
      
      1³ = 1⁰ + 1¹ + 1² + 1³ = 1 + 1 + 1 + 1 = 4
      
      
    • E: Multiply the link weights inside the loop: 1 X 4 = 4
    • F: Evaluate the loop by multiplying the link wieghts: 2 X 4 = 8.
    • G: Simpifying the loop further results in the total maximum number of paths in the flowgraph:
      
      2 X 8⁴ X 2 = 32,768.
      
      
      
  • Alternatively, you could have substituted a "1" for each link in the path expression and then simplified, as follows:
    
    a(b+c)d{e(fi)*fgj(m+l)k}*e(fi)*fgh
    = 1(1 + 1)1(1(1 x 1)³1 x 1 x 1(1 + 1)1)⁴1(1 x 1)³1 x 1 x 1
    = 2(1³1 x (2))⁴1³
    = 2(4 x 2)⁴ x 4
    = 2 x 8⁴ x 4 = 32,768
    
    
  • This is the same result we got graphically.
  • Actually, the outer loop should be taken exactly four times. That doesn't mean it will be taken zero or four times. Consequently, there is a superfluous "4" on the outlink in the last step. Therefore the maximum number of different paths is 8192 rather than 32,768.
• STRUCTURED FLOWGRAPH:
  • Structured code can be defined in several different ways that do not involve ad-hoc rules such as not using GOTOs.
  • A structured flowgraph is one that can be reduced to a single link by successive application of the transformations of Figure 5.7.
    
    
    
    

    Figure 5.7: Structured Flowgraph Transformations.

  • The node-by-node reduction procedure can also be used as a test for structured code.
  • Flow graphs that DO NOT contain one or more of the graphs shown below (Figure 5.8) as subgraphs are structured.
    1. Jumping into loops
    2. Jumping out of loops
    3. Branching into decisions
    4. Branching out of decisions
    
    
    
    
    

    Figure 5.8: Un-structured sub-graphs.

• LOWER PATH COUNT ARITHMETIC:
  • A lower bound on the number of paths in a routine can be approximated for structured flow graphs.
  • The arithmetic is as follows:
    
    
    
    
  • The values of the weights are the number of members in a set of paths.
  • EXAMPLE:
    • Applying the arithmetic to the earlier example gives us the identical steps unitl step 3 (C) as below:
      
      
      
      
      
      
    • From Step 4, the it would be different from the previous example:
      
      
      
    • If you observe the original graph, it takes at least two paths to cover and that it can be done in two paths.
    • If you have fewer paths in your test plan than this minimum you probably haven't covered. It's another check.
• CALCULATING THE PROBABILITY:
  • Path selection should be biased toward the low - rather than the high-probability paths.
  • This raises an interesting question:
    
    What is the probability of being at a certain point in a routine?
    
    This question can be answered under suitable assumptions, primarily that all probabilities involved are independent, which is to say that all decisions are independent and uncorrelated.
  • We use the same algorithm as before : node-by-node removal of uninteresting nodes.
  • Weights, Notations and Arithmetic:
    • Probabilities can come into the act only at decisions (including decisions associated with loops).
    • Annotate each outlink with a weight equal to the probability of going in that direction.
    • Evidently, the sum of the outlink probabilities must equal 1
    • For a simple loop, if the loop will be taken a mean of N times, the looping probability is N/(N + 1) and the probability of not looping is 1/(N + 1).
    • A link that is not part of a decision node has a probability of 1.
    • The arithmetic rules are those of ordinary arithmetic.
      
      
      
    • In this table, in case of a loop, P_A is the probability of the link leaving the loop and P_L is the probability of looping.
    • The rules are those of ordinary probability theory.
      1. If you can do something either from column A with a probability of P_A or from column B with a probability P_B, then the probability that you do either is P_A + P_B.
      2. For the series case, if you must do both things, and their probabilities are independent (as assumed), then the probability that you do both is the product of their probabilities.
    • For example, a loop node has a looping probability of P_L and a probability of not looping of P_A, which is obviously equal to I - PL.
      
      
      
    • Following the above rule, all we've done is replace the outgoing probability with 1 - so why the complicated rule? After a few steps in which you've removed nodes, combined parallel terms, removed loops and the like, you might find something like this:
      
      
      because P_L + P_A + P_B + P_C = 1, 1 - P_L = P_A + P_B + P_C, and
      
      
      which is what we've postulated for any decision. In other words, division by 1 - P_L renormalizes the outlink probabilities so that their sum equals unity after the loop is removed.
  • EXAMPLE:
    • Here is a complicated bit of logic. We want to know the probability associated with cases A, B, and C.
      
      
      
    • Let us do this in three parts, starting with case A. Note that the sum of the probabilities at each decision node is equal to 1. Start by throwing away anything that isn't on the way to case A, and then apply the reduction procedure. To avoid clutter, we usually leave out probabilities equal to 1.
      
      CASE A:
      
      
      
    • Case B is simpler:
      
      
      
    • Case C is similar and should yield a probability of 1 - 0.125 - 0.158 = 0.717:
      
      
      
    • This checks. It's a good idea when doing this sort of thing to calculate all the probabilities and to verify that the sum of the routine's exit probabilities does equal 1.
    • If it doesn't, then you've made calculation error or, more likely, you've left out some branching probability.
    • How about path probabilities? That's easy. Just trace the path of interest and multiply the probabilities as you go.
    • Alternatively, write down the path name and do the indicated arithmetic operation.
    • Say that a path consisted of links a, b, c, d, e, and the associated probabilities were .2, .5, 1., .01, and I respectively. Path abcbcbcdeabddea would have a probability of 5 x 10^-10.
    • Long paths are usually improbable.
• MEAN PROCESSING TIME OF A ROUTINE:
  • Given the execution time of all statements or instructions for every link in a flowgraph and the probability for each direction for all decisions are to find the mean processing time for the routine as a whole.
  • The model has two weights associated with every link: the processing time for that link, denoted by T, and the probability of that link P.
  • The arithmetic rules for calculating the mean time:
    
    
    
  • EXAMPLE:
    
    1. Start with the original flow graph annotated with probabilities and processing time.
       
       
       
    2. Combine the parallel links of the outer loop. The result is just the mean of the processing times for the links because there aren't any other links leaving the first node. Also combine the pair of links at the beginning of the flowgraph..
       
       
       
    3. Combine as many serial links as you can.
       
       
       
    4. Use the cross-term step to eliminate a node and to create the inner self - loop.
       
       
       
    5. Finally, you can get the mean processing time, by using the arithmetic rules as follows:
       
       
       
• PUSH/POP, GET/RETURN:
  
  • This model can be used to answer several different questions that can turn up in debugging.
  • It can also help decide which test cases to design.
  • The question is:
    
    Given a pair of complementary operations such as PUSH (the stack) and POP (the stack), considering the set of all possible paths through the routine, what is the net effect of the routine? PUSH or POP? How many times? Under what conditions?
    
    
  • Here are some other examples of complementary operations to which this model applies:

    GET/RETURN a resource block.
    OPEN/CLOSE a file.
    START/STOP a device or process.

  • EXAMPLE 1 (PUSH / POP):
    
    
    • Here is the Push/Pop Arithmetic:
      
      
      
    • The numeral 1 is used to indicate that nothing of interest (neither PUSH nor POP) occurs on a given link.
    • "H" denotes PUSH and "P" denotes POP. The operations are commutative, associative, and distributive.
      
      
      
    • Consider the following flowgraph:
      
      
      

      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
      

    • Below Table 5.9 shows several combinations of values for the two looping terms - M1 is the number of times the inner loop will be taken and M2 the number of times the outer loop will be taken.
      
      
      
      

      Figure 5.9: Result of the PUSH / POP Graph Analysis.

    • These expressions state that the stack will be popped only if the inner loop is not taken.
    • The stack will be left alone only if the inner loop is iterated once, but it may also be pushed.
    • For all other values of the inner loop, the stack will only be pushed.
  • EXAMPLE 2 (GET / RETURN):
    
    
    • Exactly the same arithmetic tables used for previous example are used for GET / RETURN a buffer block or resource, or, in fact, for any pair of complementary operations in which the total number of operations in either direction is cumulative.
    • The arithmetic tables for GET/RETURN are:
      
      
      "G" denotes GET and "R" denotes RETURN.
    • Consider the following flowgraph:
      
      
      
    • G(G + R)G(GR)*GGR*R
      = G(G + R)G³R*R
      = (G + R)G³R*
      = (G⁴ + G²)R*
    • This expression specifies the conditions under which the resources will be balanced on leaving the routine.
    • If the upper branch is taken at the first decision, the second loop must be taken four times.
    • If the lower branch is taken at the first decision, the second loop must be taken twice.
    • For any other values, the routine will not balance. Therefore, the first loop does not have to be instrumented to verify this behavior because its impact should be nil.
• LIMITATIONS AND SOLUTIONS:
  • The main limitation to these applications is the problem of unachievable paths.
  • The node-by-node reduction procedure, and most graph-theory-based algorithms work well when all paths are possible, but may provide misleading results when some paths are unachievable.
  • The approach to handling unachievable paths (for any application) is to partition the graph into subgraphs so that all paths in each of the subgraphs are achievable.
  • The resulting subgraphs may overlap, because one path may be common to several different subgraphs.
  • Each predicate's truth-functional value potentially splits the graph into two subgraphs. For n predicates, there could be as many as 2ⁿ subgraphs.

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REGULAR EXPRESSIONS AND FLOW ANOMALY DETECTION:

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• THE PROBLEM:
  • The generic flow-anomaly detection problem (note: not just data-flow anomalies, but any flow anomaly) is that of looking for a specific sequence of options considering all possible paths through a routine.
  • Let the operations be SET and RESET, denoted by s and r respectively, and we want to know if there is a SET followed immediately a SET or a RESET followed immediately by a RESET (an ss or an rr sequence).
  • Some more application examples:
    1. A file can be opened (o), closed (c), read (r), or written (w). If the file is read or written to after it's been closed, the sequence is nonsensical. Therefore, cr and cw are anomalous. Similarly, if the file is read before it's been written, just after opening, we may have a bug. Therefore, or is also anomalous. Furthermore, oo and cc, though not actual bugs, are a waste of time and therefore should also be examined.
    2. A tape transport can do a rewind (d), fast-forward (f), read (r), write (w), stop (p), and skip (k). There are rules concerning the use of the transport; for example, you cannot go from rewind to fast-forward without an intervening stop or from rewind or fast-forward to read or write without an intervening stop. The following sequences are anomalous: df, dr, dw, fd, and fr. Does the flowgraph lead to anomalous sequences on any path? If so, what sequences and under what circumstances?
    3. The data-flow anomalies discussed in Unit 4 requires us to detect the dd, dk, kk, and ku sequences. Are there paths with anomalous data flows?
• THE METHOD:
  • Annotate each link in the graph with the appropriate operator or the null operator 1.
  • Simplify things to the extent possible, using the fact that a + a = a and 12 = 1.
  • You now have a regular expression that denotes all the possible sequences of operators in that graph. You can now examine that regular expression for the sequences of interest.
  • EXAMPLE: Let A, B, C, be nonempty sets of character sequences whose smallest string is at least one character long. Let T be a two-character string of characters. Then if T is a substring of (i.e., if T appears within) ABⁿC, then T will appear in AB²C. (HUANG's Theorem)
  • As an example, let
    
    A = pp
    B = srr
    C = rp
    T = ss
    
    The theorem states that ss will appear in pp(srr)ⁿrp if it appears in pp(srr)²rp.
  • However, let
    
    A = p + pp + ps
    B = psr + ps(r + ps)
    C = rp
    T = P⁴
    
    Is it obvious that there is a p⁴ sequence in ABⁿC? The theorem states that we have only to look at
    
    (p + pp + ps)[psr + ps(r + ps)]²rp
    
    Multiplying out the expression and simplifying shows that there is no p⁴ sequence.
  • Incidentally, the above observation is an informal proof of the wisdom of looping twice discussed in Unit 2. Because data-flow anomalies are represented by two-character sequences, it follows the above theorem that looping twice is what you need to do to find such anomalies.
• LIMITATIONS:
  • Huang's theorem can be easily generalized to cover sequences of greater length than two characters. Beyond three characters, though, things get complex and this method has probably reached its utilitarian limit for manual application.
  • There are some nice theorems for finding sequences that occur at the beginnings and ends of strings but no nice algorithms for finding strings buried in an expression.
  • Static flow analysis methods can't determine whether a path is or is not achievable. Unless the flow analysis includes symbolic execution or similar techniques, the impact of unachievable paths will not be included in the analysis.
  • The flow-anomaly application, for example, doesn't tell us that there will be a flow anomaly - it tells us that if the path is achievable, then there will be a flow anomaly. Such analytical problems go away, of course, if you take the trouble to design routines for which all paths are achievable.

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SUMMARY:

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• A flowgraph annotated with link names for every link can be converted into a path expression that represents the set of all paths in that flowgraph. A node-by-node reduction procedure is used.
• By substituting link weights for all links, and using the appropriate arithmetic rules, the path expression is converted into an algebraic expression that can be used to determine the minimum and maximum number of possible paths in a flowgraph, the probability that a given node will be reached, the mean processing time of a routine, and other models.
• With different, suitable arithmetic rules, and by using complementary operators as weights for the links, the path expression can be converted into an expression that denotes, over the set of all possible paths, what the net effect of the routine is.
• With links annotated with the appropriate weights, the path expression is converted into a regular expression that denotes the set of all operator sequences over the set of all paths in a routine. Rules for determining whether a given sequence of operations are possible are given. In other words, we have a generalized flow-anomaly detection method that'll work for data-flow anomalies or any other flow anomaly.
• All flow analysis methods lose accuracy and utility if there are unachievable paths. Expand the accuracy and utility of your analytical tools by designs for which all paths are achievable. Such designs are always possible.