Debugging Program Slicing
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1 Debugging Franz Wotawa Institut für Informationssysteme, Database and Artificial Intelligence Group, Technische Universität Wien 1
2 What is a Slice? A Slice is a reduced program, that preserves the original program s behavior for a given set of variables at a chosen point in a program. Characteristics: Defined for variables and a given point A slice is itself a program Ignoring other statements Focus on relevant parts 2
3 Is slicing/debugging necessary? YES!? 3
4 Example A 1. z = 4; 2. y = z + 1; 3. x = 5 + z; 4. Original Program 1. z = 4; x = 5 + z; 4. Slice wrt. Deleting statements Preserving projection of programs semantics Removing irrelevant information Algorithmic debugging, testing, maintenance, and measurement 4
5 Definitions (Overview) Mark Weiser: Slicing Criterion Captures semantics projection to be preserved and a set of Varia- Consists of a line number bles : Example: The variable does not depend on, hence statement 2 can be removed. Static vs. Dynamic slicing 5
6 Formal Definitions (I) de- Definition 1.1 (REF/DEF Variable Sets) notes the set of variables defined at node. denotes the set of variables referenced at node. Example Assignment x = y + z; and. Definition 1.2 (State Trajectory) A state trajectory of length of a program is a finite list of ordered pairs where each is a node of and each is a function mapping the variables in (a subset of the variables in ) to their values. Each gives the values of variables in immediately before the execution of. 6
7 Formal Definitions (II) Definition 1.3 (Static Slicing Criterion) A static slicing criterion of a program is a pair, where is a statement in and is a subset of the variables in. Definition 1.4 (Projection) Let be a state trajectory, any node in and any function from variable names to values. Then if otherwise where is restricted to domain, and is the empty string. 7
8 Formal Definitions (III) of a pro- is any pro- Definition 1.5 (Static Slicing) A slice gram on a slicing criterion gram with the following properties: 1. can be obtained from more statements from. by deleting zero or 2. Whenever halts on an input with state trajectory, then also halts on input with state trajectory, and, where and is the nearest successor to in the original program which is also in the slice, or itself if is in the slice. 8
9 Example A (cont.) State trajectory: State trajectory for slice: Projections are equal Really a slice 9
10 Computing Slices (I) Slicing Definition does not say how to find one! Is there a minimal slice? Definition 1.6 (Minimal Slice) Let be a slicing criterion on a program. A slice of on is statement-minimal if no other slice of on has fewer statements than. Theorem 1.1 There does not exist an algorithm to find state-minimal slices for arbitrary programs. Argument: Finding minimal slices is equivalent to solving the halting problem. 10
11 Computing Slices (II) Use data flow analyses Constructs conservative slices Results have slice property Maybe to many statements included. 1. y = x; 2. z = y; 3. The value of x can affect the value of z to be directly relevant to a slice. x is said 11
12 Computing Slices (III) Definition 1.7 (Relevant Variables) Let be a slicing criterion. Then comprises all variables such that either: 1. and, or 2. is an immediate predecessor of a node such that either: (a) is in and and there is a, or in both (b) is not in and is in. Recursive definition over the length of paths to reach node. Computation of for arbitrary programs in and for structured programs in where is the number of edges in the flow-graph. 12
13 Computing Slices (IV) Statements included in the slice by. are denoted all nodes s.t. While is a function mapping statements to variables, is only a set of statements. does not include indirect effects is a sufficient but not necessary condition for including statements in a slice. 13
14 Example A (cont.) Computing for our running example 1. z = 4; 2. y = z + 1; 3. x = 5 + z; 4. Slice for Rule (1) Rule (2a) Rule (2b) Slice 14
15 Negative Example 1. read(x); 2. if (x < 1) 3. z = 1; else 4. z = 2; 5. write(z); 6. In this program statement 2 obviously has an effect on the value of z at statement 5, yet 2 is not in. Branch statements must be specially treated! Branch statements should be in the slice! 15
16 Computing Slices (IV) Avoid Problems Define Range of Influence Nearest inverse dominator of a branch Definition 1.8 is the set of statements which are on the path from to its nearest inverse dominator, excluding the endpoints of. is empty unless immediate successor. has more than one 16
17 Computing Slices (V) Definition 1.9 For all : all nodes or s.t. where. is the branch statement criterion and define nondecreasing subset bound by the set of variables and statements. Therefore, each has a least fix point denoted and. 17
18 Computing Slices (VI) The following properties hold: The upper bound on the complexity of computing is where is the number of edges in the flow-graphs and the number of nodes. Computation in practice seems much faster. is not always the smallest slice! 18
19 Example B 1. A = constant; 2. while (P(k)) 3. if (Q(C)) 4. B = A; 5. X = 1; else 6. C = B; 7. Y = 2; 8. K = K + 1; 9. Z = X + Y; 10. write(z); Slicing criterion Statement 1 will be in but cannot affect the values of in statement 10! It cannot because any path by which at 1 can influence at 3 will also execute both statements 5 and 7, resulting in a constant value for at line
20 Advantages/Disadvantages Can be computed at compile time (Offline) Fast computation of slices Provide relevant informations of a program Does not help in all cases (Neither do other techniques) Contains informations that are not relevant to a specific test-case 20
21 Extensions (Dynamic Slicing) Compute Slice for each test-case Use Execution Trace Slice is smaller Computation is fast Requires more space (for the execution trace) 21
22 Example C program sample; var n,max,min,s,i,k: integer; a: array[1..100] of integer; 1. procedure find max(s1: integer; var s2: integer); begin 2. if s1 s2 then 3. s := s1; /* correctly s2 := s1; end; 4. procedure find min(x: integer; var y: integer); begin 5. if x y then 6. y := x; end; 22
23 Example C (cont.) begin 7. readln(n); 8. readln(k); 9. i := 1; 10. while i n do begin 11. readln(a[i]); 12. i := i + 1; end; 13. max := a[1]; 14. min := a[1]; 15. s := a[1]; 16. i := k + 1; 17. while i n do begin 18. if a[i] 0 then begin 19. Find max(a[i],max); 20. Find min(a[i],min); 21. s := s + a[i]; end; 22. i := i + k; end; 23. write(max,, min,, s); end; 23
24 Example C (cont.) Flow Graph readln(n) readln(k) Flow Graph Program sample i:=1 i<=n max:=a[1] readln(a[i]) min:=a[1] i:=i+1 s:=a[1] i:=k+1 i<=n write(max,,min,,s) a[i]>0 Find_max(..) Find_min(..) s:=s+a[i] i:=i+k 24
25 Example C (cont.) Execution trace on input:,,. readln(n). readln(k). i := 1. i n. readln(a[1]). i := i + 1. i n. readln(a[1]). i := i + 1. i n. readln(a[1]). i := i + 1. i n. max := a[1]. min := a[1]. s := a[1]. i := k + 1. i n. a[i] 0. Find max(a[i],max). procedure Find max(... ). s1 s2. s := s1. Find min(a[i],min). procedure Find min(... ). x y. s := s + a[i]. i := i + k. i n. a[i] 0. i := i + k. i n. writeln(max,, min,, s) 25
26 Background Flow Graph Node for every assignment, input and output statement, procedure call, procedure entry, goto, break, continue, label, and the predicate of while loops and conditional (test node). Arc are for potential transfer of control from one node to the other. Execution Trace is a sequence of nodes that has been executed for some input. e.g.,. Node at position is written as and is referred to as an action (or test action if is a test node). is a variable at position, i.e., a variable before execution of node. 26
27 Dynamic Slice A dynamic slice is an executable part of a program whose behavior is identical, for the same input, to that of the original program wrt. a variable of interest at some execution position. Definition 1.10 (Slicing Criterion) A slicing criterion of a program executed on program input is a tuple where is a variable at execution position. A dynamic slice for preserves the value of for a given program input. Goal: Find a minimal number of statements that is a dynamic slice for. 27
28 Computing Dynamic Slices Compute the execution trace for a given input Derive the dynamic slice using data and control dependencies Data dependency: An action node assigns a value and another node uses this value, e.g., assigns a value to variable and uses that value. Control dependency: Capture the dependence between test actions and actions that have been chosen to be executed by these test actions, e.g., is depended on the outcome of test action, but is not. Use dynamic dependencies to compute the dynamic slice by tracing back the dependencies between actions in starting from action. 28
29 Computing Dynamic Slices (cont.) 1. Set all actions as unmarked and not visited. 2. Mark the last definition of. The last definition of a variable in execution trace is an action that assigns a value to and is not modified between and. E.g., last definition of is action. 3. Chose an action that is marked and not visited. Set as visited. All last definitions of all variables used in are marked, and then all actions for which there exists a control dependence between them and are marked. 4. This process continues until all marked actions are visited. 5. A dynamic slice is constructed by removing all nodes (statements) whose actions are not marked in. 29
30 Example C (cont.) Marked actions are referred as contributing actions. Dynamic slice for at position 23: program sample; var n,max,min,s,i,k: integer; a: array[1..100] of integer; begin 7. readln(n); 8. readln(k); 9. i := 1; 10. while i n do begin 11. readln(a[i]); 12. i := i + 1; end; 13. max := a[1]; 23. write(max,, min,, s); end; 30
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