1. 1 CS 201 Compiler Construction Lecture 2 Control Flow Analysis

2. 2 What is a loop ? A subgraph of CFG with the following properties: –Strongly Connected: there is a path from any node in the loop to any other node in the loop; and –Single Entry: there is a single entry into the loop from outside the loop. The entry node of the loop is called the loop header. Loop nodes: 2, 3, 5 Header node: 2 Loop back edge: 5  2 Tail  Head

3. 3 Property Given two loops: they are either disjoint or one is completely nested within the other. Loops {1,2,4} and {5,6} are Disjoint. Loop {5,6} is nested within loop {2,4,5,6}. Loop {5,6} is nested within loop {1,2,3,4,5,6}. 55555555 0 1 2 3 4 5 6

4. Identifying Loops Definitions: Dominates: node n dominates node m iff all paths from start node to node m pass through node n, i.e. to visit node m we must first visit node n. A loop has –A single entry  the entry node dominates all nodes in the loop; and –A back edge, and edge A  B such that B dominates A. B is the head & A is the tail. 4

5. Identifying Loops Algorithm for finding loops: 1.Compute Dominator Information. 2.Identify Back Edges. 3.Construct Loops corresponding to Back Edges. 5

6. Dominators: Characteristics 1.Every node dominates itself. 2.Start node dominates every node in the flow graph. 3.If N DOM M and M DOM R then N DOM R. 4.If N DOM M and O DOM M then either N DOM O or O DOM N 5.Set of dominators of a given node can be linearly ordered according to dominator relationships. 6

7. Dominators: Characteristics 6. Dominator information can be represented by a Dominator Tree. Edges in the dominator tree represent immediate dominator relationships. 7 1 is the immediate dominator of 2, 3 & 4 CFGDominator Tree

8. Computing Dominator Sets Observation: node m donimates node n iff m dominates all predecessors of n. 8 Let D(n) = set of dominators of n Where Pred(n) is set of immediate predecessors of n in the CFG

9. Computing Dominator Sets 9 Initial Approximation: D(n o ) = {n o } n o is the start node. D(n) = N, for all n!=n o N is set of all nodes. Iteratively Refine D(n)’s: Algorithm:

10. Example: Computing Dom. Sets 10 D(1) = {1} D(2) = {2} U D(1) = {1,2} D(3) = {3} U D(1) = {1,3} D(4) = {4} U (D(2) D(3) D(9)) = {1,4} D(5) = {5} U (D(4) D(10)) = {1,4,5} D(6) = {6} U (D(5) D(7)) = {1,4,5,6} D(7) = {7} U D(5) = {1,4,5,7} D(8) = {8} U (D(6) D(10)) = {1,4,5,6,8} D(9) = {9} U D(8) = {1,4,5,6,8,9} D(10)= {10} U D(8) = {1,4,5,6,8,10} Back Edges: 9  4, 10  8, 10  5

11. Loop Given a back edge N  D Loop corresponding to edge N  D = {D} + {X st X can reach N without going through D} 11 1 dominates 6  6  1 is a back edge Loop of 6  1 = {1} + {3,4,5,6} = {1,3,4,5,6}

12. Algorithm for Loop Construction Given a Back Edge N  D 12 Stack = empty Loop = {D} Insert(N) While stack not empty do pop m – top element of stack for each p in pred(m) do Insert(p) endfor Endwhile Insert(m) if m not in Loop then Loop = Loop U {m} push m onto Stack endif End Insert

13. Example Back Edge 7  2 13 Loop = {2} + {7} + {6} + {4} + {5} + {3} Stack = 7 6 4 5 3 D N

14. Examples 14 L2  B, S2 L1  A,S1,B,S2 L2 nested in L1 L1  S1,S2,S3,S4 L2  S2,S3,S4 L2 nested in L1 While A do S1 While B do S2 Endwhile ?

15. Reducible Flow Graph The edges of a reducible flow graph can be partitioned into two disjoint sets: Forward – from an acyclic graph in which every node can be reached from the initial node. Back – edges whose heads (sink) dominate tails (source). Any flow graph that cannot be partitioned as above is a non-reducible or irreducible. 15

16. Reducible Flow Graph How to check reducibility ? –Remove all back edges and see if the resulting graph is acyclic. 16 Reducible Irreducible 2  3 not a back edge 3  2 not a back edge graph is not acyclic Node Splitting Converts irreducible to reducible

17. Loop Detection in Reducible Graphs Depth-first Ordering: numbering of nodes in the reverse order in which they were last visited during depth first search. M  N is a back edge iff DFN(M) >= DFN(N) 17 -- -- -- -- -- -- -- -- Depth-first Ordering Forward edge M  N (M is descendant of N in DFST) Back edge M  N (N is ancestor of M in DFST)

18. Example 18 CFG DFST 1 2 3 4 6 7 8 7 6 4 3 5 3 2 1 1 2 3 5 4 6 7 8 Depth First Ordering Back edge Forward edge (Reverse of post-order traversal)

19. Algorithm for DFN Computation 19 Mark all nodes as “unvisited” DFST = {} // set of edges of DFST I = # of nodes in the graph; DFS(n o ); DFS(X) { mark X as “visited” for each successor S of X do if S is “unvisited” then add edge X  S to DFST call DFS(S) endif endfor DFN[X] = I; I = I – 1; }

20. 20 Sample Problems Control Flow Analysis

21. 21 Dominators 1. For the given control flow graph: (a)Compute the dominator sets and construct the dominator tree; (b)Identify the loops using the dominator information; and (c) Is this control flow graph reducible? If it is so, covert it into a reducible graph. 1 1 2 2 4 4 3 3 5 5 7 7 8 8 6 6

22. 22 Depth First Numbering 2. For the given reducible control flow graph: (a)Compute the depth first numbering; and (a)Identify the loops using the computed information. 1 1 2 2 5 5 3 3 4 4 6 6 7 7 8 8 9 9

23. 23 CS 201 Compiler Construction Lecture 3 Data Flow Analysis

24. 24 Data Flow Analysis Data flow analysis is used to collect information about the flow of data values across basic blocks. Dominator analysis collected global information regarding the program’s structure For performing global code optimizations global information must be collected regarding values of program variables. –Local optimizations involve statements from same basic block –Global optimizations involve statements from different basic blocks  data flow analysis is performed to collect global information that drives global optimizations

25. 25 Local and Global Optimization

26. Applications of Data Flow Analysis Applicability of code optimizations Symbolic debugging of code Static error checking Type inference ……. 26

27. Applications of Data Flow Analysis Reaching Definition Available Expression Live Variables Very Busy Expression 27 Definition How to compute Application

28. 1. Reaching Definitions Definition d of variable v: a statement d that assigns a value to v. (d: v = 1;) Use of variable v: reference to value of v in an expression evaluation. (u: … = v+2;) Definition d of variable v reaches a point p if there exists a path from immediately after d to p such that definition d is not killed along the path. Definition d is killed along a path between two points if there exists an assignment to variable v along the path. 28

29. Example 29 d reaches u along path 2 & d does not reach u along path 1 Since there exists a path from d to u along which d is not killed (i.e., path 2 ), d reaches u.

30. Reaching Definitions Contd. Unambiguous Definition: X = ….; Ambiguous Definition: *p = ….; p may point to X For computing reaching definitions, typically we only consider kills by unambiguous definitions. 30 X=.. *p=.. Does definition of X reach here ? Yes

31. Computing Reaching Definitions At each program point p, we compute the set of definitions that reach point p. Reaching definitions are computed by solving a system of equations (data flow equations). 31 d1 : X=… IN[B] OUT[B] GEN[B] ={ d1 } KILL[B]={ d2, d3 } d2 : X=… d3 : X=…

32. Data Flow Equations 32 GEN[B]: Definitions within B that reach the end of B. KILL[B]: Definitions that never reach the end of B due to redefinitions of variables in B. IN[B]: Definitions that reach B’s entry. OUT[B]: Definitions that reach B’s exit.

33. Reaching Definitions Contd. Forward problem – information flows forward in the direction of edges. May problem – there is a path along which definition reaches a point but it does not always reach the point. Therefore in a May problem the meet operator is the Union operator. 33

34. Applications of Reaching Definitions Constant Propagation/folding Copy Propagation 34

35. 2. Available Expressions An expression is generated at a point if it is computed at that point. An expression is killed by redefinitions of operands of the expression. An expression A+B is available at a point if every path from the start node to the point evaluates A+B and after the last evaluation of A+B on each path there is no redefinition of either A or B (i.e., A+B is not killed). 35

36. Available Expressions Available expressions problem computes: at each program point the set of expressions available at that point. 36

37. Data Flow Equations 37 GEN[B]: Expressions computed within B that are available at the end of B. KILL[B]: Expressions whose operands are redefined in B. IN[B]: Expressions available at B’s entry. OUT[B]: Expressions available at B’s exit.

38. Available Expressions Contd. Forward problem – information flows forward in the direction of edges. Must problem – expression is definitely available at a point along all paths. Therefore in a Must problem the meet operator is the Intersection operator. Application: A 38

39. 3. Live Variable Analysis A path is X-clear if it contains no definition of X. A variable X is live at point p if there exists a X- clear path from p to a use of X; otherwise X is dead at p. 39 Live Variable Analysis Computes: At each program point p identify the set of variables that are live at p.

40. Data Flow Equations 40 GEN[B]: Variables that are used in B prior to their definition in B. KILL[B]: Variables definitely assigned value in B before any use of that variable in B. IN[B]: Variables live at B’s entry. OUT[B]: Variables live at B’s exit.

41. Live Variables Contd. Backward problem – information flows backward in reverse of the direction of edges. May problem – there exists a path along which a use is encountered. Therefore in a May problem the meet operator is the Union operator. 41

42. Applications of Live Variables Register Allocation Dead Code Elimination Code Motion Out of Loops 42

43. 4. Very Busy Expressions A expression A+B is very busy at point p if for all paths starting at p and ending at the end of the program, an evaluation of A+B appears before any definition of A or B. 43 Application: Code Size Reduction Compute for each program point the set of very busy expressions at the point.

44. Data Flow Equations 44 GEN[B]: Expression computed in B and variables used in the expression are not redefined in B prior to expression’s evaluation in B. KILL[B]: Expressions that use variables that are redefined in B. IN[B]: Expressions very busy at B’s entry. OUT[B]: Expressions very busy at B’s exit.

45. Very Busy Expressions Contd. Backward problem – information flows backward in reverse of the direction of edges. Must problem – expressions must be computed along all paths. Therefore in a Must problem the meet operator is the Intersection operator. 45

46. Summary May/UnionMust/Intersecti on ForwardReaching Definitions Available Expressions BackwardLive Variables Very Busy Expressions 46

47. Conservative Analysis Optimizations that we apply must be Safe => the data flow facts we compute should definitely be true (not simply possibly true). Two main reasons that cause results of analysis to be conservative: 1. Control Flow 2. Pointers & Aliasing 47

48. Conservative Analysis 1. Control Flow – we assume that all paths are executable; however, some may be infeasible. 48 X+Y is always available if we exclude infeasible paths.

49. Conservative Analysis 2. Pointers & Aliasing – we may not know what a pointer points to. 1. X = 5 2. *p = … // p may or may not point to X 3. … = X Constant propagation: assume p does point to X (i.e., in statement 3, X cannot be replaced by 5). Dead Code Elimination: assume p does not point to X (i.e., statement 1 cannot be deleted). 49

50. Representation of Data Flow Sets Bit vectors – used to represent sets because we are computing binary information. –Does a definition reach a point ? T or F –Is an expression available/very busy ? T or F –Is a variable live ? T or F For each expression, variable, definition we have one bit – intersection and union operations can be implemented using bitwise and & or operations. 50

51. Solving Data Flow Equations 51

52. Solving Data Flow Equations 52

53. Solving Data Flow Equations 53

54. Use-Def & Def-Use Chains 54

55. 55 Sample Problems Data Flow Analysis

56. 56 Data Flow Analysis Formulate data flow equations for computing the following information: 1. Postdominators -- postdominator set of a node is the set of nodes that are encountered along all paths from the node to the end node of the control flow graph. This information is used for computing control dependence.

57. 57 2.Reachable uses -- for each definition identify the set of uses reachable by the definition. This information is used for computing def-use chains. 3.Reaching uses -- given a definition of variable x, identify the set of uses of x that are encountered prior to reaching the definition and there is no other definitions of x that intervene the use and the definition. This information is used for computing antidependences.

58. 58 4. Classify Variable Values -- classify the value of each program variable at each program point into one of the following categories: (a) the value is a unique constant -- you must also identify this constant value; (b) the value is one-of-many constants – you do not have to compute the identities of these constants as part of your solution; and (c) the value is not-a-constant, that is, it is neither a unique constant nor a one-of-many constants. This is a generalization of constant propagation.