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CS 201 Compiler Construction. Lecture 2 Control Flow Analysis. 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

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## CS 201 Compiler Construction

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**CS 201Compiler Construction**Lecture 2 Control Flow Analysis**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: 52 TailHead**Property**Given two loops: they are either disjoint or one is completely nested within the other. 0 1 2 3 4 5555 5 6 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}.**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 AB such that B dominates A. B is the head & A is the tail.**Identifying Loops**Algorithm for finding loops: • Compute Dominator Information. • Identify Back Edges. • Construct Loops corresponding to Back Edges.**Dominators: Characteristics**• Every node dominates itself. • Start node dominates every node in the flow graph. • If N DOM M and M DOM R then N DOM R. • If N DOM M and O DOM M then either N DOM O or O DOM N • Set of dominators of a given node can be linearly ordered according to dominator relationships.**Dominators: Characteristics**1 is the immediate dominator of 2, 3 & 4 CFG Dominator Tree 6. Dominator information can be represented by a Dominator Tree. Edges in the dominator tree represent immediate dominator relationships.**Computing Dominator Sets**Let D(n) = set of dominators of n Where Pred(n) is set of immediate predecessors of n in the CFG Observation: node m donimates node n iff m dominates all predecessors of n.**Computing Dominator Sets**Algorithm: Initial Approximation: D(no) = {no} no is the start node. D(n) = N, for all n!=no N is set of all nodes. Iteratively Refine D(n)’s:**Example: Computing Dom. Sets**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: 94, 108, 105**Loop**• 1 dominates 6 • 61 is a back edge • Loop of 61 • = {1} + {3,4,5,6} • = {1,3,4,5,6} Given a back edge N D Loop corresponding to edge N D = {D} + {X st X can reach N without going through D}**Algorithm for Loop Construction**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 Given a Back Edge ND**Example**Loop = {2} + {7} + {6} + {4} + {5} + {3} Stack = 7 6 4 5 3 D N Back Edge 72**Examples**While A do S1 While B do S2 Endwhile Endwhile 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 ?**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.**Reducible Flow Graph**Irreducible Node Splitting Converts irreducible to reducible 23 not a back edge 32 not a back edge graph is not acyclic How to check reducibility ? • Remove all back edges and see if the resulting graph is acyclic. Reducible**Loop Detection in Reducible Graphs**Forward edge MN (M is descendant of N in DFST) Depth-first Ordering -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- Back edge MN (N is ancestor of M in DFST) Depth-first Ordering: numbering of nodes in the reverse order in which they were last visited during depth first search. MN is a back edge iff DFN(M) >= DFN(N)**Example**CFG DFST 1 2 3 4 6 7 8764 3 5321 Forward edge Depth First Ordering 1 2 3 5 4 6 7 8 Back edge (Reverse of post-order traversal)**Algorithm for DFN Computation**DFS(X) { mark X as “visited” for each successor S of X do if S is “unvisited” then add edge XS to DFST call DFS(S) endif endfor DFN[X] = I; I = I – 1; } Mark all nodes as “unvisited” DFST = {} // set of edges of DFST I = # of nodes in the graph; DFS(no);**Sample Problems**Control Flow Analysis**1. For the given control flow graph:**• Compute the dominator sets and construct the dominator tree; • 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 Dominators 2 4 3 5 6 7 8**1**• 2. For the given reducible control flow graph: • Compute the depth first numbering; and • Identify the loops using the computed information. Depth First Numbering 2 3 4 5 6 7 8 9**CS 201Compiler Construction**Lecture 3 Data Flow Analysis**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**Applications of Data Flow Analysis**Applicability of code optimizations Symbolic debugging of code Static error checking Type inference …….**Applications of Data Flow Analysis**• Definition • How to compute • Application Reaching Definition Available Expression Live Variables Very Busy Expression**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.**Example**d reaches u along path2 & d does not reach u along path1 Since there exists a path from d to u along which d is not killed (i.e., path2), d reaches u.**Reaching Definitions Contd.**X=.. *p=.. Does definition of X reach here ? Yes Unambiguous Definition: X = ….; Ambiguous Definition: *p = ….; p may point to X For computing reaching definitions, typically we only consider kills by unambiguous definitions.**Computing Reaching Definitions**d2: X=… d3: X=… IN[B] GEN[B] ={d1} d1: X=… KILL[B]={d2,d3} OUT[B] 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).**Data Flow Equations**IN[B]: Definitions that reach B’s entry. OUT[B]: Definitions that reach B’s exit. 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.**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.**Applications of Reaching Definitions**Constant Propagation/folding Copy Propagation**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).**Available Expressions**Available expressions problem computes: at each program point the set of expressions available at that point.**Data Flow Equations**IN[B]: Expressions available at B’s entry. OUT[B]: Expressions available at B’s exit. GEN[B]: Expressions computed within B that are available at the end of B. KILL[B]: Expressions whose operands are redefined in B.**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**3. Live Variable Analysis**Live Variable Analysis Computes: At each program point p identify the set of variables that are live at p. 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.**Data Flow Equations**IN[B]: Variables live at B’s entry. OUT[B]: Variables live at B’s exit. 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.**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.**Applications of Live Variables**Register Allocation Dead Code Elimination Code Motion Out of Loops**4. Very Busy Expressions**Application: Code Size Reduction Compute for each program point the set of very busy expressions at the point. 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.**Data Flow Equations**IN[B]: Expressions very busy at B’s entry. OUT[B]: Expressions very busy at B’s exit. 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.**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.**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**Conservative Analysis**X+Y is always available if we exclude infeasible paths. 1. Control Flow – we assume that all paths are executable; however, some may be infeasible.**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).**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.

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