Introduction to Data-flow analysis Data-flow Analysis Last Time Idea – Undergraduate compilers review – Data-flow analysis derives information about the dynamic behavior of a program by only examining the static code – Assem.Instr data structure in the MiniJava compiler Example Today – How many registers do we need – Control flow graph construction a := 0 1 for the program on the right? – Liveness analysis L1: b := a + 1 2 – Easy bound: the number of c := c + b variables used (3) 3 a := b * 2 – Better answer is found by 4 considering the dynamic if a < 9 goto L1 5 requirements of the program return c 6 CS553 Lecture Introduction to Data-flow Analysis 2 CS553 Lecture Introduction to Data-flow Analysis 3 Liveness Analysis Liveness by Example What is the live range of b ? Definition – A variable is live at a particular point in the program if its value at that – Variable b is read in statement 4, a = 0 1 so b is live on the (3 → 4) edge point will be used in the future ( dead , otherwise). ∴ To compute liveness at a given point, we need to look into the future – Since statement 3 does not assign b = a + 1 2 into b , b is also live on the (2 → 3) edge Motivation: Register Allocation c = c + b 3 – Statement 2 assigns b , so any – A program contains an unbounded number of variables value of b on the (1 → 2) and (5 → 4 a = b * 2 – Must execute on a machine with a bounded number of registers 2) edges are not needed, so b is – Two variables can use the same register if they are never in use at the same dead along these edges time ( i.e, never simultaneously live). 5 a<9 ∴ Register allocation uses liveness information No Yes b ’s live range is (2 → 3 → 4) return c 6 CS553 Lecture Introduction to Data-flow Analysis 4 CS553 Lecture Introduction to Data-flow Analysis 5 1
Liveness by Example (cont) Control Flow Graphs (CFGs) Live range of a Definition – a is live from (1 → 2) and again from – A CFG is a graph whose nodes represent program statements and 1 a = 0 (4 → 5 → 2) whose directed edges represent control flow – a is dead from (2 → 3 → 4) b = a + 1 2 a = 0 1 Example Live range of b 3 c = c + b a := 0 1 b = a + 1 2 – b is live from (2 → 3 → 4) L1: b := a + 1 2 4 a = b * 2 c := c + b c = c + b 3 3 Live range of c a := b * 2 4 – c is live from a<9 5 4 a = b * 2 if a < 9 goto L1 5 (entry → 1 → 2 → 3 → 4 → 5 → 2, 5 → 6) No Yes return c 6 return c 6 5 a<9 No Yes Variables a and b are never simultaneously live, so they can share a register return c 6 CS553 Lecture Introduction to Data-flow Analysis 6 CS553 Lecture Introduction to Data-flow Analysis 7 Terminology Uses and Defs Flow Graph Terms Def (or definition) a = 0 – A CFG node has out-edges that lead to successor nodes and in-edges that – An assignment of a value to a variable come from predecessor nodes – def[v] = set of CFG nodes that define variable v – pred[n] is the set of all predecessors of node n – def[n] = set of variables that are defined at node n a = 0 1 succ[n] is the set of all successors of node n a < 9? Use 2 b = a + 1 Examples – A read of a variable’s value – Out-edges of node 5: (5 → 6) and (5 → 2) v live – use[v] = set of CFG nodes that use variable v c = c + b 3 – succ[5] = {2,6} – use[n] = set of variables that are used at node n – pred[5] = {4} ∉ def[v] 4 a = b * 2 – pred[2] = {1,5} More precise definition of liveness ∈ use[v] 5 a<9 – A variable v is live on a CFG edge if (1) ∃ a directed path from that edge to a use of v (node in use[v]), and No Yes return c 6 (2) that path does not go through any def of v (no nodes in def[v]) CS553 Lecture Introduction to Data-flow Analysis 8 CS553 Lecture Introduction to Data-flow Analysis 9 2
program points The Flow of Liveness Liveness at Nodes edges Data-flow We have liveness on edges just before computation a = 0 – Liveness of variables is a property that flows – How do we talk about just after computation through the edges of the CFG liveness at nodes? a := 0 1 Two More Definitions Direction of Flow b := a + 1 – A variable is live-out at a node if it is live on any of that node’s out- 2 – Liveness flows backwards through the CFG, edges because the behavior at future nodes 3 c := c + b n determines liveness at a given node live-out out-edges a := b * 2 4 – Consider a – A variable is live-in at a node if it is live on any of that node’s in-edges – Consider b a < 9? 5 – Later, we’ll see other properties No Yes in-edges that flow forward return c 6 n live-in CS553 Lecture Introduction to Data-flow Analysis 10 CS553 Lecture Introduction to Data-flow Analysis 11 Computing Liveness Solving the Data-flow Equations Rules for computing liveness Algorithm (1) Generate liveness: live-in n If a variable is in use[n], use for each node n in CFG initialize solutions it is live-in at node n in[n] = ∅ ; out[n] = ∅ (2) Push liveness across edges: repeat pred[n] live-out live-out live-out If a variable is live-in at a node n for each node n in CFG then it is live-out at all nodes in pred[n] n live-in in’[n] = in[n] save current results out’[n] = out[n] (3) Push liveness across nodes: If a variable is live-out at node n and not in def[n] live-in in[n] = use[n] ∪ (out[n] – def[n]) solve data-flow equations n then the variable is also live-in at n out[n] = ∪ in[s] live-out s ∈ succ[n] Data-flow equations until in’[n]=in[n] and out’[n]=out[n] for all n test for convergence in[n] = use[n] ∪ (out[n] – def[n]) (1) (3) This is iterative data-flow analysis (for liveness analysis) out[n] = ∪ in[s] (2) s ∈ succ[n] CS553 Lecture Introduction to Data-flow Analysis 12 CS553 Lecture Introduction to Data-flow Analysis 13 3
Example Liveness in the MiniJava compiler 1st 2nd 3rd 4th 5th 6th 7th node use def in out in out in out in out in out in out in out # a := 0 1 1 a a a ac c ac c ac c ac 2 a b a a bc ac bc ac bc ac bc ac bc ac bc 2 b := a + 1 3 bc c bc bc b bc b bc b bc b bc bc bc bc 3 c := c + b 4 b a b b a b a b ac bc ac bc ac bc ac a ac 5 a a a ac ac ac ac ac ac ac ac ac ac 4 a := b * 2 6 c c c c c c c c a < 9? Data-flow Equations for Liveness 5 No Yes in[n] = use[n] ∪ (out[n] – def[n]) return c 6 out[n] = ∪ in[s] s ∈ succ[n] CS553 Lecture Introduction to Data-flow Analysis 14 CS553 Lecture Introduction to Data-flow Analysis 15 Concepts Next Time Liveness Reading – Used in register allocation – Ch 11, register allocation – Generating liveness – Flow and direction Lecture – Data-flow equations and analysis – Register allocation Control flow graphs – Predecessors and successors Defs and uses CS553 Lecture Introduction to Data-flow Analysis 16 CS553 Lecture Introduction to Data-flow Analysis 17 4
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