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Liveness Analysis

Liveness Analysis. Mooly Sagiv Schrierber 317 03-640-7606 Wed 10:00-12:00 html://www.math.tau.ac.il/~msagiv/courses/wcc01.html. Source program (string). Already Studied. lexical analysis. Tokens. syntax analysis. Abstract syntax tree. semantic analysis. Abstract syntax tree.

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Liveness Analysis

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  1. Liveness Analysis Mooly Sagiv Schrierber 317 03-640-7606 Wed 10:00-12:00 html://www.math.tau.ac.il/~msagiv/courses/wcc01.html

  2. Source program (string) Already Studied lexical analysis Tokens syntax analysis Abstract syntax tree semantic analysis Abstract syntax tree Translate Tree IR Cannon Cannonical Tree IR Instruction Selection Assem (with many reg)

  3. Source program (string) Basic Compiler Phases lexical analysis Tokens syntax analysis Abstract syntax tree semantic analysis Frame Translate Intermediate representation Instruction selection Assembly Register Allocation Fin. Assembly

  4. Register Allocation • Input: • Sequence of machine code instructions(assembly) • Unbounded number of temporary registers • Output • Sequence of machine code instructions(assembly) • Machine registers • Some MOVE instructions removed • Missing prologue and epilogue

  5. LABEL(l3) CJUMP(EQ, TEMP t128, CONST 0, l0, l1) LABEL( l1) MOVE(TEMP t131, TEMP t128) MOVE(TEMP t130, CALL(nfactor, BINOP(MINUS, TEMP t128, CONST 1))) MOVE(TEMP t129, BINOP(TIMES, TEMP t131, TEMP t130)) LABEL(l2) MOVE(TEMP t103, TEMP t129) JUMP(NAME lend) LABEL(l0) MOVE(TEMP t129, CONST 1) JUMP(NAME l2) Missing updates for static link

  6. l3: beq t128, $0, l0 l1: or t131, $0, t128 addi t132, t128, -1 or $4, $0, t132 jal nfactor or t130, $0, $2 or t133, $0, t131 mult t133, t130 mflo t133 or t129, $0, t133 l2: or t103, $0, t129 b lend l0: addi t129, $0, 1 b l2 l3: beq $25, $0, l0 l1: or $30, $0, $25 addi $4, $25, -1 /* or $4, $0, $4 */ jal nfactor /* or $2, $0, $2 */ /* or $30, $0, $30 */ mult $30, $2 mflo $30 /* or $30, $0, $30 */ l2: or $2, $0, $30 b lend l0: addi $30, $0, 1 b l2

  7. .global nfactor .ent nfactor factor_framesize=40 .frame $sp,nfactor_framesize,$31 nfactor: addiu $sp,$sp,-nfactor_framesize sw $2,0+nfactor_framesize($sp) or $25, $0, $4 sw $31,-4+nfactor_framesize($sp) sw $30,-8+nfactor_framesize($sp) l3: beq $25, $0, l0 l1: or $30, $0, $25 addi $4, $25, -1 jal nfactor mult $30, $2 mflo $30 l2: or $2, $0, $30 b lend l0: addi $30, $0, 1 b l2 lend: lw $30,-8+nfactor_framesize($sp) lw $31,-4+nfactor_framesize($sp) addiu $sp,$sp,nfactor_framesize j $31 .end nfactor

  8. The need for “spilling” • The number of registers may not be enough • Spill the content of some registers into memory • Load when needed • Increase the number of instructions • Increase CPU time

  9. The Challenge • Minimize the number of spills • Minimize the number of MOVEs • Minimize CPU time

  10. Outline • Liveness Analysis • Motivation • Static Liveness • Dataflow Equations • Solutions • An Iterative Algorithm • Liveness in Tiger (Targil) • Actual Allocation

  11. Liveness Analysis • The same register may be assigned (at compile-time) to two temporaries if their “life-times” do not overlap • A variable is live a givenprogram point • its current value is used after this point prior to a definition (assignment) • v is live at a given program point • There exists an execution sequence from this point to a use of v that does not assign to v • Two variables interfere at a given point • they are simultaneously live at this point

  12. a b c A Simple Example /* c */ L0: a := 0 /* ac */ L1: b := a + 1 /* bc */ c := c + b /* bc */ a := b * 2 /* ac */ if c < N goto L1 /* c */ return c

  13. Liveness Interference Graph • For every compiled function • Nodes • Pre-colored machine registers • Temporaries • Undirected-Edges • Temporaries that are simultaneously alive • Different machine registers • Undirected MOVE edges • “Correlated” temporaries and registers

  14. Other usages of Livness

  15. a b c A Simple Example /* c */ L0: a := 0 /* ac */ L1: b := a + 1 /* bc */ c := c + b /* bc */ a := b * 2 /* ac */ if c < N goto L1 /* c */ return c

  16. t132 t130 t131 t133 t128 t129 $0 $2 $4 t103 l3: beq t128, $0, l0 /* $0, t128 */ l1: or t131, $0, t128 /* $0, t128, t131 */ addi t132, t128, -1 /* $0, t131, t132 */ or $4, $0, t132 /* $0, $4, t131 */ jal nfactor /* $0, $2, t131 */ or t130, $0, $2 /* $0, t130, t131 */ or t133, $0, t131 /* $0, t130, t133 */ mult t133, t130 /* $0, t133 */ mflo t133 /* $0, t133 */ or t129, $0, t133 /* $0, t129 */ l2: or t103, $0, t129 /* $0, t103 */ b lend /* $0, t103 */ l0: addi t129, $0, 1 /* $0, t129 */ b l2 /* $0, t129 */

  17. Undecidabily • A variable is live at a point in a givenprogram point • if its current value is used after this point prior to a definition in some execution path • It is undecidable if a variable is live at a given program location

  18. Proof Sketch Pr L: x := y Is y live at L?

  19. Conservative • The compiler need not generate the optimal code • Can use more registers (“spill code”) than necessary • Find an upper approximation of the live variables • A superset of edges in the interference graph • Not too many superfluous live variables

  20. Control Flow Graph • Nodes • Assembly instructions • Directed-Edges • If an instruction x can be immediately followed by an instruction y • A directed edge xy

  21. Static Liveness • A variable v is statically live at control flow node n • there exists a directed path p from n to a use of v such that • p does not include an assignment to v • Every live variable is statically live • Some statically live variables are not live • since some control flow paths are non-executable

  22. a := b * b ; c := a + b ; c >= b return a; return c; Example a := b * b ; c := a + b ; if (c >= b) then return c; else return a;

  23. a := 0 ; /* c */ L0: a := 0 /* ac */ L1: b := a + 1 /* bc */ c := c + b /* bc */ a := b * 2 /* ac */ if c < N goto L1 /* c */ return c b := a +1 ; c := c +b ; a := b*2 ; c <N goto L1 return c ;

  24. Computing Static Liveness • Generate a system of equations for every function • define the set of live variables recursively • Iteratively compute a minimal solution

  25. The System of Equations • For every instruction n • def[n] • The temporary and physical register(s) assigned by n • use[n] • The temporary and physical register used in n • System of equations • LiveOut[ex] =  • LiveOut[n] =  (n, m)  Edges Live[m] • Live[n] = (LiveOut[n] – def[n])  use[n]

  26. a := 0 ; 1 b := a +1 ; 2 c := c +b ; 3 a := b*2 ; 4 c <N goto L1 5 return c ; 6 LiveOut[6] =  Live[6] = (LiveOut[6] – )  {c} LiveOut[5] = Live[6]  Live[2] Live[5] = (LiveOut[5] – )  {c} LiveOut[4] = Live[5] Live[4] = (LiveOut[4] – {a}) {b} LiveOut[3] = Live[4] Live[3] = (LiveOut[3] – {c}) {c, b} LiveOut[2] = Live[3] Live[2] = (LiveOut[2] – {b}) {a} LiveOut[1] = Live[2] Live[1] = (LiveOut[1] – {a}) 

  27. a := 0 ; 1 b := a +1 ; 2 c := c +b ; 3 a := b*2 ; 4 c <N goto L1 5 return c ; 6 LiveOut[6] =  Live[6] = LiveOut[6]  {c} LiveOut[5] = Live[6]  Live[2] Live[5] = LiveOut[5]  {c} LiveOut[4] = Live[5] Live[4] = (LiveOut[4] – {a}) {b} LiveOut[3] = Live[4] Live[3] = (LiveOut[3] – {c}) {c, b} LiveOut[2] = Live[3] Live[2] = (LiveOut[2] – {b}) {a} LiveOut[1] = Live[2] Live[1] = (LiveOut[1] – {a})

  28. Fixed Points • A fixed point is a vector solution Live and LiveOut • for every instruction n • LiveOut[ex] =  • LiveOut[n] =  (n, m)  Edges Live[m] • Live[n] = (LiveOut[n] – def[n])  use[n] • There more than one fixed point • Every fixed point contains at least the statically live variables • The least fixed point (in terms of set inclusion) uniquely exists • it contains exactly the statically live variables

  29. LiveOut[6] =  Live[6] = LiveOut[6]  {c} LiveOut[5] = Live[6]  Live[2] Live[5] = LiveOut[5]  {c} LiveOut[4] = Live[5] Live[4] = (LiveOut[4] – {a}) {b} LiveOut[3] = Live[4] Live[3] = (LiveOut[3] – {c}) {c, b} LiveOut[2] = Live[3] Live[2] = (LiveOut[2] – {b}) {a} LiveOut[1] = Live[2] Live[1] = (LiveOut[1] – {a}) a := 0 ; 1 b := a +1 ; 2 c := c +b ; 3 a := b*2 ; 4 c <N goto L1 5 return c ; 6

  30. LiveOut[6] =  Live[6] = LiveOut[6]  {c} LiveOut[5] = Live[6]  Live[2] Live[5] = LiveOut[5]  {c} LiveOut[4] = Live[5] Live[4] = (LiveOut[4] – {a}) {b} LiveOut[3] = Live[4] Live[3] = (LiveOut[3] – {c}) {c, b} LiveOut[2] = Live[3] Live[2] = (LiveOut[2] – {b}) {a} LiveOut[1] = Live[2] Live[1] = (LiveOut[1] – {a}) a := 0 ; 1 b := a +1 ; 2 c := c +b ; 3 a := b*2 ; 4 c <N goto L1 5 return c ; 6

  31. LiveOut[6] =  Live[6] = LiveOut[6]  {c} LiveOut[5] = Live[6]  Live[2] Live[5] = LiveOut[5]  {c} LiveOut[4] = Live[5] Live[4] = (LiveOut[4] – {a}) {b} LiveOut[3] = Live[4] Live[3] = (LiveOut[3] – {c}) {c, b} LiveOut[2] = Live[3] Live[2] = (LiveOut[2] – {b}) {a} LiveOut[1] = Live[2] Live[1] = (LiveOut[1] – {a}) a := 0 ; 1 b := a +1 ; 2 c := c +b ; 3 a := b*2 ; 4 c <N goto L1 5 return c ; 6

  32. LiveOut[6] =  Live[6] = LiveOut[6]  {c} LiveOut[5] = Live[6]  Live[2] Live[5] = LiveOut[5]  {c} LiveOut[4] = Live[5] Live[4] = (LiveOut[4] – {a}) {b} LiveOut[3] = Live[4] Live[3] = (LiveOut[3] – {c}) {c, b} LiveOut[2] = Live[3] Live[2] = (LiveOut[2] – {b}) {a} LiveOut[1] = Live[2] Live[1] = (LiveOut[1] – {a}) a := 0 ; 1 b := a +1 ; 2 c := c +b ; 3 a := b*2 ; 4 c <N goto L1 5 return c ; 6

  33. Computing Least Fixed Points • Start with an empty set of Live and LiveOut for every instruction • Repeatedly add new variables according to the equations • The sets of LiveOut and Live variables must monotonically increase • The process must terminate • Unique least solution

  34. An Iterative Algorithm WL := ; for each instruction n LiveOut[n] :=  Live[n] :=  WL := WL  {n} while WL !=  select and remove n from WL new := (LiveOut[n] –def[n])  use[n] if new != Live[n] then Live[n] := new for all predecessors m of n do LiveOut[m] := LiveOut[m]  Live[n] WL := WL {m}

  35. a := 0 ; 1 b := a +1 ; 2 c := c +b ; 3 a := b*2 ; 4 c <N goto L1 5 return c ; 6

  36. Representation of Sets • Bit-Vectors • Var bits for every n • Live[n][v] = 1 • the variable v is live before n • Cost of set operation is • O(Vars/word-size) • Ordered Elements • Linear time for set operations

  37. Time Complexity • Parameters • N number of nodes (instructions) • Assume that pred[n] is constant • V Number of variables • d Number of loop nesting level • DFS back edges • Initialization NV • Inner-Most Iteration V • For-Loop N • Repeat • Worst-Case NV • Worst-Case-DFS d + 1 • Total-Worst-Case (NV)2 • Total-DFS NVd • Single-variable N

  38. An Interference Graph for every instruction n for every variable a def[n] for every variable b  LiveOut[n] Create an interference edge b a May introduce too many edges for move instructions

  39. Example t := s … x := … s … … y := t

  40. An Interference Graph for every non move instruction n for every variable a def[n] for every variable b  LiveOut[n] Create an interference edge b a for every move instruction n a:= c for every variable b  LiveOut[n] – {c} Create an interference edge b a

  41. a b c A Simple Example /* c */ L0: a := 0 /* ac */ L1: b := a + 1 /* bc */ c := c + b /* bc */ a := b * 2 /* ac */ if c < N goto L1 /* c */ return c

  42. t132 t130 t131 t133 t128 t129 $0 $2 $4 t103 l3: beq t128, $0, l0 /* $0, t128 */ l1: or t131, $0, t128 /* $0, t128, t131 */ addi t132, t128, -1 /* $0, t131, t132 */ or $4, $0, t132 /* $0, $4, t131 */ jal nfactor /* $0, $2, t131 */ or t130, $0, $2 /* $0, t130, t131 */ or t133, $0, t131 /* $0, t130, t133 */ mult t133, t130 /* $0, t133 */ mflo t133 /* $0, t133 */ or t129, $0, t133 /* $0, t129 */ l2: or t103, $0, t129 /* $0, t103 */ b lend /* $0, t103 */ l0: addi t129, $0, 1 /* $0, t129 */ b l2 /* $0, t129 */

  43. Summary • The compiler can statically predict liveness of variables • May be expensive • Other useful static information • Constant expressions • Common sub-expression • Loop invariant • Liveness inference graph will be colored next week

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