Liveness Analysis Mooly Sagiv Schrierber Wed 10:00-12:00 html://

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

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

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

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 t128$25 t129$30 t130$2 t131$30 t132$4 t133$30 t103$2

.globalnfactor.entnfactor 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) lend:lw$30,-8+nfactor_framesize($sp) lw$31,-4+nfactor_framesize($sp) addiu$sp,$sp,nfactor_framesize j$31.endnfactor 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

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

The Challenge Minimize the number of spills Minimize the number of MOVEs Minimize CPU time

Outline Liveness Analysis –Motivation –Static Liveness –Dataflow Equations –Solutions –An Iterative Algorithm –Liveness in Tiger (Targil) Actual Allocation –Next week

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 given program point –its current 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

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 ab c

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

Other usages of Livness

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 ab c

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 */ t133 $2 $0 $4 t128 t129 t130 t131 t132 t103

Undecidabily A variable is live at a given point in the program –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

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

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

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

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

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

/* 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 a := 0 ; b := a +1 ; c := c +b ; a := b*2 ; c <N goto L1 return c ;

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

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]

a := 0 ; b := a +1 ; c := c +b ; a := b*2 ; c <N goto L1 return c ; insdef[n]use[n] 1{a}  2{b}{a} 3{c}{c, b} 4{a}{b} 5  {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})  

a := 0 ; b := a +1 ; c := c +b ; a := b*2 ; c <N goto L1 return c ; insdef[n]use[n] 1{a}  2{b}{a} 3{c}{c, b} 4{a}{b} 5  {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})

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

a := 0 ; b := a +1 ; c := c +b ; a := b*2 ; c <N goto L1 return c ; 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}) insLiveOutLive 1{c, a}{c} 2{c, b}{c, a} 3{c, b} 4{c, a}{c, b} 5{c,a} 6  {c}

a := 0 ; b := a +1 ; c := c +b ; a := b*2 ; c <N goto L1 return c ; 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}) insLiveOutLive 1{c, a, d}{c, d} 2{c, b, d}{c, a, d} 3{c, b, d} 4{c, a, d}{c, b, d} 5{c,a, d} 6  {c}

a := 0 ; b := a +1 ; c := c +b ; a := b*2 ; c <N goto L1 return c ; 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}) insLiveOutLive 1{c, a, b}{c, b} 2 {c, a} 3{c, b} 4{c, a}{c, b} 5{c, a} 6  {c}

a := 0 ; b := a +1 ; c := c +b ; a := b*2 ; c <N goto L1 return c ; 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}) insLiveOutLive 1{c, a}{c} 2{c, b}{c, a} 3{c, b} 4{c}{c, b} 5{c} 6 

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

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} An Iterative Algorithm

a := 0 ; b := a +1 ; c := c +b ; a := b*2 ; c <N goto L1 return c ; nLive[n]LiveOutWL {6, 5, 4, 3, 2, 1} 6{c}LiveOut[5]={c}{5, 4, 3, 2, 1} 5{c}LiveOut[4]={c}{4, 5, 2, 1} 4{c, b}LiveOut[3]={c,b}{3, 2, 1} 3{c, b}LiveOut[2]={c,b}{2, 1} 2{c, a}LiveOut[1]={c,a} LiveOut[5]={c,a} {5, 1} 5{c, a}LiveOut[4]={c,a}{4, 1} 4{c, b}{1} 1{c} 

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

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

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

t := s … x := … s … … y := t Example

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

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 ab c

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 */ t133 $2 $0 $4 t128 t129 t130 t131 t132 t103

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