Lecture Outline. Global flow analysis. Global Optimization. Global constant propagation. Liveness analysis. Local Optimization. Global Optimization

Transcription

1 Lecture Outline Global flow analyi Global Optimization Global contant propagation Livene analyi Adapted from Lecture by Prof. Alex Aiken and George Necula (UCB) CS781(Praad) L27OP 1 CS781(Praad) L27OP 2 Local Optimization Recall the imple baic-block optimization Contant propagation Dead code elimination Global Optimization Thee optimization can be extended to an entire control-flow graph Y := Z * W Y := Z * W Y := Z * W Q := 3 + Y Q := X + Y Q := 3 + Y CS781(Praad) L27OP 3 CS781(Praad) L27OP 4 Global Optimization Thee optimization can be extended to an entire control-flow graph Global Optimization Thee optimization can be extended to an entire control-flow graph A := 2 * 3 CS781(Praad) L27OP 5 CS781(Praad) L27OP 6

2 Correctne How do we know it i OK to globally propagate contant? There are ituation where it i incorrect: Correctne (Cont.) To replace a ue of x by a contant k we mut know that: On every path to the ue of x, the lat aignment to x i x := k ** Y := Z + W X := 4 Y := 0 CS781(Praad) L27OP 7 CS781(Praad) L27OP 8 Example 1 Reviited Example 2 Reviited Y := Z + W X := 4 Y := 0 CS781(Praad) L27OP 9 CS781(Praad) L27OP 10 Dicuion The correctne condition i not trivial to check All path include path around loop and through branche of conditional Checking the condition require global analyi An analyi of the entire control-flow graph CS781(Praad) L27OP 11 Java Example: Reachability and Initialization cla Flow { int f() { int j; int k = 50; // while (fale) ; // unreachable tatement // while (true) ; // unreachable tatement if (true) j = 5; // if (fale) j = 5; // j uninitialized // if (k < 150 k >= 150) j = 5; // j uninitialized return j; } public tatic void main(string[] arg) { Sytem.out.println((new Flow()). f()); }} CS781(Praad) L27OP 12

3 Global Analyi Global optimization tak hare everal trait: The optimization depend on knowing a property X at a particular point in program execution Proving X at any point require knowledge of the entire program It i OK to be conervative. If the optimization require X to be true, then want to know either X i definitely true Don t know if X i true It i alway afe to ay don t know Global Analyi (Cont.) Global dataflow analyi i a tandard technique for olving problem with thee characteritic Global contant propagation i one example of an optimization that require global dataflow analyi Global contant propagation can be performed at any point where ** hold Conider the cae of computing ** for a ingle variable X at all program point CS781(Praad) L27OP 13 CS781(Praad) L27OP 14 Global Contant Propagation Example To make the problem precie, we aociate one of the following value with X at every program point value # c * interpretation Thi tatement never execute ontant c X i not a contant Y := Z + W X := 4 X = 4 Y := 0 CS781(Praad) L27OP 15 CS781(Praad) L27OP 16 Uing the Information Given global contant information, it i eay to perform the optimization Simply inpect the x =? aociated with a tatement uing x If x i contant at that point replace that ue of x by the contant But how do we compute the propertie x =? The Idea The analyi of a complicated program can be expreed a a combination of imple rule relating the change in information between adjacent tatement CS781(Praad) L27OP 17 CS781(Praad) L27OP 18

4 Explanation The idea i to puh or tranfer information from one tatement to the next For each tatement, we compute information about the value of x immediately before and after C(x,,in) = value of x before C(x,,out) = value of x after Tranfer Function Define a tranfer function that tranfer information from one tatement to another Recall alo that in the context of a baic block, the firt tatement in a baic block can be reached by multiple path, typically. In the following rule, let tatement have immediate predeceor tatement p 1,,p n CS781(Praad) L27OP 19 CS781(Praad) L27OP 20 Rule 1 Rule 2 X = d if C(p i, x, out) = * for any i, then C(, x, in) = * C(p i, x, out) = c & C(p j, x, out) = d & d <> c then C(, x, in) = * CS781(Praad) L27OP 21 CS781(Praad) L27OP 22 Rule 3 Rule 4 if C(p i, x, out) = c or # for all i, then C(, x, in) = c if C(p i, x, out) = # for all i, then C(, x, in) = # CS781(Praad) L27OP 23 CS781(Praad) L27OP 24

5 The Other Half Rule 5 Rule 1-4 relate the out of one tatement to the in of the next tatement Now we need rule relating the in of a tatement to the out of the ame tatement C(, x, out) = # if C(, x, in) = # CS781(Praad) L27OP 25 CS781(Praad) L27OP 26 Rule 6 Rule 7 x := c x := f( ) C(x := c, x, out) = c if c i a contant C(x := f( ), x, out) = * CS781(Praad) L27OP 27 CS781(Praad) L27OP 28 Rule 8 An Algorithm y :=... X = a X = a 1. For every entry to the program, et C(, x, in) = * 2. Set C(, x, in) = C(, x, out) = # everywhere ele C(y :=, x, out) = C(y :=, x, in) if x <> y 3. Repeat until all point atify 1-8: Pick not atifying 1-8 and update uing the appropriate rule CS781(Praad) L27OP 29 CS781(Praad) L27OP 30

6 The Value # To undertand why we need #, look at a loop Dicuion Conider the tatement Y := 0 To compute whether X i contant at thi point, we need to know whether X i contant at the two predeceor But info for depend on it predeceor, including Y := 0! CS781(Praad) L27OP 31 CS781(Praad) L27OP 32 The Value # (Cont.) Example Becaue of cycle, all point mut have value at all time Intuitively, aigning ome initial value allow the analyi to break cycle The initial value # mean So far a we know, control never reache thi point CS781(Praad) L27OP 33 CS781(Praad) L27OP 34 Example Example CS781(Praad) L27OP 35 CS781(Praad) L27OP 36

7 Example Ordering We can implify the preentation of the analyi by ordering the value # < c < * Drawing a picture with lower value drawn lower, we get * CS781(Praad) L27OP 37 CS781(Praad) L27OP 38 # Ordering (Cont.) * i the greatet value, # i the leat All contant are in between and incomparable Let lub be the leat-upper bound in thi ordering Rule 1-4 can be written uing lub: C(, x, in) = lub { C(p, x, out) p i a predeceor of } Termination Simply aying repeat until nothing change doen t guarantee that eventually nothing change The ue of lub explain why the algorithm terminate Value tart a # and only increae # can change to a contant, and a contant to * Thu, C(, x, _) can change at mot twice CS781(Praad) L27OP 39 CS781(Praad) L27OP 40 Termination (Cont.) Thu the algorithm i linear in program ize Number of tep = Number of C(.) value computed * 2 = Number of program tatement * 4 Livene Analyi Once contant have been globally propagated, we would like to eliminate dead code After contant propagation, i dead (auming X not ued elewhere) CS781(Praad) L27OP 41 CS781(Praad) L27OP 42

8 Live and Dead Livene The firt value of x i dead (never ued) The econd value of x i live (may be ued) X := 4 A variable x i live at tatement if There exit a tatement that ue x Livene i an important concept Y := X There i a path from to That path ha no intervening aignment to x CS781(Praad) L27OP 43 CS781(Praad) L27OP 44 Global Dead Code Elimination A tatement x := i dead code if x i dead after the aignment Dead tatement can be deleted from the program Computing Livene We can expre livene in term of information tranferred between adjacent tatement, jut a in copy propagation Livene i impler than contant propagation, ince it i a boolean property (true or fale) But we need livene information firt... CS781(Praad) L27OP 45 CS781(Praad) L27OP 46 Livene Rule 1 Livene Rule 2 p X = true X = true X = true E.g., := f(x) L(p, x, out) = { L(, x, in) a ucceor of p } L(, x, in) = true if refer to x on the rh CS781(Praad) L27OP 47 CS781(Praad) L27OP 48

9 Livene Rule 3 Livene Rule 4 x := e X = fale X = a X = a L(x := e, x, in) = fale if e doe not refer to x L(, x, in) = L(, x, out) if doe not refer to x CS781(Praad) L27OP 49 CS781(Praad) L27OP 50 Algorithm 1. Let all L( ) = fale initially 2. Repeat until all tatement atify rule 1-4 Pick where one of 1-4 doe not hold and update uing the appropriate rule Termination A value can change from fale to true, but not the other way around Each value can change only once, o termination i guaranteed Once the analyi i computed, it i imple to eliminate dead code CS781(Praad) L27OP 51 CS781(Praad) L27OP 52 Forward v. Backward Analyi We ve een two kind of analyi: Contant propagation i a forward analyi: information i puhed from input to output Livene i a backward analyi: information i puhed from output back toward input Analyi There are many other global flow analye Mot can be claified a either forward or backward Mot alo follow the methodology of local rule relating information between adjacent program point CS781(Praad) L27OP 53 CS781(Praad) L27OP 54