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20101017 program analysis_for_security_livshits_lecture02_compilers


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20101017 program analysis_for_security_livshits_lecture02_compilers

  1. 1. Introduction to Compilers<br />Ben Livshits<br />Based in part of Stanford class slides from <br /><br />
  2. 2. Organization<br />Really basic stuff<br />Flow Graphs<br />Constant Folding<br />Global Common Subexpressions<br />Induction Variables/Reduction in Strength<br />Data-flow analysis<br />Proving Little Theorems<br />Data-Flow Equations<br />Major Examples<br />Pointer analysis<br />
  3. 3. Compiler Organization<br />
  4. 4. Dataflow Analysis Basics<br />L2: <br />Compiler Organization<br />Dataflow analysis basics<br />L3:<br />Dataflow lattices <br />Integrative dataflow solution<br />Gen/kill frameworks<br />
  5. 5. Pointer Analysis<br />L10:<br />Pointer analysis<br />L11<br />Pointer analysis and bddbddb<br />
  6. 6. 6<br />Really Basic Stuff<br /><ul><li>Flow Graphs
  7. 7. Constant Folding
  8. 8. Global Common Subexpressions
  9. 9. Induction Variables/Reduction in Strength</li></li></ul><li>7<br />Dawn of Code Optimization<br />A never-published Stanford technical report by Fran Allen in 1968<br />Flow graphs of intermediate code<br />Key things worth doing<br />
  10. 10. 8<br />Intermediate Code<br />for (i=0; i<n; i++)<br /> A[i] = 1;<br />Intermediate code exposes optimizable constructs we cannot see at source-code level.<br />Make flow explicit by breaking into basic blocks = sequences of steps with entry at beginning, exit at end.<br />
  11. 11. 9<br />i = 0<br />if i>=n goto …<br />t1 = 8*i <br />A[t1] = 1<br />i = i+1 <br />Basic Blocks <br />for (i=0; i<n; i++)<br /> A[i] = 1;<br />
  12. 12. 10<br />Induction Variables<br />x is an induction variable in a loop if it takes on a linear sequence of values each time through the loop.<br />Common case: loop index like i and computed array index like t1.<br />Eliminate “superfluous” induction variables.<br />Replace multiplication by addition (reduction in strength ). <br />
  13. 13. 11<br />Example<br />i = 0<br />if i>=n goto …<br />t1 = 8*i <br />A[t1] = 1<br />i = i+1 <br />t1 = 0 <br />n1 = 8*n<br />if t1>=n1 goto …<br />A[t1] = 1 <br />t1 = t1+8 <br />
  14. 14. 12<br />Loop-Invariant Code Motion<br />Sometimes, a computation is done each time around a loop.<br />Move it before the loop to save n-1 computations.<br />Be careful: could n=0? I.e., the loop is typically executed 0 times.<br />
  15. 15. 13<br />Example<br />i = 0<br />i = 0 <br />t1 = y+z<br />if i>=n goto …<br />if i>=n goto …<br />t1 = y+z <br />x = x+t1 <br />i = i+1 <br />x = x+t1<br />i = i+1 <br />
  16. 16. 14<br />Constant Folding<br />Sometimes a variable has a known constant value at a point.<br />If so, replacing the variable by the constant simplifies and speeds-up the code.<br />Easy within a basic block; harder across blocks.<br />
  17. 17. 15<br />Example<br />i = 0 <br />n = 100<br />if i>=n goto …<br />t1 = 8*i <br />A[t1] = 1<br />i = i+1 <br />t1 = 0 <br />if t1>=800 goto …<br />A[t1] = 1 <br />t1 = t1+8 <br />
  18. 18. 16<br />Global Common Subexpressions<br />Suppose block B has a computation of x+y.<br />Suppose we are sure that when we reach this computation, we are sure to have:<br />Computed x+y, and<br />Not subsequently reassigned x or y.<br />Then we can hold the value of x+y and use it in B.<br />
  19. 19. 17<br />Example<br />a = x+y <br />t = x+y<br />a = t <br />b = x+y <br />t = x+y<br />b = t <br />c = x+y <br />c = t <br />
  20. 20. 18<br />Example --- Even Better<br />t = x+y<br />a = t <br />t = x+y<br />b = t <br />c = t <br />t = x+y<br />a = t <br />b = t <br />t = x+y<br />b = t <br />c = t <br />
  21. 21. 19<br />Data-Flow Analysis<br /><ul><li>Proving Little Theorems
  22. 22. Data-Flow Equations
  23. 23. Major Examples</li></li></ul><li>20<br />An Obvious Theorem<br />boolean x = true;<br />while (x) {<br /> . . . // no change to x<br />}<br />Doesn’t terminate.<br />Proof: only assignment to x is at top, so x is always true.<br />
  24. 24. 21<br />As a Flow Graph<br />x = true<br />if x == true<br />“body”<br />
  25. 25. 22<br />Formulation: Reaching Definitions<br />Each place some variable x is assigned is a definition.<br />Ask: for this use of x, where could x last have been defined.<br />In our example: only at x=true.<br />
  26. 26. 23<br />d1<br />d2<br />Example: Reaching Definitions<br />d1: x = true<br />d1<br />if x == true<br />d2<br />d1<br />d2: a = 10<br />
  27. 27. 24<br />Clincher<br />Since at x == true, d1 is the only definition of x that reaches, it must be that x is true at that point.<br />The conditional is not really a conditional and can be replaced by a branch.<br />
  28. 28. 25<br />Not Always That Easy<br />int i = 2; int j = 3;<br />while (i != j) {<br /> if (i < j) i += 2;<br /> else j += 2;<br />}<br />We’ll develop techniques for this problem, but later …<br />
  29. 29. 26<br />d1<br />d2<br />d3<br />d4<br />d2, d3, d4<br />d1, d3, d4<br />d1, d2, d3, d4<br />d1, d2, d3, d4<br />The Flow Graph<br />d1: i = 2<br />d2: j = 3<br />if i != j<br />d1, d2, d3, d4<br />if i < j<br />d4: j = j+2<br />d3: i = i+2<br />
  30. 30. 27<br />DFA Is Sometimes Insufficient<br />In this example, i can be defined in two places, and j in two places.<br />No obvious way to discover that i!=j is always true.<br />But OK, because reaching definitions is sufficient to catch most opportunities for constant folding (replacement of a variable by its only possible value).<br />
  31. 31. 28<br />Be Conservative!<br />(Code optimization only)<br />It’s OK to discover a subset of the opportunities to make some code-improving transformation.<br />It’s notOK to think you have an opportunity that you don’t really have.<br />
  32. 32. 29<br />Example: Be Conservative<br />boolean x = true;<br />while (x) {<br /> . . . *p = false; . . .<br />}<br />Is it possible that p points to x?<br />
  33. 33. 30<br />Another<br />def of x<br />d2<br />As a Flow Graph<br />d1: x = true<br />d1<br />if x == true<br />d2: *p = false<br />
  34. 34. 31<br />Possible Resolution<br />Just as data-flow analysis of “reaching definitions” can tell what definitions of x might reach a point, another DFA can eliminate cases where p definitely does not point to x.<br />Example: the only definition of p is p = &y and there is no possibility that y is an alias of x.<br />
  35. 35. 32<br />Reaching Definitions Formalized<br />A definition d of a variable x is said to reach a point p in a flow graph if:<br />Every path from the entry of the flow graph to p has d on the path, and<br />After the last occurrence of d there is no possibility that x is redefined.<br />
  36. 36. 33<br />Data-Flow Equations --- (1)<br />A basic block can generate a definition.<br />A basic block can either<br />Kill a definition of x if it surely redefines x.<br />Transmit a definition if it may not redefine the same variable(s) as that definition.<br />
  37. 37. 34<br />Data-Flow Equations --- (2)<br />Variables:<br />IN(B) = set of definitions reaching the beginning of block B.<br />OUT(B) = set of definitions reaching the end of B.<br />
  38. 38. 35<br />Data-Flow Equations --- (3)<br />Two kinds of equations:<br />Confluence equations : IN(B) in terms of outs of predecessors of B.<br />Transfer equations : OUT(B) in terms of of IN(B) and what goes on in block B. <br />
  39. 39. 36<br />Confluence Equations<br />IN(B) = ∪predecessors P of B OUT(P)<br />{d2, d3}<br />{d1, d2}<br />P2<br />P1<br />{d1, d2, d3}<br />B<br />
  40. 40. 37<br />Transfer Equations<br />Generate a definition in the block if its variable is not definitely rewritten later in the basic block.<br />Kill a definition if its variable is definitely rewritten in the block.<br />An internal definition may be both killed and generated.<br />
  41. 41. 38<br />Example: Gen and Kill<br />IN = {d2(x), d3(y), d3(z), d5(y), d6(y), d7(z)} <br />d1: y = 3 <br />d2: x = y+z<br />d3: *p = 10<br />d4: y = 5 <br />Kill includes {d1(x), d2(x),<br />d3(y), d5(y), d6(y),…} <br />Gen = {d2(x), d3(x),<br /> d3(z),…, d4(y)} <br />OUT = {d2(x), d3(x), d3(z),…, d4(y), d7(z)} <br />
  42. 42. 39<br />Transfer Function for a Block<br />For any block B:<br />OUT(B) = (IN(B) – Kill(B)) ∪Gen(B)<br />
  43. 43. 40<br />Iterative Solution to Equations<br />For an n-block flow graph, there are 2n equations in 2n unknowns.<br />Alas, the solution is not unique.<br />Use iterative solution to get the least fixed-point.<br />Identifies any def that might reach a point.<br />
  44. 44. 41<br />Iterative Solution --- (2)<br />IN(entry) = ∅;<br />for each block B do OUT(B)= ∅;<br />while (changes occur) do<br /> for each block B do {<br />IN(B) = ∪predecessors P of B OUT(P);<br /> OUT(B) = (IN(B) – Kill(B)) ∪Gen(B);<br /> }<br />
  45. 45. 42<br />IN(B1) = {}<br />OUT(B1) = {<br />IN(B2) = {<br />d1,<br />OUT(B2) = {<br />IN(B3) = {<br />d1,<br />OUT(B3) = {<br />Example: Reaching Definitions<br />d1: x = 5<br />B1<br />d1}<br />d2}<br />if x == 10<br />B2<br />d1,<br />d2}<br />d2}<br />d2: x = 15<br />B3<br />d2}<br />
  46. 46. 43<br />Aside: Notice the Conservatism<br />Not only the most conservative assumption about when a def is killed or gen’d.<br />Also the conservative assumption that any path in the flow graph can actually be taken.<br />
  47. 47. 44<br />Everything Else About Data Flow Analysis<br /><ul><li>Flow- and Context-Sensitivity Logical Representation
  48. 48. Pointer Analysis
  49. 49. Interprocedural Analysis</li></li></ul><li>45<br />Three Levels of Sensitivity<br />In DFA so far, we have cared about where in the program we are.<br />Called flow-sensitivity.<br />But we didn’t care how we got there.<br />Called context-sensitivity.<br />We could even care about neither.<br />Example: where could x ever be defined in this program?<br />
  50. 50. 46<br />Flow/Context Insensitivity<br />Not so bad when program units are small (few assignments to any variable).<br />Example: Java code often consists of many small methods.<br />Remember: you can distinguish variables by their full name, e.g., class.method.block.identifier.<br />
  51. 51. 47<br />Context Sensitivity<br />Can distinguish paths to a given point.<br />Example: If we remembered paths, we would not have the problem in the constant-propagation framework where x+y = 5 but neither x nor y is constant over all paths.<br />
  52. 52. 48<br />The Example Again<br />x = 3<br />y = 2<br />x = 2<br />y = 3<br />z = x+y<br />
  53. 53. 49<br />An Interprocedural Example<br />int id(int x) {return x;}<br />void p() {a=2; b=id(a);…}<br />void q() {c=3; d=id(c);…}<br />If we distinguish p calling id from q calling id, then we can discover b=2 and d=3.<br />Otherwise, we think b, d = {2, 3}.<br />
  54. 54. 50<br />Context-Sensitivity --- (2)<br />Loops and recursive calls lead to an infinite number of contexts.<br />Generally used only for interprocedural analysis, so forget about loops.<br />Need to collapse strong components of the calling graph to a single group.<br />“Context” becomes the sequence of groups on the calling stack.<br />
  55. 55. 51<br />Example: Calling Graph<br />t<br />Contexts:<br />Green<br />Green, pink<br />Green, yellow<br />Green, pink, yellow<br />s<br />r<br />p<br />q<br />main<br />
  56. 56. 52<br />Comparative Complexity<br />Insensitive: proportional to size of program (number of variables).<br />Flow-Sensitive: size of program, squared (points times variables).<br />Context-Sensitive: worst-case exponential in program size (acyclic paths through the code).<br />
  57. 57. 53<br />Logical Representation<br />We have used a set-theoretic formulation of DFA.<br />IN = set of definitions, e.g.<br />There has been recent success with a logical formulation, involving predicates.<br />Example: Reach(d,x,i) = “definition d of variable x can reach point i.”<br />
  58. 58. 54<br />Comparison: Sets Vs. Logic<br />Both have an efficiency enhancement.<br />Sets: bit vectors and boolean ops.<br />Logic: BDD’s, incremental evaluation.<br />Logic allows integration of different aspects of a flow problem.<br />Think of PRE as an example. We needed 6 stages to compute what we wanted.<br />
  59. 59. 55<br />Datalog --- (1)<br />Predicate<br />Arguments:<br />variables or constants<br />The body :<br />For each assignment of values<br />to variables that makes all these<br />true …<br />Make this<br />atom true<br />(the head ).<br />Atom = Reach(d,x,i)<br />Literal = Atom or NOT Atom<br />Rule = Atom :- Literal & … & Literal<br />
  60. 60. 56<br />Example: Datalog Rules<br />Reach(d,x,j) :- Reach(d,x,i) &<br /> StatementAt(i,s) &<br /> NOT Assign(s,x) &<br /> Follows(i,j)<br />Reach(s,x,j) :- StatementAt(i,s) &<br /> Assign(s,x) &<br /> Follows(i,j)<br />
  61. 61. 57<br />Datalog --- (2)<br />Intuition: subgoals in the body are combined by “and” (strictly speaking: “join”).<br />Intuition: Multiple rules for a predicate (head) are combined by “or.”<br />
  62. 62. 58<br />Datalog --- (3)<br />Predicates can be implemented by relations (as in a database).<br />Each tuple, or assignment of values to the arguments, also represents a propositional (boolean) variable.<br />
  63. 63. 59<br />Iterative Algorithm for Datalog<br />Start with the EDB predicates = “whatever the code dictates,” and with all IDB predicates empty.<br />Repeatedly examine the bodies of the rules, and see what new IDB facts can be discovered from the EDB and existing IDB facts.<br />
  64. 64. 60<br />Example: Seminaive<br />Path(x,y) :- Arc(x,y)<br />Path(x,y) :- Path(x,z) & Path(z,y)<br />NewPath(x,y) = Arc(x,y); Path(x,y) = ∅;<br />while (NewPath != ∅) do {<br /> NewPath(x,y) = {(x,y) | NewPath(x,z)<br /> && Path(z,y) || Path(x,z) &&<br /> NewPath(z,y)} – Path(x,y);<br /> Path(x,y) = Path(x,y) ∪ NewPath(x,y);<br />}<br />
  65. 65. Pointer analysis<br />61<br />
  66. 66. 62<br />New Topic: Pointer Analysis<br />We shall consider Andersen’s formulation of Java object references.<br />Flow/context insensitive analysis.<br />Cast of characters:<br />Local variables, which point to:<br />Heap objects, which may have fields that are references to other heap objects.<br />
  67. 67. 63<br />Representing Heap Objects<br />A heap object is named by the statement in which it is created.<br />Note many run-time objects may have the same name.<br />Example: h: T v = new T;says variable v can point to (one of) the heap object(s) created by statement h.<br />v<br />h<br />
  68. 68. 64<br />Other Relevant Statements<br />v.f = w makes the f field of the heap object h pointed to by v point to what variable w points to.<br />f<br />v<br />w<br />f<br />h<br />g<br />i<br />
  69. 69. 65<br />Other Statements --- (2)<br />v = w.f makes v point to what the f field of the heap object h pointed to by w points to.<br />v<br />w<br />i<br />f<br />h<br />g<br />
  70. 70. 66<br />Other Statements --- (3)<br />v = w makes v point to whatever w points to.<br />Interprocedural Analysis : Also models copying an actual parameter to the corresponding formal or return value to a variable.<br />v<br />w<br />h<br />
  71. 71. 67<br />Datalog Rules<br />Pts(V,H) :- “H: V = new T”<br />Pts(V,H) :- “V=W” & Pts(W,H)<br />Pts(V,H) :- “V=W.F” & Pts(W,G) & Hpts(G,F,H)<br />Hpts(H,F,G) :- “V.F=W” & Pts(V,H) & Pts(W,G)<br />
  72. 72. 68<br />Example<br />T p(T x) {<br /> h: T a = new T;<br /> a.f = x;<br /> return a;<br />}<br />void main() {<br /> g: T b = new T;<br /> b = p(b);<br /> b = b.f;<br />}<br />
  73. 73. 69<br />Apply Rules Recursively --- Round 1<br />Pts(a,h)<br />Pts(b,g)<br />T p(T x) {h: T a = new T;<br /> a.f = x; return a;}<br />void main() {g: T b = new T;<br /> b = p(b); b = b.f;}<br />
  74. 74. 70<br />Apply Rules Recursively --- Round 2<br />Pts(x,g)<br />Pts(b,h)<br />T p(T x) {h: T a = new T;<br /> a.f = x; return a;}<br />void main() {g: T b = new T;<br /> b = p(b); b = b.f;}<br />Pts(a,h)<br />Pts(b,g)<br />
  75. 75. 71<br />Apply Rules Recursively --- Round 3<br />Hpts(h,f,g)<br />Pts(x,h)<br />T p(T x) {h: T a = new T;<br /> a.f = x; return a;}<br />void main() {g: T b = new T;<br /> b = p(b); b = b.f;}<br />Pts(a,h)<br />Pts(b,g)<br />Pts(x,g)<br />Pts(b,h)<br />
  76. 76. 72<br />Apply Rules Recursively --- Round 4<br />Hpts(h,f,h)<br />T p(T x) {h: T a = new T;<br /> a.f = x; return a;}<br />void main() {g: T b = new T;<br /> b = p(b); b = b.f;}<br />Pts(a,h)<br />Pts(b,g)<br />Pts(x,g)<br />Pts(b,h)<br />Pts(x,h)<br />Hpts(h,f,g)<br />
  77. 77. 73<br />Adding Context Sensitivity<br />Include a component C = context.<br />C doesn’t change within a function.<br />Call and return can extend the context if the called function is not mutually recursive with the caller.<br />
  78. 78. 74<br />Example of Rules: Context Sensitive<br />Pts(V,H,B,I+1,C) :- “B,I: V=W” & Pts(W,H,B,I,C)<br />Pts(X,H,B0,0,D) :- Pts(V,H,B,I,C) & “B,I: call P(…,V,…)” & “X is the corresponding actual to V in P” & “B0 is the entry of P” & “context D is C extended by P”<br />