Course Outline

1. Course Outline • Traditional Static Program Analysis • Theory • Compiler Optimizations; Control Flow Graphs, • Data-flow Analysis • Data-flow Frameworks --- today’s class • Specific Analyses, Applications, etc. • Software Testing • Dynamic Program Analysis

2. Announcement • Handout • Homework 1, due February 17th

3. Outline • The four classical data-flow problems, continue • Solving data-flow problems • Data-flow frameworks • Reading: Compilers: Principles, Techniques and Tools, by Aho, Lam, Sethi and Ullman, Chapter 9.2 and 9.3

4. Dataflow Problems

5. Similarities • There is a finite set Uof dataflow facts: • Reaching Definitions: the set of all definitions in program • Live Uses of Variables: the set of all variables • Available Expressions and Very Busy Expressions: the set of all expressions in program • The solution at a program point i (i.e., in(i), out(i)) is a subset of U (e.g., for each definition it either reaches program point i or does not).

6. Similarities • Dataflow equations are of the form: out(i) = (in(i)-kill(i)) gen(i) • Dataflow equations are transfer functions: • Transfer function Fi takes in(i) and computes the out(i): out(i) = Fi(in(i))

7. The Worklist Algorithm /* initially all inRD sets are empty */ for m := 2 to n do inRD(m) := Ø; inRD(1) = UNDEF W := {1,2,…,n} /* put every node on the worklist */ while W ≠ Ø do { remove k from W; new = { inRD(m) pres(m) gen(m) }; if new ≠ inRD (k) then { inRD (k) = new; for j succ(k) do add j to W } out(m) or Fm(in(m)

8. Dataflow Frameworks • Lattices • Partial ordering • Meet, Join, Lattice, and Chain • Monotone functions • The “Maximal Fixed Point” (MFP) solution • The “Meet Over all Paths” (MOP) solution

9. Lattice Theory • Partial ordering (denoted by ≤ or ) • Relation between pairs of elements • Reflexive x ≤ x • Anti-symmetric x ≤ y, y ≤ x implies x=y • Transitive x ≤ y, y ≤ z implies x ≤ z • Poset (set S, ≤) • 0 Element 0 ≤ x, for every x in S • 1 Element x ≤ 1, for every x in S We don’t necessarily need 0 and 1 element.

10. Poset Example {a,b,c} U = {a,b,c}The poset is 2U, ≤ is set inclusion {a,b} {b,c} {a,c} {a} {b} {c} {}

11. Lattice Theory • Greatest lower bound (glb) l1,l2 in poset S, a in poset S is the glb(l1,l2) If a ≤ l1 and a ≤ l2 then for any b in S, b ≤ l1, b ≤ l2 implies b ≤ a If glb exists, it is unique. Why? It is called the meet (denoted by Λ or┌┐) of l1 and l2. • Least upper bound (lub) l1, l2 in poset S, c in poset S is the lub(l1,l2) If c ≥ l1 and c ≥ l2 then for any d in S, d ≥ l1, d ≥ l2 implies d ≥ c If lub exists, it is unique. It is called the join (denoted by V or└┘) of l1 and l2.

12. Definition of a Lattice (L, Λ, V) • L, a poset under ≤ such that every pair of elements has a glb (meet) and lub (join) • A lattice need not contain a 0 or 1 element • A finite lattice must contain 0 and 1 elements • Not every poset is a lattice • If a ≤ x for every x in L, then a is the 0 element of L • If x ≤ a for every x in L, then a is the 1 element of L

13. 5 A poset but not a lattice 4 3 1 2 0 There is no lub(3,4) in this poset so it is not a lattice. Even if we put a lub(3,4), is it going to be a lattice?

14. Examples of Lattices • H = (2U, ∩, U) where U is a finite set • glb(s1,s2) is (s1Λs2) which is s1∩s2 • lub(s1,s2) is (s1Vs2) which is s1Us2 • J = (N1, gcd, lcm) • Partial order is integer divide on N1 • lub(n1,n2) is (n1Vn2) which is lcm(n1,n2) • glb(n1,n2) is (n1Λn2) which is gcd(n1,n2)

15. Chain • A poset C where for every pair of elements c1, c2 in C, either c1 ≤ c2 or c2 ≤ c1. • E.g., {} ≤ {a} ≤ {a,b} ≤ {a,b,c} And from the lattice J as shown here, 1 ≤ 2 ≤ 6 ≤ 30 1 ≤ 3 ≤ 15 ≤ 30 30 6 15 10 Lattices are used in dataflow analysis to reason about the solution obtainable through fixed-point iteration. 2 5 3 1

16. Dataflow Lattices: Reaching Definitions U = all definitions:{(x,1),(x,4),(a,3)}The poset is 2U, ≤ is the subset relation {(x,1),(x,4),(a,3)} 1 1. x:=a*b 2. if y<=a*b {(x,1),(x,4)} {(x,4),(a,3)} {(x,1),(a,3)} 3. a:=a+1 {(x,1)} {(x,4)} {(a,3)} 4. x:=a*b 5. goto 3 0 {}

17. Dataflow Lattices: Available Expressions U = all expressions: {(a*b),(a+1),(y*z)}The poset is 2U, ≤ is the superset relation {} 1 1. x:=a*b 2. if y*z<=a*b {(a*b)} {(a+1)} {(y*z)} 3. a:=a+1 {(a*b),(y*z)} {(a*b),(a+1)} {(a+1),(y*z)} 4. x:=a*b 5. goto 2 0 {(a*b),(a+1),(y*z)}

18. Monotone Dataflow Frameworks • Framework parameters in(i)= V out(j) out(i)=Fi(in(i)) where: • in(i), out(i) are elements of a property space • combination operator Vis U for the may problems and ∩for the must problems • Fiis the transfer function associated with node i • 2 other parameters: the set of initial/final CFG nodes, and the initial analysis information at them! j in pred(i)

19. Monotone Frameworks (cont.) • The property space: • A complete lattice (L, ≤ ) • L satisfies the Ascending Chain Condition (i.e., all ascending chains are finite) • The combination operator: V, and lub(Y) = V Y • Reaching Definitions: L = P(Var i), and ≤ is set inclusion.Thus,Vis U. The lattice has finite height, therefore it satisfies the ACC. • Available Expressions: What is (L, ≤)? What is V? ACC? • Live Uses: What is (L, ≤)? What is V? ACC?

20. Monotone Frameworks (cont.) • The transfer functions: Fi: L L. Formally, there is space F such that • F contains all Fi, • F contains the identity function id(x) = x • F is closed under composition. • Each Fiis monotone.

21. Monotonicity • It is defined as (1) a ≤ b f(a) ≤ f(b) • An equivalent definitions is (2) f(x) V f(y) ≤ f(x V y) • Lemma: The two definitions are equivalent. First, we show that (1) implies (2). Second, we show that (2) implies (1).

22. Distributivity • A distributive framework: A monotone framework with distributive transfer functions: f(xV y) = f(x)Vf(y).