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CSE 231 : Advanced Compilers Building Program Analyzers
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dataflow analysis
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http://blog.ezyang.com/2011/04/hoopl-dataflow-lattices/
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http://lambda-the-ultimate.org/node/3557
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http://blog.ezyang.com/2011/04/hoopl-dataflow-lattices/ http://lambda-the-ultimate.org/node/3557 http://research.microsoft.com/en- us/um/people/simonpj/papers/c--/dfopt.pdf
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http://blog.ezyang.com/2011/04/hoopl-dataflow-lattices/ http://lambda-the-ultimate.org/node/3557 http://research.microsoft.com/en- us/um/people/simonpj/papers/c--/dfopt.pdf
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Now where were we…
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for edge e in CFG: m[e] = EMPTY for node n in CFG: q.push(n) while not q.empty(): n = q.pop() info_in = m[n.in_edges] info_out = F(n, info_in) for i from 0 to len(info_out): e = n.out_edges[i] new_info = m[e] UNION info_out[i] if m[e] != new_info: m[e] = new_info q.push(e.dest) Started with sets. Termination worries.
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Foundations : Lattices A lattice is (S, ⊑, ⊥, ⊤, ⊔, ⊓ ) where: (S, ⊑ ) is a poset ⊥ is the smallest thing in S ⊤ is the biggest thing in S lub(a, b) and glb(a, b) always exist a ⊔ b = lub(a, b) a ⊓ b = glb(a, b)
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Foundations : Lattices (Formally) A lattice is (S, ⊑, ⊥, ⊤, ⊔, ⊓ ) where: (S, ⊑ ) is a poset ∀ a ∈ S. ⊥ ⊑ a ∀ a ∈ S. a ⊑ ⊤ ∀ a, b ∈ S. ∃ c. c = lub(a, b) /\ a ⊔ b = c ∀ a, b ∈ S. ∃ c. c = glb(a, b) /\ a ⊓ b = c
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Foundations : Fancy Lattice Names ⊥ is “botom” ⊤ is “top” ⊔ is “join” ⊓ is “meet”
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for edge e in CFG: m[e] = BOTTOM for node n in CFG: q.push(n) while not q.empty(): n = q.pop() info_in = m[n.in_edges] info_out = F(n, info_in) for i from 0 to len(info_out): e = n.out_edges[i] new_info = m[e] JOIN info_out[i] if m[e] != new_info: m[e] = new_info q.push(e.dest) Port to lattices. Small patch set.
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while not q.empty(): n = q.pop() info_in = m[n.in_edges] info_out = F(n, info_in) for i from 0 to len(info_out): e = n.out_edges[i] new_info = m[e] JOIN info_out[i] if m[e] != new_info: m[e] = new_info q.push(e.dest) Termination. Finite lattice height implies termination.
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while not q.empty(): n = q.pop() info_in = m[n.in_edges] info_out = F(n, info_in) for i from 0 to len(info_out): e = n.out_edges[i] new_info = m[e] JOIN info_out[i] if m[e] != new_info: m[e] = new_info q.push(e.dest) Termination. But can we do better? Finite lattice height implies termination. Get rid of that JOIN right in the middle?
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while not q.empty(): n = q.pop() info_in = m[n.in_edges] info_out = F(n, info_in) for i from 0 to len(info_out): e = n.out_edges[i] new_info = m[e] JOIN info_out[i] if m[e] != new_info: m[e] = new_info q.push(e.dest) Termination. But can we do better? Get rid of that JOIN right in the middle? Yes. The trick is in the flow functions.
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In general, can’t remove join and have termination. So, when is it OK? Safely Getting Rid Of The Join
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In general, can’t remove join and have termination. So, when is it OK? Port our algorithm to math to figure it out. Build in terms of fixpoints. Safely Getting Rid Of The Join
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Fixpoint: an input equal the output. A fixpoint of F is any X such that F(X) = X. “Best” answer since repeating yields same result. Fixpoints Are Easy
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Using Fixpoints In Analysis Goal: compute map m from CFG edges to dataflow information Strategy: define a global flow function F as follows: F takes a map m as a parameter and returns a new map m’, in which individual local flow functions have been applied
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The Big F F mm’
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Just Regular Flow Funcs Inside mm’ f1 f2 f3
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Goal: Find Fixpoint of F F m F F m’ …
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Fixpoint of F Goal: a fixed point of F, i.e. m where m = F(m) How should we do this?
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Fixpoint of F Goal: a fixed point of F, i.e. m where m = F(m) How should we do this? Let ⊥ be ⊥ lifted to a map: ⊥ = e. ⊥ Compute F( ⊥ ), then F(F( ⊥ )), then F(F(F( ⊥ ))),... until the result doesn’t change
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Fixpoint of F Goal: a fixed point of F, i.e. m where m = F(m) How should we do this? Let ⊥ be ⊥ lifted to a map: ⊥ = e. ⊥ Compute F( ⊥ ), then F(F( ⊥ )), then F(F(F( ⊥ ))),... until the result doesn’t change … but how long could that take ???
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Fixpoint of F : Formal Solution Solution: ⊔ i = 0 Fi(⊥)Fi(⊥)
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Fixpoint of F : Formal Solution Solution: We want F 1 ( ⊥ ) ⊑ F 2 ( ⊥ ) ⊑ F 3 ( ⊥ ) … ⊑ F k ( ⊥ ) Allows us to eliminate the big join. Just require F to be monotonic: ∀ a b, a ⊑ b ➞ F(a) ⊑ F(b) ⊔ i = 0 Fi(⊥)Fi(⊥)
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Back To Termination Solution: if F is monotonic, we have it. Finite lattice height termination w/out joins! OK. But how do we know F is monotonic? F is monotonic if flow functions monotonic.
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Another benefit of monotonicity Suppose Marsians came to earth, and miraculously give you a fixed point of F, call it fp. Then:
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Another benefit of monotonicity Suppose Marsians came to earth, and miraculously give you a fixed point of F, call it fp. Then:
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Another benefit of monotonicity We are computing the least fixed point...
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Recap Let’s do a recap of what we’ve seen so far Started with worklist algorithm for reaching definitions
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Worklist algorithm for reaching defns let m: map from edge to computed value at edge let worklist: work list of nodes for each edge e in CFG do m(e) := ∅ for each node n do worklist.add(n) while (worklist.empty.not) do let n := worklist.remove_any; let info_in := m(n.incoming_edges); let info_out := F(n, info_in); for i := 0.. info_out.length do let new_info := m(n.outgoing_edges[i]) ∪ info_out[i]; if (m(n.outgoing_edges[i]) new_info]) m(n.outgoing_edges[i]) := new_info; worklist.add(n.outgoing_edges[i].dst);
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Generalized algorithm using lattices let m: map from edge to computed value at edge let worklist: work list of nodes for each edge e in CFG do m(e) := ⊥ for each node n do worklist.add(n) while (worklist.empty.not) do let n := worklist.remove_any; let info_in := m(n.incoming_edges); let info_out := F(n, info_in); for i := 0.. info_out.length do let new_info := m(n.outgoing_edges[i]) ⊔ info_out[i]; if (m(n.outgoing_edges[i]) new_info]) m(n.outgoing_edges[i]) := new_info; worklist.add(n.outgoing_edges[i].dst);
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Next step: removed outer join Wanted to remove the outer join, while still providing termination guarantee To do this, we re-expressed our algorithm more formally We first defined a “global” flow function F, and then expressed our algorithm as a fixed point computation
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Guarantees If F is monotonic, don’t need outer join If F is monotonic and height of lattice is finite: iterative algorithm terminates If F is monotonic, the fixed point we find is the least fixed point. Any questions?
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What about if we start at top? What if we start with ⊤, F( ⊤ ), F(F( ⊤ )), F(F(F( ⊤ )))
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What about if we start at top? What if we start with ⊤, F( ⊤ ), F(F( ⊤ )), F(F(F( ⊤ ))) We get the greatest fixed point Why do we prefer the least fixed point? –More precise
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Graphically x y 10
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Graphically x y 10
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Graphically x y 10
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Another example: constant prop Set D = x := N in out F x := N (in) = x := y op z in out F x := y op z (in) =
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Another example: constant prop Set D = 2 { x ! N | x 2 Vars Æ N 2 Z } x := N in out F x := N (in) = in – { x ! * } [ { x ! N } x := y op z in out F x := y op z (in) = in – { x ! * } [ { x ! N | ( y ! N 1 ) 2 in Æ ( z ! N 2 ) 2 in Æ N = N 1 op N 2 }
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Another example: constant prop *x := y in out F *x := y (in) = x := *y in out F x := *y (in) =
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Another example: constant prop *x := y in out F *x := y (in) = in – { z ! * | z 2 may-point(x) } [ { z ! N | z 2 must-point-to(x) Æ y ! N 2 in } [ { z ! N | (y ! N) 2 in Æ (z ! N) 2 in } x := *y in out F x := *y (in) = in – { x ! * } [ { x ! N | 8 z 2 may-point-to(x). (z ! N) 2 in }
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Another example: constant prop x := f(...) in out F x := f(...) (in) = *x := *y + *z in out F *x := *y + *z (in) =
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Another example: constant prop x := f(...) in out F x := f(...) (in) = ; *x := *y + *z in out F *x := *y + *z (in) = F a := *y;b := *z;c := a + b; *x := c (in)
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Another example: constant prop s: if (...) in out[0]out[1] merge out in[0]in[1]
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Another example: constant prop Set D = 2 { x ! N | x 2 Vars Æ N 2 Z } x := N in out F x := N (in) = in – { x ! * } [ { x ! N } x := y op z in out F x := y op z (in) = in – { x ! * } [ { x ! N | ( y ! N 1 ) 2 in Æ ( z ! N 2 ) 2 in Æ N = N 1 op N 2 }
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Another example: constant prop *x := y in out F *x := y (in) = in – { z ! * | z 2 may-point(x) } [ { z ! N | z 2 must-point-to(x) Æ y ! N 2 in } [ { z ! N | (y ! N) 2 in Æ (z ! N) 2 in } x := *y in out F x := *y (in) = in – { x ! * } [ { x ! N | 8 z 2 may-point-to(x). (z ! N) 2 in }
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Another example: constant prop x := f(...) in out F x := f(...) (in) = ; *x := *y + *z in out F *x := *y + *z (in) = F a := *y;b := *z;c := a + b; *x := c (in)
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Another example: constant prop s: if (...) in out[0]out[1] merge out in[0]in[1]
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Lattice (D, v, ?, >, t, u) =
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