Equailty Saturation for Translation Validation

Xingyu Xie

References

(POPL'09) Ross Tate, Michael Stepp, Zachary Tatlock, and Sorin Lerner. Equality Saturation: A New Approach to Optimization.
(CAV'11) Michael Stepp, Ross Tate, and Sorin Lerner. Equality-based translation validator for LLVM. (tool)

Approach

CFG ⇒ PEG (Program Expression Graph)
saturation: PEG ⇒ E-PEG
check: PEG ∈ E-PEG

Program Experssion Graph

operator node & dataflow edge

referentially transparent: value of an expression depends only on its constituent expressions, no side effect

complete: any source program could be represented as PEG

Loop in PEG

θ(init, v): the sequence of values that the variable takes throughout the loop.
φ(cond, t, f): gated SSA
eval(seq, n): the n-th element of the sequence
pass(cond): the index of the first true element in the sequence

≥

Validation Algorithm

function validate(cfg1, cfg2: CFG): CFG
  let peg1 = ConvertToPEG(cfg1)
  let peg2 = ConvertToPEG(cfg2)
  // A is the set of axioms
  let epeg = Saturate(peg1, A)
  return isInstantiation(peg2, epeg)

E-PEG

E-PEG is PEG with a set of equalities over nodes of PEG.

Ax1

Ax2

Ax3

Ax2

Ax3

Ax1

Ax1

Saturation

saturation: repeatedly apply axioms to infer new equalities, until no axiom could be applied or given bounds are reached.

Formalization

Saturation

Saturation

Axiom application:

Partial order (subset):

Axiom a is monotonic:

Normal form

Generally, there maybe not a normal form such as

axiom A = (A + 1) - 1
the inlining axiom applied to a recursive function

When the normal form doesn't exist, saturation may not terminate. Thus, we bound the number of times of same application.

My thought: there must be a normal sequence , where , s.t.

We define ir2 to be a normal form of ir1 if:

ir_1 \xrightarrow{*} ir_2 \land \neg (\exists ir_3 \exists a . ir_2 \xrightarrow{a} ir_3)

\{ ir_n \}

ir_{n - 1} \to ir_n

\forall g \exists n \ge 1 , g \xrightarrow{*} ir_n

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By Xingyu Xie

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Xingyu Xie