Provably Correct Compilers (Part 2) Nazrul Alam and Krishnaprasad Vikram April 21, 2005

Automated Soundness Proofs for Dataflow Analyses and Transformations via Local Rules (Rhodium Paper) Sorin Lerner, Todd Millstein, Erika Rice, Craig Chambers Today’s Focus…

Rhodium: Successor of Cobalt Increased expressiveness –New model for expressing opts: local propagation rules with explicit dataflow facts –Heap summaries –Infinite analysis domains –Flow-sensitive and -insensitive –Intraprocedural and interprocedural Some Rhodium opts not expressible in Cobalt: –Arithmetic invariant detection, integer range analysis, loop-induction-variable strength reduction, Andersen's may-point-to analysis with allocation-site summaries

Recap from Last Class Optimization needs analysis. Each analysis is formulated as a global path condition in Cobalt. In Rhodium it is done with local rules.

Similarities with Cobalt… (Also helpful for refreshing your memory..:)

Checker Written by programmer Given Rhodium Execution engine Rdm Opt Rdm Opt Rdm Opt

Checker Written by programmer Given Rhodium Execution engine Rdm Opt Rdm Opt Rdm Opt

Rdm Opt Rdm Opt Rdm Opt Checker

Exec Compiler Rhodium Execution engine Rdm Opt Rdm Opt Rdm Opt if (…) { x := …; } else { y := …; } …; Checker

Verification Task Automatic Theorem Prover Rdm Opt Verification Task Checker Show that for any original program: behavior of original program = behavior of optimized program Verification Task

Automatic Theorem Prover Rdm Opt Verification Task

Automatic Theorem Prover Rdm Opt Verification Task

Three techniques to simplify Verification Task 1.Rhodium is declarative no loops, no branches, no program counter declare intent using rules execution engine takes care of the rest Automatic Theorem Prover Rdm Opt

Three techniques to simplify Verification Task 1.Rhodium is declarative no loops, no branches, no program counter declare intent using rules execution engine takes care of the rest Automatic Theorem Prover Rdm Opt

Three techniques to simplify Verification Task 1.Rhodium is declarative 2.Factor out heuristics –legal transformations –vs. profitable transformations Automatic Theorem Prover Rdm Opt Heuristics not affecting correctness Part that must be reasoned about

Three techniques to simplify Verification Task Automatic Theorem Prover 1.Rhodium is declarative 2.Factor out heuristics –legal transformations –vs. profitable transformations Heuristics not affecting correctness Part that must be reasoned about

Three techniques to simplify Verification Task 1.Rhodium is declarative 2.Factor out heuristics 3.Split verification task –opt-dependent –vs. opt-independent Automatic Theorem Prover opt- dependent opt- independent

Three techniques to simplify Verification Task 1.Rhodium is declarative 2.Factor out heuristics 3.Split verification task –opt-dependent –vs. opt-independent Automatic Theorem Prover

Three techniques to simplify Verification Task Automatic Theorem Prover 1.Rhodium is declarative 2.Factor out heuristics 3.Split verification task –opt-dependent –vs. opt-independent

Three techniques to simplify Verification Task Automatic Theorem Prover 1.Rhodium is declarative 2.Factor out heuristics 3.Split verification task Result: Expressive language Automated correctness checking

Where is the difference? Rohodium’s local rules are different from Cobalt’s global condition Then how exactly Rhodium works?

MustPointTo analysis c = a a = &b d = *c ab c ab d = b

MustPointTo info in Rhodium c = a a = &b mustPointTo ( a, b ) c ab mustPointTo ( c, b ) ab d = *c

MustPointTo info in Rhodium c = a a = &b d = *c c ab mustPointTo ( a, b ) mustPointTo ( c, b ) c = a a = &b d = *c c ab mustPointTo ( a, b ) mustPointTo ( c, b ) mustPointTo ( a, b ) abab

MustPointTo info in Rhodium define fact mustPointTo(X:Var,Y:Var) with meaning σ( X)== σ( &Y) c = a a = &b d = *c c ab mustPointTo ( a, b ) mustPointTo ( c, b ) mustPointTo ( a, b ) ab Fact correct on edge if: whenever program execution reaches edge, meaning of fact evaluates to true in the program state

Propagating facts c = a a = &b d = *c c ab mustPointTo ( a, b ) mustPointTo ( c, b ) define fact mustPointTo(X:Var,Y:Var) with meaning σ( X ) == σ( &Y ) mustPointTo ( a, b ) ab

a = &b if currStmt == [X = &Y] then Propagating facts c = a a = &b d = *c c ab mustPointTo ( a, b ) mustPointTo ( c, b ) if currStmt == [X = &Y] then define fact mustPointTo(X:Var,Y:Var) with meaning σ( X) == σ( &Y) mustPointTo ( a, b ) ab

if currStmt == [X = &Y] then Propagating facts c = a a = &b d = *c c ab mustPointTo ( a, b ) mustPointTo ( c, b ) define fact mustPointTo(X:Var,Y:Var) with meaning σ( X) == σ(&Y) mustPointTo ( a, b ) ab

c = a Propagating facts a = &b d = *c c ab mustPointTo ( a, b ) mustPointTo ( c, b ) if  currStmt == [Z = X] then mustPointTo ( c, b ) define fact mustPointTo(X:Var,Y:Var) with meaning σ( X) == σ(&Y) if currStmt == [X = &Y] then mustPointTo ( a, b ) ab

Propagating facts c = a a = &b d = *c c ab mustPointTo ( a, b ) mustPointTo ( c, b ) define fact mustPointTo(X:Var,Y:Var) with meaning σ( X ) == σ( &Y ) if currStmt == [X = &Y] then if  currStmt == [Z = X] then mustPointTo ( a, b ) ab

d = *c Transformations c = a a = &b c ab mustPointTo ( a, b ) mustPointTo ( c, b ) if  currStmt == [Z = *X] then transform to [Z = Y] mustPointTo ( c, b ) d = b define fact mustPointTo(X:Var,Y:Var) with meaning σ( X )== σ( &Y ) if currStmt == [X = &Y] then if  currStmt == [Z = X] then mustPointTo ( a, b ) ab

d = *c Transformations c = a a = &b c ab mustPointTo ( a, b ) mustPointTo ( c, b ) if  currStmt == [Z = *X] then transform to [Z = Y] d = b define fact mustPointTo(X:Var,Y:Var) with meaning σ( X )== σ( &Y ) if currStmt == [X = &Y] then if  currStmt == [Z = X] then mustPointTo ( a, b ) ab

Semantics of a Rhodium opt Run propagation rules in a loop until there are no more changes (optimistic iterative analysis) Then run transformation rules Then run profitability heuristics For better precision, combine propagation rules and transformations rules.

Rhodium is more expressive

Cobalt: Simple Pointer Analysis

Rhodium version..

And Rhodium can do more… …..Can not be expressed in Cobalt

Arithmetic Simplification Optimization

Arithmetic Simplification Optimization (Cont.)

Loop induction-variable strength reduction

Loop induction-variable strength reduction (cont.)

Checking Rhodium optimizations

Exec Compiler Rhodium Execution engine Rdm Opt Rdm Opt if (…) { x := …; } else { y := …; } …; Checker Rhodium correctness checker Rdm Opt Checker

Rhodium correctness checker Checker Rdm Opt

Checker Rhodium correctness checker Automatic theorem prover Rdm Opt Checker

Rhodium correctness checker Automatic theorem prover define fact … if … then transform … if … then … Checker Profitability heuristics Rhodium optimization

Rhodium correctness checker Automatic theorem prover Rhodium optimization define fact … if … then transform … if … then … Checker

Rhodium correctness checker Automatic theorem prover Rhodium optimization define fact … VCGen Local VC Lemma For any Rhodium opt: If Local VCs are true Then opt is correct Proof   «  ¬   $  \ r t  l Checker Opt- dependent Opt- independent VCGen if … then … if … then transform …

Local correctness of prop. rules currStmt == [Z = X] then if  define fact mustPointTo(X,Y) with meaning σ( X) == σ(&Y ) Fact correct on edge iff: whenever program execution reaches edge, meaning of fact evaluates to true in the program state

Local correctness of prop. rules currStmt == [Z = X] then Local VC (generated and proven automatically) if  define fact mustPointTo(X,Y) with meaning σ( X)==σ(&Y ) Assume: Propagated fact is correct Show: All incoming facts are correct Show: Z == &Y (  out ) X == &Y (  in )   out  = step (  in, [Z = X] ) Assume:

Local correctness of prop. rules Show: Z == &Y (  out ) X == &Y (  in )   out  = step (  in, [Z = X] ) Assume: Local VC (generated and proven automatically) define fact mustPointTo(X,Y) with meaning X == &Y currStmt == [Z = X] then if 

Evaluation

Dimensions of evaluation Correctness guarantees Usefulness of the checker Expressiveness

Correctness guarantees Once checked, optimizations are guaranteed to be correct Caveat: trusted computing base –execution engine –checker implementation –proofs done by hand once by Lerner. Adding a new optimization does not increase the size of the trusted computing base Ease of use Guarantees Usefulness Expressiveness

Usefulness of the checker Found subtle bugs in Lerner’s initial implementation of various optimizations define fact equals(X:Var, E:Expr) with meaning σ( X) == σ(E) if currStmt == [X = E] then x := x + 1x = x + 1 equals ( x, x + 1 ) Ease of use Guarantees Usefulness Expressiveness

if currStmt == [X = E] then if currStmt == [X = E]  “X does not appear in E” then Usefulness of the checker Found subtle bugs in Lerner’s initial implementation of various optimizations define fact equals(X:Var, E:Expr) with meaning σ( X) == σ(E ) x := x + 1x = x + 1 equals ( x, x + 1 ) Ease of use Guarantees Usefulness Expressiveness

x = x + 1 x = *y + 1 Usefulness of the checker Found subtle bugs in Lerner’s initial implementation of various optimizations define fact equals(X:Var, E:Expr) with meaning σ( X) == σ(E ) if currStmt == [X = E] Æ “X does not appear in E” then equals ( x, x + 1 ) equals ( x, *y + 1 ) if currStmt == [X = E]  “E does not use X” then Ease of use Guarantees Usefulness Expressiveness

Rhodium expressiveness Traditional optimizations: –const prop and folding, branch folding, dead assignment elim, common sub-expression elim, partial redundancy elim, partial dead assignment elim, arithmetic invariant detection, and integer range analysis. Pointer analyses –must-point-to analysis, Andersen's may-point-to analysis with heap summaries Loop opts –loop-induction-variable strength reduction, code hoisting, code sinking Array opts –constant propagation through array elements, redundant array load elimination Ease of use Guarantees Usefulness Expressiveness

Expressiveness limitations May not be able to express your optimization in Rhodium –opts that build complicated data structures –opts that perform complicated many-to-many transformations (e.g.: loop fusion, loop unrolling) A correct Rhodium optimization may be rejected by the correctness checker –limitations of the theorem prover –limitations of first-order logic Ease of use Guarantees Usefulness Expressiveness

Summary Rhodium system –makes it easier to write optimizations –provides correctness guarantees –is expressive enough for realistic optimizations Rhodium system provides a foundation for safe extensible program manipulators

Future works Overcome the limitation of Rhodium –opts that build complicated data structures –opts that perform complicated many-to-many transformations (e.g.: loop fusion, loop unrolling) Overcoming the limitation of theorem prover –Simplify is conservative –Using higher order logic?

Future works.. Automatically infer the whole compiler from a high level specification –Infer analyses –Infer transformations –Program optimizations by demonstration –Automatically pick good data representations –Automatically explore the tradeoffs between scalability and precision

References Correctness –By hand [Cousot and Cousot 77, 79, Benton 04, Lacey et al. 02] –With interactive theorem prover [Cachera et al. 04] –One compilation at a time [Pnueli et al. 98, Necula 00, Rinard 99] Declarative languages for writing transformations –Attribute grammars [Reps and Teitelbaum 88] –Temporal logic [Steffen 91, Lacey et al. 02] Execution engines –Incremental execution of transformations [Sittampalam et al. 04] –Running opts specified with temporal logic [Steffen 91]

More facts define fact mustNotPointTo(X:Var,Y:Var) with meaning σ ( X)  σ ( &Y) define fact hasConstantValue(X:Var,C:Const) with meaning σ( X) == C define fact doesNotPointIntoHeap(X:Var) with meaning Y:Var and σ ( X) == σ ( &Y)

More rules if currStmt == [X = *A]    B:Var.  mustNotPointTo(B,Y) then