Contification Using Dominators

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Contification Using Dominators

• Matthew Fluet
• Cornell University

Stephen Weeks InterTrust STAR Lab
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Motivation

• use traditional optimizations in
  functional-language compilers
• traditional optimizations require intraprocedural
  control-flow information
• in functional languages, most control-flow
  information is interprocedural
• contification a technique for turning function
  calls into local control-flow transfers
• interprocedural ? intraprocedural

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An Example
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MLton

• a whole-program optimizing compiler for Standard
  ML

SML
defunctorize
polymorphic, higher-order IL
monomorphise
simply-typed, higher-order IL
closure convert
optimize
simply-typed, first-order IL
generate code
native x86
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Contification

• when a function g is contified within a function
  f
• intraprocedural control-flow is exposed
• g will share the same stack frame as f
• invocations of g can be optimized
• goal contify as many functions as possible
• components of contification
• analysis what functions are contified and where
• transformation single-pass rewrite of the
  program

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Contification Analysis I

• represent a program by its call graph
• functions m, f, g, h and continuations k, l
• a distinguished main function m
• nontail calls
• tail calls

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Contification Analysis II

• an analysis maps functions to abstract return
  locations
• Return ? ? Cont ? Func
• A ? Analysis Func ? Return
• A(g) f g always returns to function f g is
  contified in the body of f
• A(g) k g always returns to continuation k g
  is contified where k is defined and g is
  transformed to transfer control to k
• A(g) ? g is not contified in the transformed
  program

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Safety

• safety ensures a well-defined and correct
  transformation
• an analysis A is safe if the following hold
• A(m) ?
• if then A(g) ? k, ?
• if then A(g) ? f, A(f), ?

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The Acall and Acont Analyses

• Acall a simple syntactic analysis
• Acont a least fixed-point analysis Reppy 2001

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The Adom Analysis I

• build a directed graph G, similar to the call
  graph
• Node Return ? ? Cont ? Func
• each edge (l, f) indicates that f returns to
  location l
• if l ?, then f has no return location

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The Adom Analysis II

• build the dominator tree D of G
• l dominates f if f always returns to l in any
  execution of the program
• define Adom(f) to be the dominator of f closest
  to ?
• if the immediate dominator of f is ?, then
  Adom(f) ?

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The Adom Analysis III

• Theorem Adom is safe.
• Theorem Adom is maximal.
• for all safe analyses B and all functions f,if
  B(f) ? ?, then Adom(f) ? ?

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Compile-time Performance

• three rounds of contification in the optimizing
  cycle
• contification quiesces after two rounds
• contification takes less than 4 of total compile
  time
• typically less than one second, with a maximum of
  13 seconds

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Run-time Performance

• anomalies
• contification may disable size-based inlining
• MLtons optimizer evolved around Acall

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Conclusions

• a simple, yet general, framework for expressing
  contification analyses
• single transformation
• safety condition
• maximality criterion
• contification is efficient and improves running
  times
• Got MLton?
• You should.
• http//www.sourcelight.com/MLton

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Related Transformations

• Local CPS conversion Reppy 2001
• analyzes a module in isolation
• operates over a higher-order IL
• escaping functions with unknown control-flow
  cause imprecision
• requires local CPS conversion to apply
  transformation
• Lambda dropping Danvy and Schultz 2000
• does not approximate the returns of a function
• does not change calls from tail to nontail, or
  vice versa
• Loop headers Appel 1994
• transformation local to a particular function
• relies on inlining to expose new control-flow

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References

• A. W. Appel. Loop headers in ?-calculus or CPS.
  Lisp and Symbolic Computation, 7337-343, 1994.
• O. Danvy and U. P. Schultz. Lambda-dropping
  Transforming recursive equations into programs
  with block structure. Theoretical Computer
  Science, 248(1-2)243-287, 2000.
• J. Reppy. Local CPS conversion in a direct-style
  compiler. In Workshop on Continuations, pages
  1-5, Jan. 2001.

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Comparison to Inlining

• no substitution of actual arguments for function
  parameters
• no duplication of code

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The Acall Analysis

• intuition a function f has one return location
  if
• there is exactly one call to f from outside its
  body and
• there are only tail calls to f within its body
• a simple syntactic analysis
• original contification analysis used in MLton

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But

• f, g, and h always return to continuation k
• but, the call analysis fails to contify any of
  the functions

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The Acont Analysis

• intuition a function f returns to continuation k
  if
• all nontail calls to f use k and
• all tail callers of f also return to k
• a least fixed point ties the recursion
• based on a similar analysis in Moby Reppy 2001

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But

• g1, g2, and h all return to function f
• but, the call analysis fails to contify h
• and, the continuation analysis fails to contify
  any function

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The Adom Analysis IV

• setting Adom(f) equal to the immediate dominator
  of f violates safety
• Adom(h1) f ? g3, A(g3), ? g3, g2, ?

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An Example I
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An Example II
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An Example III
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An Example IV