A Symbolic Approach to Bernstein Expansion for Program Analysis and Optimization

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A Symbolic Approachto Bernstein Expansionfor
Program Analysis and Optimization

• Philippe Clauss Irina Tchoupaeva
• ICPS/LSIIT Université Louis Pasteur
• Strasbourg - France

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ANALYSIS OPTIMIZATION
Programs
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Static Analysis

• Gives the most accurate and general results
• But is not always possible
• Non-predictable behavior
• dynamic approach
• Complex models and expressions
• Adapted mathematical frameworks
• Usable in optimizing tools or compilers
• Giving the most accurate and error-free results
• Example the polytope model

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The Polytope Model

• Geometric transformations translate to nested
  loops transformations
• Some characteristics translate to quantitative
  evaluations

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Example live range of an array element axy

• Number of iterations where any axy was
  accessed and will be accessed again?
• Number of iterations between the first and last
  access to axy?

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• For each reference aij, aj-1Ni-j,
  ai-1j-i
• Computation of the number of iterations occurring
  before the iteration that references axy

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• Example for reference aij
• axy is referenced when ix and jy, i.e., at
  iteration (x,y)
• Number of iterations occurring before (x,y)

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• and
• are parameterized polytopes
• Number of integer points of each polytope
• Ehrhart polynomial of each polytope
• Implemented in Polylib

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• Computed Ehrhart polynomials

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Linear to non-linear model

• How can such a non-linear expression be used?
• Maximum value maximum number of simultaneously
  alive elements
• Would be useful for array contraction
• Cannot be handled by the polytope model
• Limited to linear (in-)equalities
• Need of additionnal tools

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Non-linear cases in program analysis

• Can this loop be
• parallelized?
• Does reference aq
• overlap reference
• aialpha1?

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• Conventional value range analysis
• The value range of ialpha1 is entirely included
  in the range of -i210ialpha-25
• The loop cannot be parallelized
• Is this true?

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• Better value range analysis using the Horner form
  of -i210ialpha-25 alpha-25(10-i)i
• More accurate bound
• Same conclusion
• Is this true?

NO! The exact value range is alpha-25,alpha
and the loop CAN be parallelized!
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Bernstein Expansion

• Gives an enclosure of the range of a polynomial
  over a given interval
• Avoids function evaluation
• Gives more accurate results than classic interval
  methods
• Applies for any multi-variate polynomial without
  restriction
• Exact range can be obtained

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• Implicit power form linear combination in the
  power basis
• Bernstein form linear combination in the
  Bernstein basis

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Basis conversion

• Univariate case
• i-th Bernstein polynomial
• i-th Bernstein coefficient

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Basis conversion

• Multivariate case notations

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Basis conversion

• Multivariate polynomial
• I-th Bernstein polynomial
• I-th Bernstein coefficient

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Important theorems

• THEOREM 1 the minimum and maximum Bernstein
  coefficients give an enclosure of the range of a
  polynomial over the interval 0,1
• THEOREM 2 - VERTEX CONDITION - if the minimum and
  maximum Bernstein coefficients occur at the
  vertices of the Bernstein coefficients array,
  then the enclosure is the exact range.

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?
!
Minimum bernstein coefficient Maximum bernstein
coefficient
Exact range
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From 0,1 to any interval

• Can any interval a,b be considered?
• Any interval can be mapped linearly onto 0,1
• The Bernstein coefficients of p(x) on a,b are
  equal to the Bernstein coefficients of the
  corresponding polynomial p(y) on 0,1
• The Bernstein coefficients of p(y) give an
  enclosure of the range of p(x) on a,b

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From 0,1 to any interval

• Linear mapping

where
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Symbolic Bernstein expansion

• To handle parameterized polynomials over
  intervals with bounds depending on variables and
  parameters
• quite more powerful for general program
  analysis
• Bernstein expansion allows it!

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Symbolic Bernstein expansion

• By considering polynomial coefficients on
  parameters
• The same formulas apply!

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Symbolic Bernstein expansion

• Mapping onto 0,1 can be done for parameterized
  boxes by evicting some values

must be ? 0
where
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Example (with parameterized coefficients)
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Example (with parameterized coefficients)
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Example (with intervals depending on parameters
and variables)

• Not valid for N0 or i0
• (0,0) excluded
• New mapping
• ? Consider Ngt1

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Example (contd.) array contraction

• Consider a as a temporary array
• Transform a into a smaller array

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• Minimum memory amount needed
• maximum number of simultaneously
• alive array elements
• maximum live range for all array elements
• Live range of any array element axy

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• Not valid for N5 and N3
• Let us consider N gt 5

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• Bernstein coefficients
• Maximum coefficient

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• Array a contracted from to
• elements

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Related works

• Decomposition of polynomials into affine
  constraints and hyperbolic/elliptical equations
  (Maslov Pugh)
• Analysis of monotonic polynomials (Blume
  Eigenmann)
• Univariate and monotonic polynomials (Van Engelen)

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Complexity

• Primitive algorithm
• O(nv), vnumber of variables
• Linear complexity algorithm proposed by Delgado
  Peña

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Conclusion

• Many other application fields
• Many static analysis issues data reuse distance
  in loops, memory consumption,
• Symbolic cache behavior modeling
• Program trace analysis (polynomial interpolation)
• Partially implemented
• Parameterized intervals not yet considered

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Conclusion

• Further developments
• Iterative Bernstein expansion when Bernstein
  coefficients are not linear
• Tighter bounds through degree elevation
• Parameterized polynomial inequations solving for
  the general case
• Symbolic number of solutions for systems of
  parameterized polynomial inequations

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THANK YOU !