Formal Bit With Determination for Nested Loop Programs

1
Formal Bit With Determination for Nested Loop
Programs

• David Cachera,
• Tanguy Risset,
• Djamel Zegaoui

2
Outline

• Introduction/motivation
• Explaining the methodology
• Solving the Bit Width equation with (max,)

3
Context and Motivations

• Context
• High level synthesis (hardware compilation from
  functional specification)
• How to go (safely) from algorithmic description
  to finite precision implementation
• Specific motivations
• Parameterized loop nests programs
• MMAlpha methodology

4
Context and Motivations MMAlpha

• Provide a formal methodology based on the strong
  semantic properties of the Alpha language
• But still ! Keep applicability for effective VHDL
  generation

5
BW determination state of the art

• Formal methods
• Provide abstract framework for solving the
  problem (Gaut, Ptolemy, DeepC)
• Limited applicability
• Simulation based methods
• Based on probabilistic models for input data
  (Ptolemy, Imec,etc.)
• Time consuming processes
• Ideally provide formal methods to speed up the
  simulation.

6
Our methodology

• Start from loop nest specification (in Alpha)
• Schedule and Place (SIMD-like specification)
• Bit Width determination
• problem modeling
• BW equation generation
• BW equation solving
• Hardware generation (VHDL)

7
Example the FIR !
system fir N,M 3ltNltM-1 (x n
1ltnltM of integer w i 0ltiltN-1 of
integer) returns (res n NltnltM of
integer) var Y n,i NltnltM -1ltiltN-1
of integer let Yn,i case i-1
0 0lti Yn,i-1 wi xn-i
esac resn Yn,N-1 tel
8
Problem modeling error signal

•  Formal  signal s(n), implementation (n)
• Noise signal e(n)s(n)- (n)
• Noise Standard deviation
• Signal to Noise ratio (SNR)
• Good bit width if Rs is greater than a given
  value

9
Operators modeling Tou99

• Let X be a signal encoded on mn1 bits
• Generated error where q2-n
• Error propagation
• Addition
• Multiplication

10
Architectural description in Alpha
Wt,p case tp1 wt-1
p2ltt Wt-1,p esac XPt,p
case p0 xtN-1
1ltp XPt-2,p-1 esac Yt,p case
p-1 0 0ltp
Yt-1,p-1 Wt-1,p XPt-1,p esac
11
Generation of BW equation

• Simple projection of Alpha equation on space (p
  index) (BWA?A2)

Wt,p case tp1 wt-1
p2ltt Wt-1,p esac XPt,p
case p0 xtN-1
1ltp XPt-2,p-1 esac Yt,p case
p-1 0 0ltp
Yt-1,p-1 Wt-1,p XPt-1,p esac
BWWp Max( BWw BWWp) BWXPp case
p0 BWx 1ltp BWXPp-1
esac BWYp case p-1 0
0ltp q2/12max(BWYp-1 q2/12,
BWWXPpq2/12) esac
12
Solving the BW equations (FIR)

• Here the solution can be easily provided by a
  symbolic solver (q2-n)

13
Solving the BW equations...

• In general, we solve successively the strongly
  connected component of the reduced dependence
  graph

input
input
W
X
V2
V1
Y
V3
Fir (3 SCC)
Other example 1 SCC
14
Solving BW Eq for 1 SCC
BWV1p case p0 ?0 pgt1
max(BWV1p-1 ?, BWV3p-1 ?)
esac BWV2p case p0 ?0
1ltp max(BWV2p-1 ?, BWV3p-1 ?)
esac BWV3p case p0 ?0
1ltp max(BWV1p-1 ?, BWV2p-1
?) esac
V1t,p case p0 Input
pgt1 V1t-1,p-1- V3t-2,p-1 esac
V2t,p case p0 Input
1ltp V2t-2,p-1 V3t-1,p-1 esac
V3t,p case p0 Input
1ltp V1t-1,p-1 V2t-3,p-1 esac
15
Solving the BW equations...

• General form (under some assumptions) of the BW
  equation for one SCC with k variables (for
  i1..k)
• Example

16
Using (max,) notations

• ? is the max and ? is the addition
• Or

17
Perron-Frobenius for (max,)

• Let M?Rmaxn?n be an irreducible matrix in (max,)
  with spectral ray ?M and cyclicity c(M), there
  exist an integer N such that
• Here c(M)1, ?M ? and N1

18
Result

• If we respect our restrictions, we are able to
  solve, in a parametric way the bit Width
  equations for a loop nest program.
• This is the only method that solves this problem
  in a parametric way (MIT did something with DeepC
  but they do not handle symbolic parameters)

19
Restrictions of our methodology

• Linear array architecture
• BW equation solvable (i.e. no auto-adaptive
  mechanism or complicated convergence property)
• No multiplication in strongly connected component
  of the graph

a0x Do i1,N aiai-1ai-1 Enddo
20
Conclusion

• First method for parameterized loop nest bit
  width determination
• Allow reducing the time needed for simulation
  (probably not much more than previous methods
  did)
• New typing mechanism introduced in Alpha
• IntegerS,8
• IntegerS,3,6

• C Mul8x8-12(A,B)
• B Trunc(C,11)

21
Processor variable dependent BW