1. Tanguy Risset 1 Formal Bit With Determination for Nested Loop Programs David Cachera, Tanguy Risset, Djamel Zegaoui

2. Tanguy Risset 2 Outline Introduction/motivation Explaining the methodology Solving the Bit Width equation with (max,+)

3. Tanguy Risset 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. Tanguy Risset 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 FPGA ASIC Uniformization RTL Derivation Scheduling/Mapping Alpha VHDL

5. Tanguy Risset 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. Tanguy Risset 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. Tanguy Risset 7 Example: … the FIR ! system fir : {N,M | 3<=N<=M-1} (x : {n | 1<=n<=M} of integer; w : {i | 0<=i<=N-1} of integer) returns (res : {n | N<=n<=M} of integer); var Y : {n,i | N<=n<=M; -1<=i<=N-1} of integer; let Y[n,i] = case { | i=-1} : 0[]; { | 0<=i} : Y[n,i-1] +w[i] * x[n-i]; esac; res[n] = Y[n,N-1]; tel;

8. Tanguy Risset 8 Problem modeling: error signal « Formal » signal s(n), implementation š(n) Noise signal: e(n)=s(n)- š(n) Noise Standard deviation: Signal to Noise ratio (SNR): Good bit width if R s is greater than a given value

9. Tanguy Risset 9 Operators modeling [Tou99] Let X be a signal encoded on m+n+1 bits Generated error: where q=2 -n Error propagation: –Addition: –Multiplication: bmbm...b0b0 b -1...b -n b -n+1....

10. Tanguy Risset 10 Architectural description in Alpha W[t,p] = case { | t=p+1} : w[t-1]; { | p+2<=t} : W[t-1,p]; esac; XP[t,p] = case { | p=0} : x[t+N-1]; { | 1<=p} : XP[t-2,p-1]; esac; Y[t,p] = case { | p=-1} : 0[]; { | 0<=p} : Y[t-1,p-1] + W[t-1,p] * XP[t-1,p]; esac;

11. Tanguy Risset 11 Generation of BW equation Simple projection of Alpha equation on space (p index) (BWA=  A 2 ): BWW [p] = Max( BWw[] BWW[p]) BWXP[p] =case { | p=0} : BWx[] { | 1<=p} : BWXP[p-1] esac BWY[p] = case { | p=-1} : 0[]; { | 0<=p} : q 2 /12+max(BWY[p-1] + q 2 /12, BWW*XP[p]+q 2 /12) esac; W[t,p] = case { | t=p+1} : w[t-1]; { | p+2<=t} : W[t-1,p]; esac; XP[t,p] = case { | p=0} : x[t+N-1]; { | 1<=p} : XP[t-2,p-1]; esac; Y[t,p] = case { | p=-1} : 0[]; { | 0<=p} : Y[t-1,p-1] + W[t-1,p] * XP[t-1,p]; esac;

12. Tanguy Risset 12 Solving the BW equations (FIR) Here the solution can be easily provided by a symbolic solver (q=2 -n ):

13. Tanguy Risset 13 Solving the BW equations... In general, we solve successively the strongly connected component of the reduced dependence graph Y WX Fir (3 SCC) V3 V2V1 input Other example: 1 SCC

14. Tanguy Risset 14 Solving BW Eq for 1 SCC V1[t,p] = case { | p=0} : Input[] { | p>=1} : V1[t-1,p-1]- V3[t-2,p-1]; esac; V2[t,p] = case { | p=0} : Input[]; { | 1<=p} : V2[t-2,p-1]+ V3[t-1,p-1]; esac; V3[t,p] = case { | p=0} : Input[]; { | 1<=p} : V1[t-1,p-1]+ V2[t-3,p-1] esac; V3 V2V1 input BWV1[p] = case { | p=0} : 0 { | p>=1} : max(BWV1[p-1]+, BWV3[p-1] ]+ ); esac; BWV2[p] = case { | p=0} : 0 { | 1<=p} : max(BWV2[p-1]+, BWV3[p-1] ]+ ); esac; BWV3[p] = case { | p=0} : 0 { | 1<=p} : max(BWV1[p-1]+, BWV2[p-1] ]+ ); esac;

15. Tanguy Risset 15 Solving the BW equations... General form (under some assumptions) of the BW equation for one SCC with k variables (for i=1..k): Example :

16. Tanguy Risset 16 Using (max,+) notations  is the max and  is the addition Or:

17. Tanguy Risset 17 Perron-Frobenius for (max,+) Let M  R max n  n be a n 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 N=1:

18. Tanguy Risset 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. Tanguy Risset 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: a[0]=x Do i=1,N a[i]=a[i-1]*a[i-1] Enddo

20. Tanguy Risset 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: –Integer[S,8] –Integer[S,3,6] – C = Mul8x8-12(A,B) – B = Trunc(C,11)

21. Tanguy Risset 21 Processor variable dependent BW