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Penn ESE353 Spring 2015 -- DeHon 1 ESE535: Electronic Design Automation Day 22: April 15, 2015 Multi-level Synthesis.

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Presentation on theme: "Penn ESE353 Spring 2015 -- DeHon 1 ESE535: Electronic Design Automation Day 22: April 15, 2015 Multi-level Synthesis."— Presentation transcript:

1 Penn ESE353 Spring 2015 -- DeHon 1 ESE535: Electronic Design Automation Day 22: April 15, 2015 Multi-level Synthesis

2 Penn ESE353 Spring 2015 -- DeHon 2 Today Multilevel Synthesis/Optimization –Why –Transforms -- defined –Division/extraction How we support transforms Behavioral (C, MATLAB, …) RTL Gate Netlist Layout Masks Arch. Select Schedule FSM assign Two-level Multilevel opt. Covering Retiming Placement Routing

3 Penn ESE353 Spring 2015 -- DeHon 3 Multi-level Logic General circuit netlist May have –sums within products –products within sum –arbitrarily deep y=((a (b+c)+e)fg+h)i

4 Penn ESE353 Spring 2015 -- DeHon 4 Why Multi-level Logic? ab(c+d+e)(f+g) abcf+abdf+abef+abcg+abdg+abeg 6 product terms  23 2-input gates vs. 3 gates: and4,or3,or2  6 2-input gates Aside from Pterm sharing between outputs, –two level cannot share sub-expressions

5 Penn ESE353 Spring 2015 -- DeHon 5 Why Multi-level Logic a xor b –a/b+/ab a xor b xor c –a/bc+/abc+/a/b/c+ab/c a xor b xor c xor d –a/bcd+/abcd+/a/b/cd+ab/cd+/ab/c/d+a/b/c/d+abc/d+/a/bc/d

6 Penn ESE353 Spring 2015 -- DeHon 6 Why Multilevel a xor b –a/b+/ab a xor b xor c –a/bc+/abc+/a/b/c+ab/c a xor b xor c xor d –a/bcd+/abcd+/a/b/cd+ab/cd +/ab/c/d+a/b/c/d+abc/d+/a/ bc/d Compare a xor b –x1=a/b+/ab a xor b xor c –x2=x1/c+/x1*c a xor b xor c xor d –x3=x2/d+x2*d

7 Penn ESE353 Spring 2015 -- DeHon 7 Why Multilevel a xor b –x1=a/b+/ab a xor b xor c –x2=x1/c+/x1*c a xor b xor c xor d –x3=x2/d+x2*d Multi-level –exploit common sub-expressions –linear complexity Two-level –exponential complexity

8 Harder than Two-Level Two-level already NP-hard –has canonical representation –clean formulation –observed can limit to Primes –identified opportunities for pruning Multi-level has more flexibility –  larger space to explore –Not formulated cleanly Solution more heuristic … art –…all problems start this way, some stay… Penn ESE353 Spring 2015 -- DeHon 8

9 9 Goal Find the structure Exploit to minimize gates –Total (area) –In path (delay)

10 Penn ESE353 Spring 2015 -- DeHon 10 Multi-level Transformations Decomposition Extraction Factoring Substitution Collapsing [copy these to board so stay up as we move forward]

11 Penn ESE353 Spring 2015 -- DeHon 11 Decomposition F=abc+abd+/a/c/d+/b/c/d F=XY+/X/Y X=ab Y=c+d

12 Penn ESE353 Spring 2015 -- DeHon 12 Decomposition F=abc+abd+/a/c/d+/b/c/d –4 3-input + 1 4-input  11 2-input gates F=XY+/X/Y X=ab Y=c+d –5 2-input gates Note: use X and /X, use at multiple places

13 Penn ESE353 Spring 2015 -- DeHon 13 Extraction F=(a+b)cd+e G=(a+b)/e H=cde F=XY+e G=X/e H=Ye X=a+b Y=cd

14 Penn ESE353 Spring 2015 -- DeHon 14 Extraction F=(a+b)cd+e G=(a+b)/e H=cde 2-input: 4 3-input: 2  8 2-input gates F=XY+e G=X/e H=Ye X=a+b Y=cd 2-input: 6 Common sub-expressions over multiple output

15 Penn ESE353 Spring 2015 -- DeHon 15 Factoring F=ac+ad+bc+bd+e F=(a+b)(c+d)+e

16 Penn ESE353 Spring 2015 -- DeHon 16 Factoring F=ac+ad+bc+bd+e –4 2-input, 1 5-input  8 2-input gates –9 literals F=(a+b)(c+d)+e –4 2-input –5 literals

17 Penn ESE353 Spring 2015 -- DeHon 17 Substitution G=a+b F=a+bc Substitute G into F F=G(a+c) –(verify) F=(a+b)(a+c)=aa+ab+ac+bc=a+bc useful if also have H=a+c, then F=GH

18 Penn ESE353 Spring 2015 -- DeHon 18 Collapsing F=Ga+/Gb G=c+d F=ac+ad+b/c/d opposite of substitution –sometimes want to collapse and refactor –especially for delay optimization [next lecture]

19 Penn ESE353 Spring 2015 -- DeHon 19 Moves These transforms define the “moves” we can make to modify our network. Goal is to apply, usually repeatedly, to minimize gates –…then apply as necessary to accelerate design MIS/SIS –Applies to canonical 2-input gates –Then covers with target gate library Day 3

20 Penn ESE353 Spring 2015 -- DeHon 20 Division

21 Penn ESE353 Spring 2015 -- DeHon 21 Division Given: function (f) and divisor (p) Find: quotient (q) and remainder (r) f=pq+r E.g. f=abc+abd+ef, p=ab q=c+d, r=ef

22 Penn ESE353 Spring 2015 -- DeHon 22 Algebraic Division Use basic rules of algebra, rather than full boolean properties Computationally simple Weaker than boolean division f=a+bc p=(a+b) Algebra: not divisible Boolean: q=(a+c), r=0

23 Penn ESE353 Spring 2015 -- DeHon 23 Algebraic Division Given: function (f) and divisor (p) Find: quotient and remainder f=pq+r f and p are expressions (lists of cubes) –p={a 1,a 2,…} Define: h i ={c j | a i * c j  f} f/p = h 1  h 2  h 3 …

24 Penn ESE353 Spring 2015 -- DeHon 24 Algebraic Division Example (adv to alg.; work ex on board) f=abc+abd+de p=ab+e

25 Penn ESE353 Spring 2015 -- DeHon 25 Algebraic Division f and p are expressions (lists of cubes) p={a 1,a 2,…} h i ={c j | a i * c j  f} f/p = h 1  h 2  h 3 …

26 Penn ESE353 Spring 2015 -- DeHon 26 Algebraic Division Example f=abc+abd+de, p=ab+e p={ab,e} h1={c,d} h2={d} h1  h2={d} f/p=d r=f- p *(f/p) r=abc+abd+de-(ab+e)d r=abc

27 Penn ESE353 Spring 2015 -- DeHon 27 Algebraic Division Time O(|f||p|) as described –compare every cube pair Sort cubes first –O((|f|+|p|)log(|f|+|p|)

28 Penn ESE353 Spring 2015 -- DeHon 28 Primary Divisor f/c such that c is a cube f =abc+abde f/a=bc+bde is a primary divisor

29 Penn ESE353 Spring 2015 -- DeHon 29 Cube Free The only cube that divides p is 1 c+de is cube free bc+bde is not cube free

30 Penn ESE353 Spring 2015 -- DeHon 30 Kernel Kernels of f are –cube free primary divisors of f –Informally: sums w/ cubes factored out f=abc+abde f/ab = c+de is a kernel ab is cokernel of f to (c+de) –cokernels always cubes

31 Penn ESE353 Spring 2015 -- DeHon 31 Factoring Gfactor(f) if (terms==1) return(f) p=CHOOSE_DIVISOR(f) (h,r)=DIVIDE(f,p) f=Gfactor(h)*Gfactor(p)+Gfactor(r) return(f) // factored

32 Penn ESE353 Spring 2015 -- DeHon 32 Factoring Trick is picking divisor –pick from kernels –goal minimize literals after resubstitution Re-express design using new intermediate variables Variable and complement

33 Penn ESE353 Spring 2015 -- DeHon 33 Kernel Extraction Find c f = largest cube factor of f K=Kernel1(0,f/c f ) if (f is cube-free) –return(f  K) else –return(K) Kernel1(j,g) –R=g –N max index in g –for(i=j+1 to N) if (l i in 2 or more cubes) –c f =largest cube divide g/l i –if (forall k  i, l k  c f ) »R=R  KERNEL1(i,g/(l i  c f )) –return(R) Must be to Generate Non-trivial kernel Consider each literal for cokernel once (largest cokernels will already have been found)

34 Penn ESE353 Spring 2015 -- DeHon 34 Kernel Extract Example (ex. on board; adv to return to alg.) f=abcd+abce+abef

35 Penn ESE353 Spring 2015 -- DeHon 35 Kernel Extraction Find c f = largest cube factor of f K=Kernel1(0,f/c f ) if (f is cube-free) –return(f  K) else –return(K) Kernel1(j,g) –R=g –N max index in g –for(i=j+1 to N) if (l i in 2 or more cubes) –c f =largest cube divide g/l i –if (forall k  i, l k  c f ) »R=R  KERNEL1(i,g/(l i  c f )) –return(R) Must be to Generate Non-trivial kernel Consider each literal for cokernel once (largest cokernels will already have been found)

36 Penn ESE353 Spring 2015 -- DeHon 36 Kernel Extract Example (stay on prev. slide, ex. on board) f=abcd+abce+abef c f =ab f/c f =cd+ce+ef R={cd+ce+ef} N=6 a,b not present (cd+ce+ef)/c=e+d largest cube 1 Recurse  e+d R={cd+ce+ef,e+d} only 1 d (d+ce+ef)/e=c+f Recurse  c+f R={cd+ce+ef,e+d,c+f}

37 Penn ESE353 Spring 2015 -- DeHon 37 Extraction Identify cube-free expressions in many functions (common sub expressions) 1.Generate kernels for each function 2.select pair such that k1  k2 is not a cube Note: k1=k2 is simplest case of this …but intersection case is more powerful Example to come 3.new variable from intersection –v= k1  k2 4.update functions (resubstitute) –f i = v*(f i /v)+ r i –(similar for common cubes)

38 Penn ESE353 Spring 2015 -- DeHon 38 Extraction Example X=ab(c(d+e)+f+g)+g Y=ai(c(d+e)+f+j)+k

39 Penn ESE353 Spring 2015 -- DeHon 39 Extraction Example X=ab(c(d+e)+f+g)+g Y=ai(c(d+e)+f+j)+k d+e kernel of both L=d+e X=ab(cL+f+g)+h Y=ai(cL+f+j)+k

40 Penn ESE353 Spring 2015 -- DeHon 40 Extraction Example L=d+e X=ab(cL+f+g)+h Y=ai(cL+f+j)+k kernels:(cL+f+g), (cL+f+j) extract: M=cL+f X=ab(M+g)+h Y=ai(M+f)+h

41 Penn ESE353 Spring 2015 -- DeHon 41 Extraction Example L=d+e M=cL+f X=ab(M+g)+h Y=ai(M+j)+h no kernels common cube: aM N=aM M=cL+f L=d+e X=b(N+ag)+h Y=i(N+aj)+k

42 Penn ESE353 Spring 2015 -- DeHon 42 Extraction Example N=aM M=cL+f L=d+e X=b(N+ag)+h Y=I(N+aj)+k Can collapse –L into M into N –Only used once Get larger common kernel N –maybe useful if components becoming too small for efficient gate implementation

43 Penn ESE353 Spring 2015 -- DeHon 43 Resubstitution Also useful to try complement on new factors f=ab+ac+/b/cd X=b+c f=aX+/b/cd /X=/b/c f=aX+/Xd …extracting complements not a direct target

44 Multilevel Optimization Unlike Two-level, very heuristic Not clear when done Goal find common terms to share Often start with two-level optimization –Identifies product term sharing Identify kernels and cubes Factor them out If can be used many places, get benefit SIS included common recipes More after timing analysis Penn ESE353 Spring 2015 -- DeHon 44

45 Penn ESE353 Spring 2015 -- DeHon 45 Summary Want to exploit structure in problems to reduce (contain) size –common sub-expressions Identify component elements –decomposition, factoring, extraction Division key to these operations Kernels give us divisors

46 Penn ESE353 Spring 2015 -- DeHon 46 Big Ideas Exploit freedom –form Exploit structure/sharing –common sub expressions Techniques –Iterative Improvement –Refinement/relaxation

47 Penn ESE353 Spring 2015 -- DeHon 47 Admin Reading for Monday on canvas Updated energy model: update_model.tar Milestone Thursday

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