EDA CS286'5b

1
EDA (CS286.5b)

• Day 16
• Logic Synthesis
• Multi-Level

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Today

• Multilevel Synthesis/Optimization
• Why
• Transforms
• Division/extraction
• Boolean Division
• Dont care

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Multi-level

• General circuit netlist
• May have
• sums within products
• products within sum
• arbitrarily deep
• y((a (bc)e)fgh)i

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Why multi-level?

• ab(cde)(fg)

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Why multi-level?

• ab(cde)(fg)
• abcfabdfabefabcgabdgabeg
• 6 product terms
• 3 gates and4,or3,or2
• Cannot share sub-expressions

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Why Multilevel

• a?b
• a/b/ab
• a?b?c
• a/bc/abc/a/b/cab/c
• a?b?c?d
• a/bcd/abcd/a/b/cdab/cd/ab/c/da/b/c/dabc/d/a
  /bc/d

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Why Multilevel

• a?b
• x1a/b/ab
• a?b?c
• x2x1/c/x1c
• a?b?c?d
• x3x2/dx2d

• Multi-level
• exploit common sub-expressions
• linear complexity
• Two-level
• exponential complexity

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Multi-level Transformations

• Decomposition
• Extraction
• Factoring
• Substitution
• Collapsing

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Decomposition

• Fabcabd/a/c/d/b/c/d
• FXY/X/Y
• Xab
• Ycd

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Decomposition

• Fabcabd/a/c/d/b/c/d
• 4 3-input 1 4-input
• FXY/X/Y
• Xab
• Ycd
• 5 2-input gates
• Note use X and /X, use at multiple places

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Extraction

• F(ab)cde
• G(ab)/e
• Hcde

• FXYe
• Gx/e
• HYe
• Xab
• Ycd

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Extraction

• F(ab)cde
• G(ab)/e
• Hcde
• 2-input 4
• 3-input 2

• FXYe
• Gx/e
• HYe
• Xab
• Ycd
• 2-input 6

Common sub-expressions over multiple output
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Factoring

• Facadbcbde
• F(ab)(cd)e

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Factoring

• Facadbcbde
• 4 2-input, 1 5-input
• 9 literals
• F(ab)(cd)e
• 4 2-input
• 5 literals

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Substitution

• Gab
• Fabc
• Substitute G into F
• FG(ac)
• (verify) F(ab)(ac)aaabacbcabc
• useful if also have Hac? (FGH)

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Collapsing

• FGa/Gb
• Gcd
• Facadb/c/d
• opposite of substitution
• sometimes want to collapse and refactor
• especially for delay optimization

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Division

• Given function (f) and divisor (p)
• Find quotient and remainder
• fpqr
• E.g.
• fabcabdef, pab
• qcd, ref

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Algebraic Division

• Use basic rules of algebra, rather than full
  boolean properties
• Computationally simple
• Weaker than boolean division
• fabc p(ab)
• Algebra not divisible
• Boolean q(ac), r0

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Algebraic Division

• f and p are expressions (lists of cubes)
• pa1,a2,
• hi cj ai cj?f
• f/p h1?h2 ?h3

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Algebraic Division Example

• fabcabdde
• gabe

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Algebraic Division Example

• fabcabdde
• pabe
• pab,e
• h1c,d h2d
• h1?h2d
• f/gd
• rf-f/gabcabdde-(abe)dabc

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Algebraic Division Time

• O(fp) as described
• compare every cube pair
• Sort cubes first
• O((fp)log(fp)

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Primary Divisor

• f/c such that c is a cube
• f abcabde
• f/abcbde is a primary divisor

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Cube Free

• The only cube that divides p is 1
• cde is cube free
• bcbde is not cube free

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Kernel

• Kernels of f are
• cube free primary divisors of f
• fabcabde
• f/ab cde is a kernel
• ab is cokernel of f to (cde)
• cokernels always cubes

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Kernel Extraction

• Find c largest cube factor of f
• KKernel1(0,f/c)
• if (f is cube-free)
• return(f?K)
• else
• return(K)

• Kernel1(j,g)
• Rg
• N max index in g
• for(ij1 to N)
• if (li in 2 or more cubes)
• clargest cube divide g/li
• if (forall k?I,lk?c)
• RR ? KERNEL1(I,g/(li?c))
• return(R)

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Kernel Extract Example

• fabcdabceabef
• cab
• f/ccdceef
• Rcdceef
• N6
• a,b not present
• (cdceef)/ced
• largest cube 1
• recurse-gted

• Rcdceef,ed
• only 1 d
• (dceef)/ecf
• recurse-gtcf
• Rcdceef,ed,cf

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Factoring

• Gfactor(f)
• if (terms1) return(f)
• pCHOOSE_DIVISOR(f)
• (h,r)DIVIDE(f,p)
• fGfactor(h)Gfactor(p)Gfactor(r)
• return(f) // factored

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Factoring

• Trick is picking divisor
• pick from kernels
• goal minimize literals after resubstitution

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Extraction

• Identify cube-free expressions in many functions
  (common sub expressions)
• Generate kernels for each function
• select pair such that k1?k2 is not a cube
• new variable from intersection
• update functions (resubstitute)
• (similar for common cubes)

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Extraction Example

• Xab(c(de)fg)g
• Yai(c(de)fj)k

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Extraction Example

• Xab(c(de)fg)g
• Yai(c(de)fj)k
• de kernel of both
• Lde
• Xab(cLfg)h
• Yai(cLfj)k

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Extraction Example

• Lde
• Xab(cLfg)h
• Yai(cLfj)k
• kernels(cLfg), (cLfj)
• extract McLf
• Xab(Mg)h
• Yai(Mf)h

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Extraction Example

• Lde
• McLf
• Xab(Mg)h
• Yai(Mj)h
• no kernels
• common cube aM

• NaM
• McLf
• Lde
• Xb(Nag)h
• YI(Naj)k

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Extraction Example

• NaM
• McLf
• Lde
• Xb(Nag)h
• YI(Naj)k

• Can collapse
• L into M into N
• Get larger common kernel N
• maybe useful if components becoming too small for
  efficient gate implementation

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Resubstitution

• Also useful to try complement on new factors
• fabac/b/cd
• Xbc
• faX/b/cd
• /X/b/c
• faX/Xd
• extracting complements not a direct target

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Good Divisors?

• Key to CHOOSE_DIVISOR in GFACTOR
• Variations to improve
• e.g. rectangle covering

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Boolean Division

• f//p assuming sup(p)?sup(f)
• add input P to f (for p)
• new function F
• DC-set DP/p/Pp
• ON-set f?/D
• OFF-set /(f?D)
• minimize F for minimum literals in sum-of-products

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Boolean Division (check)

• Sanity (FPpF/P/p)pFp

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Boolean Division Example

• fabca/b/c/ab/c/a/bc
• gab/a/b
• F-DCab/G/a/b/G/abGa/bG
• F-OnabcG/a/bcG/ab/cGa/b/c/G
• minimize
• FcG/c/G

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Boolean Division

• Trick is finding boolean divisors
• Not as clear how to find
• (not as much work on)
• Can always use boolean division on algebraic
  divisors
• give same or better reuslts

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Dont Care

• Mentioned last time
• Structure of logic restricts values intermediates
  can take on

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Dont Care Example

• e/a/b
• g/c/d
• f/e/g
• e0, a0, b0 cannot occur
• DC e(ab)/e/a/b

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Satisfiability DC

• In general
• yF(x,y)
• y/F(x,y) /yF(x,y) is dont care

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Observability DC

• If F output not differentiated by input y,
• then dont care value of y
• FyF/y
• ODC ?(FyF/y)
• (product over all outputs)

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Summary

• Want to exploit structure in problems to reduce
  (contain) size
• common subexpressions
• structural dont cares
• Identify component elements
• decomposition, factoring, extraction
• Division

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Todays Big Ideas

• Exploit freedom
• form
• dont care
• Exploit structure/sharing
• common sub expressions