1. Multilevel Logic Synthesis -- Introduction

2. ENEE 6442 Multilevel Logic Synthesis: Outline > Introduction =What is multilevel logic? =Why we need it? =Problems and challenges. > Multilevel Logic Synthesis and Minimization =Restructuring =Technology Independent Local Optimization > Technology Mapping > Delay Analysis and Optimization (*)

3. ENEE 6443 Multilevel Logic vs. Two-Level Logic Example: Let F=a’b+ab’, define G and H as follows: if F is true, then G=cd+a’d’, H=cd+a’d’+e(f+b), else G=e(f+b), H=(cd+a’d’)e(f+b). else G=e(f+b), H=(cd+a’d’)e(f+b).cd a’ d’ f b F F’ e G Multi-Level Implementation: G=F(cd+a’d’)+F’e(f+b)4-level H=F(cd+a’d’+e(f+b))+F’(cd+a’d’)e(f+b)5-level

4. ENEE 6444 Multilevel Logic vs. Two-Level Logic Example: Let F=a’b+ab’, define G and H as follows: if F is true, then G=cd+a’d’, H=cd+a’d’+e(f+b), else G=e(f+b), H=(cd+a’d’)e(f+b). else G=e(f+b), H=(cd+a’d’)e(f+b). G = F(cd+a’d’)+F’e(f+b) = (a’b + ab’)(cd + a’d’) + (a’b + ab’)’(e)(f + b) = a’bcd + a’bd’ + acdb’ + e(f+b)(a+b’)(a’ +b) = a’bcd + a’bd’ + acdb’ + (efa + efb’ + eba)(a’+b) = a’bcd + a’bd’ + acdb’ + efab + efb’a’ + eba THIS IS MORE COMPLICATED TO IMPLEMENT

5. ENEE 6445 Multilevel Logic vs. Two-Level Logic Two-level: > At most two gates between a primary input and a primary output. > Real life circuits: PLA. > Exact optimization methods: well-developed, feasible. > Heuristics. Multi-level: > Any number of gates between a primary input and a primary output. > Most circuits in real life are multilevel (e.g. standard cells, FPGA). > Smaller, less power, and (in many cases) faster. > Exact optimization methods: few, high complexity, impractical. > Heuristics.

6. ENEE 6446 Optimization Objectives 1.Area : occupied by the logic gates and interconnect, e.g. measured by “one literal = one transistor” in technology independent optimization. 2.Delay : measured by the longest path (critical path) through the logic. 3.Power Consumption Optimization is performed while simultaneously satisfying upper/lower bound constraints placed on these physical quantities.

7. ENEE 6447 Optimization Cost Criteria (Figure source: Prof. Brayton’s lecture notes.)

8. ENEE 6448 > A design is technology independent when the formula (function, system) has no connection with the building blocks in the implementation. > Advantage: cost can be simply estimated by the number of logic symbols (I.e. one literal = one transistor), delay is the length of the “critical path”, better for local optimization. > Example : Full Adder =z(x,y,c) = XOR(x,y,c) = xyc+x’y’c+x’yc’+xy’c’ =c new (x,y,c) = Majority(x,y,c) = xy+(x+y)c =The literal count is 12+5=17 =The critical path delay is 3: x  x+y  (x+y)c  c new Technology Independent > A design is technology independent when the formula (function, system) has no connection with the building blocks in the implementation. > Advantage: cost can be simply estimated by the number of logic symbols (I.e. one literal = one transistor), delay is the length of the “critical path”, better for local optimization. > A design is technology independent when the formula (function, system) has no connection with the building blocks in the implementation.

9. ENEE 6449 > A design is technology dependent if the formula (function, circuit, system) is implemented by one or more logic gates in a pre-designed set of gates (called technology library or cell library). > Advantage: gates in the cell library have a highly optimized, pre-defined path to silicon, so that the area and delay parameters are known and accurate. > Example : Full Adder Technology Dependent 2NOTj’ c new 4NOR(a+f)’j 2NOTh’f 4NAND(dc)’h 2NOTb’d 8XORe’c+ec’z 4NOR(a+b)’e 4NOR(x+y)’b 2NOTg’a 4NAND(xy)’g transi stors gateformulasignal

10. ENEE 64410 Multilevel Logic Synthesis Problem > Goal: =Model the multilevel logic =Optimize the logic based on the cost criteria > Difficulty: =Multilevel gives more design freedom and increases the design complexity =Technology dependent/independent views > Models: =Algebraic forms =Boolean networks > Optimization Techniques: HARD =Exact method: few, exponential complexity, impractical =Approximation method: heuristic algorithm, rules-based methods

11. ENEE 64411 Sum of Products (SOP) > Example: abc’+a’bd+b’d’+b’e’f > Advantages: =Easy to manipulate and minimize. =Algorithms available (e.g. AND, OR, TAUTOLOGY). =Two-level theory applies. > Disadvantages: =Not representative of logic complexity. For example f=ad+ae+bd+be+cd+cef’=a’b’c’+d’e’ which differ in their implementation by an inverter. =Difficult to estimate logic and to estimate progress during logic manipulation.

12. ENEE 64412 Factored Forms: Overview > Example: (ad+b’c)(c+d’(e+ac’))+(d+e)fg > Advantages: =Good representative of logic complexity f=ad+ae+bd+be+cd+ce=(a+b+c)(d+e) f’=a’b’c’+d’e’ =Good estimator of logic implementation complexity =Implicitly Imply Multi-Level Computation (Computation using tree) > Disadvantages: =Not many algorithms available for manipulation =Not canonical: ab+c(a+b)=bc+a(b+c)=ac+b(a+c)

13. ENEE 64413 Factored Forms: Definition A factored form is defined recursively by the rules: 1.A product is either a single literal or a product of factored form. 2.A sum is either a single literal or a sum of factored forms. 3.A factored form is either a product or a sum Example: =Factored forms: x, y’, abc’, ab+c’, ((a’+b)cd+e)(a+b’)+e’. =However, (a’+b’)’c’ is not, although it equals abc’. This is because internal complement in not allowed Another point of view: =p  l (1) (base case) =p  f*f(2) =s  l(3) (base case) =s  f+f(4) =f  p(5) =f  s(6)

14. ENEE 64414 Factoring Tree > Factored forms can be graphically represented as labeled trees, called factoring trees, in which each internal node including the root is labeled with either + or ·, and each leaf has a label of either a variable or its complement. > Example: ((a’+b)cd+e)(a+b’)+e’ + · ++ · + e’ b’a ba’ e cd

15. ENEE 64415 Equivalent Factorizations > Two factored forms are equivalent if they represent the same logic function. =Example:a(b+c)+bcandab+c(a+b) > Two factored forms are syntactically equivalent if their factoring tree are isomorphic. (WHAT IS ISOMOSRPHIC?) =Example: (a+b)(c+d)eand(c+d)e(a+b) a(b+c)+bcandab+c(a+b) + cb + · cb a · ba + · cb a · +

16. ENEE 64416 Maximal Factorizations > A factored form is maximally factored if =For every sum of products, there are no two syntactically equivalent factors in the products; =For every product of sums, there are no two syntactically equivalent factors in the sums. > Example: =ab+ac is not maximally factored, a(b+c) is. =(a+b)(a+c) is not maximally factored, a+bc is. b + · ca a ·

17. ENEE 64417 Optimum Factored Forms > The size of a factored form f,  (f), is the number of literals in the factored form. > Example: =  ((a+b)ca’) = 4 =  ((a+b+cd)(a’+b’)) = 6 > A factored form is optimum if no other equivalent factored form has fewer literals. > Example: =ab+a’c+bc is not optimum (ab+a’c+bc = ab+a’c).

18. ENEE 64418 Unate Factored Forms > A factored form F is positive unate in x, if x appears in F, but x’ does not. A factored form is negative unate in x, if x’ appears in F, but x does not. F is unate in either case, and is binate if it is not unate. > Example: =(a+b’)c+a’ is (positive) unate in c, (negative) unate in b, and binate in a.

19. ENEE 64419 Cofactor of Factored Forms > The cofactor of a factored form F with respect to a literal x (or x’ ) is the factored form F x = F x=1 (x) (or F x’ =F x=0 (x) ) obtained by: =replacing all occurrences of x by 1, and x’ by 0 =simplifying the factored form using the Boolean algebra identities 1y=y 1+y=1 0y=0 0+y=y =after constant propagation (all constants are removed), part of the factored form may appear as G + G. In general, G is another factored form, and the G’s may have different factored forms.

20. ENEE 64420 Cofactor of Factored Forms > The cofactor of a factored form F with respect to a cube c is a factored form obtained by successively cofactoring F with each literal in c. > Example: =F = (x+y’+z)(x’u+z’y’(v+u’)) and c = vz’ F z’ = (x+y’)(x’u+y’(v+u’)) F z’ v = (x+y’)(x’u+y’)

21. ENEE 64421 Algebraic and Boolean Expressions > f is an algebraic expression if f is a set of cubes (SOP), such that no single cube contains another (minimal with respect to single cube containment). Otherwise, f is called a Boolean expression. =Example: a+bc is algebraic, a+ab is Boolean. (ab+a’c+bc???) > Special Properties of Boolean Algebra: =a+a = a·a = a =a+bc = (a+b)(a+c) =a+ab = a > The support of an expression f, supp(f), is the set of variables that f explicitly depends on. Two expressions f and g are said to be orthogonal if supp(f)  supp(g)= , denoted by f  g. =Example: a+b  c+d

22. ENEE 64422 Algebraic and Boolean Factored Forms > A factored form f is said to be algebraic if the SOP expression obtained by multiplying f out directly (I.e. without using aa=a+a=a, aa’=0, a+ab=a) is algebraic, otherwise it is Boolean. > Example: abg+acg+adf+aef+afg+bd+ce+be+cd = (b+c)(d+e)+((d+e+g)f+(b+c)g)a =(bd+be+cd+ce)+(df+ef+gf+bg+cg)a = bd+ce+be+cd+abg+acg+adf+aef+afg(algebraic) = (b+c)(d+e+ag)+(d+e+g)af(algebraic) = (af+b+c)(ag+d+e) =afag+afd+afe+bag+bd+be+cag+cd+ce =afg+adf+aef+abg+acg+bd+be+cd+ce(Boolean)

23. ENEE 64423 Value of a Factorization > Given an algebraic factorization F=G 1 G 2 +R, its factorization value is given by: fact_val(F,G 2 ) = lits(F)-(lits(G 1 )+lits(G 2 )+lits(R)) where lits(F) and cubes(F) are the number of literals and cubes in SOP form of F respectively, G 1,G 2, and R are algebraic expressions. > Example: The algebraic expression F = ae+af+ag+bce+bcf+bcg+bde+bdf+bdg can be expressed in the form F = (a+b(c+d))(e+f+g), which requires 7 literals, rather than 24, a save of 17 literals. If G 1 =(a+bc+bd) and G 2 =(e+f+g), then R=  and fact_val(F,G 2 ) = 2  3+2  5=16. Where the extra literal saving comes from?