Document Sample

Multilevel Logic Synthesis -- Introduction 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 (*) ENEE 644 2 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). 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 c d a’ F G d’ F’ e f b ENEE 644 3 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). 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 ENEE 644 4 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. ENEE 644 5 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. ENEE 644 6 Optimization Cost Criteria (Figure source: Prof. Brayton’s lecture notes.) ENEE 644 7 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. Example: Full Adder z(x,y,c) = XOR(x,y,c) = xyc+x’y’c+x’yc’+xy’c’ cnew(x,y,c) = Majority(x,y,c) = xy+(x+y)c The literal count is 12+5=17 The critical path delay is 3: xx+y(x+y)ccnew ENEE 644 8 Technology Dependent A design is technology signal formula gate transi dependent if the formula stors (function, circuit, system) is g (xy)’ NAND 4 implemented by one or more a g’ NOT 2 logic gates in a pre-designed b (x+y)’ NOR 4 set of gates (called technology library or cell library). e (a+b)’ NOR 4 Advantage: gates in the cell library z e’c+ec’ XOR 8 have a highly optimized, pre-defined d b’ NOT 2 path to silicon, so that the area and delay parameters are known and h (dc)’ NAND 4 accurate. f h’ NOT 2 Example: Full Adder j (a+f)’ NOR 4 cnew j’ NOT 2 ENEE 644 9 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 ENEE 644 10 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+ce f’=a’b’c’+d’e’ which differ in their implementation by an inverter. Difficult to estimate logic and to estimate progress during logic manipulation. ENEE 644 11 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) ENEE 644 12 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 Another point of view: Example: p l (1) (base case) Factored forms: x, y’, abc’, p f*f (2) ab+c’, ((a’+b)cd+e)(a+b’)+e’. s l (3) (base case) s f+f (4) However, (a’+b’)’c’ is not, f p (5) although it equals abc’. This is because internal f s (6) complement in not allowed ENEE 644 13 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’ + + · e a b’ + c d a’ b ENEE 644 14 Equivalent Factorizations Two factored forms are equivalent if they represent the same logic function. Example: a(b+c)+bc and ab+c(a+b) Two factored forms are syntactically equivalent if their factoring tree are isomorphic. (WHAT IS ISOMOSRPHIC?) Example: (a+b)(c+d)e and (c+d)e(a+b) a(b+c)+bc and ab+c(a+b) + + · · · · a + b c a b c + b c a b ENEE 644 15 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 b a c a+bc is. ENEE 644 16 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). ENEE 644 17 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. ENEE 644 18 Cofactor of Factored Forms The cofactor of a factored form F with respect to a literal x (or x’ ) is the factored form Fx= Fx=1(x) (or Fx’=Fx=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. ENEE 644 19 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’ Fz’ = (x+y’)(x’u+y’(v+u’)) Fz’ v = (x+y’)(x’u+y’) ENEE 644 20 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 fg. Example: a+b c+d ENEE 644 21 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) ENEE 644 22 Value of a Factorization Given an algebraic factorization F=G1G2+R, its factorization value is given by: fact_val(F,G2) = lits(F)-(lits(G1)+lits(G2)+lits(R)) where lits(F) and cubes(F) are the number of literals and cubes in SOP form of F respectively, G1,G2, 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 G1=(a+bc+bd) and G2=(e+f+g), then R= and fact_val(F,G2) = 23+25=16. Where the extra literal saving comes from? ENEE 644 23

DOCUMENT INFO

Shared By:

Categories:

Tags:
Boolean algebra, truth table, Boolean expression, truth tables, logic gates, boolean function, boolean logic, Boolean expressions, two elements, Boolean ring

Stats:

views: | 73 |

posted: | 5/12/2011 |

language: | English |

pages: | 23 |

OTHER DOCS BY wanghonghx

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.