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Multi-Level Optimization

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- 1. Reduce number of literals
- fewer literals means less transistors (less space)
- fewer inputs implies faster gates (less switches in series)
- fan-ins (# of gate inputs) are limited in some technologies

- 2. Reduce number of gates
- number of gates (or gate packages) influences manufacturing costs

- 3. Reduce number of levels of gates
- fewer levels of gates implies reduced signal propagation delays
- minimum delay configuration typically requires more gates (wider less deep circuits)

- Explore tradeoffs between increased circuit delay and reduced gate count
- automated tools to optimize logic and explore possibilities

CSE 567 - Autumn 1998 - Combinational Logic - 1

- Exploit common subexpressions (less gates)
- Minimize number of literals rather than terms
- Trade more levels of logic for reduced fan-in (may also be faster)
- No systematic minimization procedure exists as in the two-level case

X = AC'D + BC'D + ACD' + BCD'(12 literals and 4 wires, max fan-in = 4)

X = (A+B)C'D + (A+B)CD'(8 literals and 6 wires, max fan-in = 2)

X = (A+B)(C xor D) (4 literals and 2 wires, max fan-in = 2)

CSE 567 - Autumn 1998 - Combinational Logic - 2

- Operations on factored forms
- elimination
- decomposition
- extraction
- simplification
- substitution

manipulate network via a collection of transformations

there exists no algorithm that guarantees an "optimal" multi-level network will be obtained

outputs

inputs

each node is an arbitrarily complex gate

CSE 567 - Autumn 1998 - Combinational Logic - 3

- Division with Boolean functionsF = DQ + RD = divisorQ = quotientR = remainder
- Example:X = ac + ad + bc + bd + eY = a + bX/Y = c + dX = Y (c + d) + e

interesting divisors are called kernelsand cubes

remainder

divisor

quotient

CSE 567 - Autumn 1998 - Combinational Logic - 4

- Algebraic division – use rules of algebra (see previous example)
- Boolean division – use rules of Boolean algebraF = ad + bcd + eG = a + bF/G = (a + c) dF = GQ + R =[G (a + c) d] + e(a + b) (a + c) d + e(aa + ac + ab + bc) d + e(a + bc) d + ead + bcd + e

G does not divide F under algebraic rulesG does divide F under Boolean rules (very large number of these)

the key here is the absorption theorem of Boolean algebra

CSE 567 - Autumn 1998 - Combinational Logic - 5

- Kernel: cube-free factor of an expression (no cube can factor it evenly)kernels:a + b, a + cdnon-kernels:a, abc, a(c + d)
- Co-kernel: quotient resulting from dividing the expression by the kernele.g., F = a c + b c + b’ d’kernels:a + bco-kernels: cG = (a + b + c) (d + e) f + gkernels:a + b + c; d + eco-kernels: de, df; af, bf, cf

CSE 567 - Autumn 1998 - Combinational Logic - 6

- Multi-cube algebraic divisors (only other divisors are cubes)
- Can be partitioned into a hierarchy (efficient extraction algorithms)
- level-0 kernel: cannot be divided evenly by a kernel
- level-n kernel: can be divided evenly only by level-(n-1) kernels and itselfF = (a (b + c) + d) (eg’ + g (f + e’)) + (b + c) (h + i)level-0 (among others):b + clevel-1 (among others):a (b + c) + dlevel-2:FF = j (a (b + c) + d) (eg’ + g (f + e’)) + (b + c) (h + i)F is level-3 because it contains a level-2 kernel: (a (b + c) + d) (eg' + g (f + e'))

CSE 567 - Autumn 1998 - Combinational Logic - 7

- Use a cube-literal matrix
- Rectangles represent a cube
- The co-rectangle represents a kernel
- e.g. g = abe + acd + bcd
- cube = cd
- kernel = a+b

CSE 567 - Autumn 1998 - Combinational Logic - 8

- Find the cubes common two several expressions
- Useful for extracting the cubes (factoring)
- e.g. F = abc + abd +egG = abfgH = bd + ef

CSE 567 - Autumn 1998 - Combinational Logic - 9

- First find the kernels and co-kernels (cubes)
- e.g.F = af + bf + ag + cg + ade + bde + cdeG = af + bf + ace + bceH = ade + cde
- (Number these cubes in order of appearance)

CSE 567 - Autumn 1998 - Combinational Logic - 10

- The cokernel-cube matrix
- A column for each cube
- A row for each cube in each function
- Numbers indicate which cubes in the corresponding kernel
- Rectangles in this matrix correspond to common kernels

CSE 567 - Autumn 1998 - Combinational Logic - 11

- Unoptimized logic network

a

b

c

d

e

v = a’d + bd + c’d + ae’

w

x

y

z

p = ce + de

r = p + a’

s = r + b’

t = ac + ad + bc + bd + e

q = a + b

u = q’c + qc’ + qc

CSE 567 - Autumn 1998 - Combinational Logic - 12

- Optimized network

a

b

c

d

e

j = a’ + b + c’

v = jd + ae’

w

x

y

z

s = ke + a’ + b’

k = c + d

t = kq + e

q = a + b

u = q + c

CSE 567 - Autumn 1998 - Combinational Logic - 13

- Removing a node (too simple a function, better to absorb into other gates)

a

b

c

d

e

v = a’d + bd + c’d + ae’

w

x

y

z

p = ce + de

s = p + a’ + b’

t = ac + ad + bc + bd + e

q = a + b

u = q’c + qc’ + qc

CSE 567 - Autumn 1998 - Combinational Logic - 14

- Break a complex node into simpler ones (too complex for a single gate, create opportunities for sharing sub-expressions)

a

b

c

d

e

j = a’ + b + c’

v = jd + ae’

w

x

y

z

p = ce + de

r = p + a’

s = r + b’

t = ac + ad + bc + bd + e

q = a + b

u = q’c + qc’ + qc

CSE 567 - Autumn 1998 - Combinational Logic - 15

- Finding common sub-expressions and pulling them out into their own node(most important and complex function in multi-level optimization)

a

b

c

d

e

v = a’d + bd + c’d + ae’

w

x

y

z

p = ke

r = p + a’

s = r + b’

k = c + d

t = ka + kb + e

q = a + b

u = q’c + qc’ + qc

CSE 567 - Autumn 1998 - Combinational Logic - 16

- Two-level minimization applied to a node (exploit structural don't cares)

a

b

c

d

e

v = a’d + bd + c’d + ae’

w

x

y

z

p = ce + de

r = p + a’

s = r + b’

t = ac + ad + bc + bd + e

q = a + b

u = q + c

CSE 567 - Autumn 1998 - Combinational Logic - 17

- Reuse existing nodes to make others simpler (closely linked to extraction and decomposition)

a

b

c

d

e

v = a’d + bd + c’d + ae’

w

x

y

z

p = ke

r = p + a’

s = r + b’

k = c + d

t = kq + e

q = a + b

u = q’c + qc’ + qc

CSE 567 - Autumn 1998 - Combinational Logic - 18

a

a =1, b = 1, x =1 can never occur

x

b

- Don't cares come from two sources in multi-level circuits
- From specification (external explicit don't cares)
- in terms of circuit inputs and outputs

- From structure of circuit graph (internal implicit don't cares)
- a combination of input and internal values cannot occur or
- an internal node output is irrelevant for some input combinations depending on how it is used by its fanout

- Both are critical in arriving at minimal circuits
- Must be maintained throughout all graph operations

CSE 567 - Autumn 1998 - Combinational Logic - 19

- Decrease fanout of nodes
- more destinations for a signal implies slower transmission
- elimination

- Decrease fanin of nodes
- gate speed proportional to square of number of inputs (1st order)
- decomposition, simplification

- Move late input closer to outputs
- make path to output shorter, pre-compute other logic
- Shannon decomposition (f = a fa + a’ fa’)

A

A

A is a late arriving inputthat is moved closer to the output by restructuring the logic(i.e., changing DAG structure)

CSE 567 - Autumn 1998 - Combinational Logic - 20

- Minimization procedures
- heuristic application of the operations we just listed
- no guarantee of finding an optimal realization
- does quite well in a practical amount of time (with algebraic division)

- Everything up to this point has been technology independent
- just considering literal count or depth of circuit
- not the types of elements available to actually implement the circuit

- Technology mapping
- process of converting circuit graph into one where each node is directly implementable with an available gate or function block

CSE 567 - Autumn 1998 - Combinational Logic - 21

- Process of transforming logic network so that all nodes can be directly implemented with an available component directed toward area or speed optimization
- Requires library of available gates
- permutations of inputs (e.g., a•b + c – a and b can be switched)
- area and delay for each library gate

- Example:

NAND4area: 8delay: 8

NAND2area: 4delay: 2

AOI21area: 6delay: 5

XOR2area: 16delay: 6

CSE 567 - Autumn 1998 - Combinational Logic - 22

- Represent function in terms of 2-input NAND gates
- Not a unique representation
- library must represent all non-isomorphic possibilities

- Example:
- F = (ABCD)'has two representations

CSE 567 - Autumn 1998 - Combinational Logic - 23

cell in library

node in graph

- Dynamic programming algorithm
- taken from code generation – Aho and Johnson's TWIG

- DAG is viewed as a forest of trees (two options)
- 1. partition into trees (break graph at fanout nodes)
- 2. duplicate logic in common sub-trees

- Consider adding inverter pairs along any arc of original DAG

CSE 567 - Autumn 1998 - Combinational Logic - 24