Logic Synthesis Optimization Lect18: Multi Level Logic Minimization

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Logic Synthesis OptimizationLect18 Multi Level
Logic Minimization
State university of New York at New
Paltz Electrical and Computer Engineering
Department
Dr. Yaser Khalifa Electrical and Computer
Engineering Department State University of New
York at New Paltz
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Irredundant

• The IRREDUNDANT procedure generates an ISOP with
  as few products as possible from the given set of
  PIs.

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• Let f c1, c2, c3, , cn be the given set of
  PIs. Partition the PIs of F into three sets
• Relatively essential set
• Totally redundant set
• Partially redundant set
• Find a subset of RP that together with ER covers
  F. For each ci in the RP, find the set H(ci) of
  minimum sets MS RP, such that ci (RP MS) ER

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• Algorithm
• Let f c1, c2, c3, , cn be the given set of
  PIs. Partition the PIs of F into three sets
• Relatively essential set
• Totally redundant set
• Partially redundant set
• Find a subset of RP that together with ER covers
  F. For each ci in the RP, find the set H(ci) of
  minimum sets MS RP, such that ci (RP MS) ER

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Why Multi-Level

• Two-level may need a large number of product
  terms.
• Some primary outputs may have huge fanouts.
  Example XOR logic
• Number of product terms of n inputs is
  exponential in n in 2-level
• Number of product terms of n inputs is linear in
  n in Multi-level

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• Multi-level logic optimzation is the process of
  trying to reduce the cost of the representations
  for the given logic functions.
• The cost criteria for optimization can be
• Number of literals (targeting areas of the
  circuits)
• Numbers of levels (targeting delays in the
  circuits)
• Power
• wireability

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• If the specification is given in sum of product
  forms, the first step is to make it more compact
  by introducing more logic levels in the
  representation.
• This is typially done by factoring a logic
  expression and dividing a logic expression with
  another logic expression
• For example, acf bcf adf bdf ef acg
  bcg adg bdg eg is an optimum 2-level
  representation. There are 28 literals.
• It can be factored to (( a b)(c d) e)(f
  g) which needs only 7 literals.

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Optimization Methods

• Boolean Methods Optimization methods that
  consider logic functions and their
  representations.
• Algebric Methods Optimization methods that
  consider only logic representations.
• Algebric methods treat logic functions as
  polynomial expressions.
• Boolean methods recognize Boolean algebra rules
  that can transform one logic representation to
  another.

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Algebric Methods

• We use graph model called Boolean network to
  represent a technology independent multi level
  circuit structure.
• Primary input nodes represent the primary input
  to the circuit
• Primary output nodes nodes that have no outing
  edges.
• Intermediate nodes internal structure of the
  circuit.

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• Each intermediate node is associated with a
  Boolean function called the local function of a
  node.
• All nodes are associated with Boolean variables.
  We treat a node and a variable associated with
  the node interchangeably.
• An edge from node ni to node nj represents that
  the local function of node nj directly depends on
  ni.
• Since Boolean networks are not bound to any
  particular circuit technology, it is very generic
  and is widely used in various logic systhesis
  applications.

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Optimization Methods

• Boolean Methods Optimization methods that
  consider logic functions and their
  representations.
• Algebric Methods Optimization methods that
  consider only logic representations.
• Algebric methods treat logic functions as
  polynomial expressions.
• Boolean methods recognize Boolean algebra rules
  that can transform one logic representation to
  another.

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Models used in Algebric Methods

• There are several ways to represent a local
  function of a node
• SOP expression is the most popular form to
  manipulate
• Factored form is a good measurement for area
  estimation by counting literals. But manipulation
  is difficult.
• Simple gate (AND/OR/NAND/NOR) this mode is easy
  to manipulate. But it is not suitable for
  representing structure containing more complex
  gates.

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• In algebric methods, we treat a SOP expression in
  a different manner from the Boolean method
• A literal is a variable or its negation.
• A positive and negative literal (e.g. a and a/)
  have no relation to each other and are treated as
  mutually independent.
• A cube is a set of literals
• An expression is a set of cubes
• For example F ab c is represented as F1, F2
  a,b,c.
• A capital letter (C) denotes a cube, and a small
  letter (a) denotes a literal.

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• A cube is considered a set of literals, not a
  product of literals. A cube ab is contained in a
  cube abc as a set of literals, i.e.
• An expression is called algebric if no cube in
  the expression contains another.
• The support is the set of variables that appear
  in a given expression. SUP(F)
• If two expressions FF1, F2, and g G1, G2,
  have no common support variables, i.e.
  f, we can define the
  algebric product of two expressions f and g as
  follows,
• For example the product of a bc and d f is ad
  af bcd bcf

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Algebric Division

• Algebric Division is an operation to compute a
  quotient expression Q and a reminder expression R
  from a given expression F and divisor expression
  D such that F and Q.D R are the same
  expressions.
• Before describing the weak division algorithm we
  define the cube division operation, F/D, where F
  is an expression and D is a cube.

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We say that g divides f weakly if there exist h,r
such that Fgh r and g is orthogonal on
h Example fab ac d g b c
f a(b c) d ha rd We say
that g divides f evenly if rf
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Pseudo code for the weak division algorithm
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Kernels

• An expression is called cube-free if no cube
  divides the expression evenly (no reminder)
• The primary divisors of an expression F, denoted
  by D(F), are the set of quotients that are
  derived by dividing F by all possible cubes.
• The Kernels of an expressions F, denoted by K(F),
  is the set of cube free primary divisors of F.
• A cube that is used to drive a kernel is called
  co-kernel.

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• Each kernel has a level
• If a kernel does not contain any othe kernel, the
  level of the kernel is 0.
• If the kernel contains kernels of level n-1 but
  does not contain kernels of whose level is more
  than n-1, the level of the kernel is n.
• Example
• F adf aef bdf bef cdf cef bfg h
• (a b c)(d e)f bfg h

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