A Theory of Non-Deterministic Networks

1
A Theory of Non-Deterministic Networks
R. K. Brayton EECS Dept. University of
California Berkeley
2
Overview

• What is an ND network
• Motivation
• Basic definitions
• Defining and comparing ND behaviors
• ND network operations and how they change
  behaviors
• Experimental observations
• Conclusions and future work

3
What is an ND network?
PO

• Similar to a Boolean network except
• Each node has a single output multi-valued
  variable
• Each node has a non-deterministic relation
  relating its input and output values.

F
MV
ND
PI
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Why consider ND networks?

1. Dont cares are a form of non-determinism. They
   generalize to non-determinism when considering
   multi-valued logic
2. Multi-valued domains can be used to explore
   larger optimization spaces.
3. ND arises naturally when considering the
   flexibility of implementing a node in a network
4. Given a ND relation, the minimum well-defined ND
   sub-relation is always smaller than the minimum
   deterministic one
5. Can be used to compile software to evaluate logic

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Definition Boolean Network

• Directed Acyclic Graph
• Each node represents a Boolean function
• Edge from node j to node k if the function at k
  depends syntactically on the variable yj at the
  output of j
• Primary inputs (PI) X and outputs (PO) Z
• All signals are binary
• External specification provides allowed input and
  output combinations (external dont cares)

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Definition ND Multi-Valued Network

• Network of MV-nodes (PI, PO, internal)
• Each node is represented by an MV variable yn
  with its own range0, 1,, pn-1
• Internal node is represented by an MV
  non-deterministic relation

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Example Ternary Relations
All relations are well-defined, i.e. for each
input minterm there exists at least one output
value
R1
R2
R3
b/a 0 1
0 0 2
1 0 0
2 1 0
b/a 0 1
0 0,1,2 2
1 0 0
2 0,1,2 0
b/a 0 1
0 0,1 1,2
1 0,1 0
2 0,1,2 0,1
2
R1 is completely specified (deterministic) R2 is
incompletely specified R3 is partially specified,
or non-deterministic R1 is contained in R2 R2 is
not contained in R3
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ND Network Behavior

• Given an ND network, what is its behavior, i.e.
  what is the set of all PI/PO pairs that are
  related?
• this question is not straightforward.
• For a deterministic, well-defined network, there
  is exactly one PO vector for each PI vector
• however, if there are some external dont cares,
  then there may be several PO vectors for a PI
  vector,
• but dont cares are well understood.

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ND Network Behaviors (PI/PO Pairs)

• Normal Simulation (NS)
• Normal Simulation made Compatible (NSC)
• Set Simulation (SS)

Note all these become the same when the network
is deterministic.
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Normal Simulation

• Network is evaluated in topological order
• At each node its fanins have a specific vector of
  values.
• The relation at the node determines a set of
  possible output values of that node
• One of these is chosen randomly and broadcast to
  the fanouts

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Normal Simulation
0,2
2
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NSC Simulation

• Like normal simulation except that each PO is
  handled separately.
• Take one PO, j, and look at the transitive fanin
  cone.
• Compute (X, zj ) pairs using NS.
• Repeat for all PO

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NSC Simulation
1
3
PO2
PO2
PO1
2
4
PO1
2
3
PI/PO relation contains 3 1 1 / 2 1 3 1 1 / 2
3 3 1 1 / 4 1 3 1 1 / 4 3 It is the cross
product of all PO sets
1
0
0,2
2
0
0
2
fanins
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Set Simulation

• Done in topological order.
• On each signal a set of values is obtained
• At each node a vector of fanin sets is known.
• The output set of values for a node is the union
  of the sets obtained for all fanin vectors in the
  cross product of the fanin sets

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Set Simulation
PO2
PO1
1,2,4
1,4
0,1
PI/PO relation contains 3 1 1 / 1 1 3 1 1 / 1
3 3 1 1 / 2 1 3 1 1 / 2 3 3 1 1 / 4 1 3 1 1 / 4
3 It is the cross product of all PO sets
0,2
fanins
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SS is not Scattered Simulation

• Scattered simulation
• for each fanout one value is selected from the
  acceptable set.
• each fanout may have different values
• Example

0
0,1
0,1
0
1
0,1
0
1
0,1
1
0
0,1
0
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Comparisons

• is a general MV Boolean relation
• relatively hard to compute and store
• and can be
  computed for each output and
  They are symmetric Boolean relations.
• can be obtained by elimination
  in reverse topological order
• can be obtained by elimination in
  topological order

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Computing RNS input determinization

• At each ND node introduce one MV parameter pi
  with the same range as the node output.
• Relation at node i is replaced by
• pi controls the output value of node i
• the operator m is a special BDD projection
  operator, defined by Bill Lin, that projects onto
  the smallest allowed output value.
• RNS can be obtained by eliminating all internal
  nodes and existentially quantifying all
  parameters.

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External Specification

• Can be specified by
• The initial network plus dont cares
• e.g. in Boolean networks, we can give external
  dont cares, one set for each output.
• A separate specification (network or BDD or
  other)
• Notation
• Requirement conformity

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Conformity with External Specification

• Can use any one of the behaviors
• Just be consistent
• For example, we may have but
  If we use consistently there
  is no problem.
• Ultimately, in most applications we want a final
  deterministic network.
• If any behavior conforms, then it contains only
  correct deterministic ones

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Goal of this paper

• Develop a theory of ND networks where
• a network can be manipulated using classical
  operations,
• eliminate, optimize, decompose,
• all the intermediate networks conform to the
  external specification.
• we need to understand when a particular network
  operation may increase a particular type behavior
• this might cause the network to not conform

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Network Manipulations

• Eliminate a node
• Optimize a node
• Decompose a node
• Substitute one node into another
• Partially Encode a node
• Merge nodes

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Eliminating a node into a fanout
i is eliminated into k
yi
If i has been eliminated in all of its fanouts,
it can be removed from the network
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Elimination can increase behavior
Theorem 10 Eliminating a node can increase the
NS or NSC behaviors of a network only if the node
is ND and has more than one fanout.
Increased behavior because there are now two
copies in the network
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Elimination and NSC Behavior
Theorem 11 Eliminating a node can increase a
networks NSC behavior if and only if the node is
ND and has reconvergent fanout.
ND node
4
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Elimination and SS behavior
Corollary 1 Eliminating a node can never
increase the SS behavior of a network.

• Corollary 2 Eliminating a node A can decrease
  the SS behavior of a network only if
• A has an ND node B in its TFI and
• a fanin C of A in the TFO of B is also a fanin
  of a fanout of A

C
B
A
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Minimizing a Node Computing the Flexibility at
a Node

• Definition. A flexibility at node ? is a relation
  Rf? such that placing at ? any well-defined
  deterministic relation contained in Rf? leads to
  a network that conforms to the external
  specification.
• Definition. The complete flexibility (CF) is the
  maximum flexibility at a node.

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Computing the Global CF
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Imaging into the Local Space
Yi
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Properties of Flexibilities

• If a network conforms, then any well-defined
  deterministic function contained in
  is acceptable at node j, for
• For NS or NSC, any ND relation will also be
  acceptable
• But for SS, it is possible that an ND relation
  contained in can cause the
  network to not conform

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Flexibility
M is the number of the input minterms V is the
size of the output range. ti is the number of
output values in the relation for input minterm
mi
The amount of flexibility is equal to 0 for
completely specified functions and 100 for
relations that take all values in any minterm.
Examples M6, V3
R1
R2
R3
b/a 0 1
0 0,1 1,2
1 0,1 0
2 0,1,2 0,1
b/a 0 1
0 0 2
1 0 0
2 1 0
b/a 0 1
0 0,1,2 2
1 0 0
2 0,1,2 0
T 10
T 12
T 6
F 0
F 33
F 50
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Amount of Flexibility (SDC, CODC, CF)
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Node Simplification

• Compute and use complete flexibility (CF) to
  simplify the node. Recall
• CF in global space
• CF in local space
• Use to optimize MV-SOP (heuristic,
  exact) at node j

We will look at how to find the smallest
well-defined SOP representation contained in a
given ND relation
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Finding minimum deterministic SOP representation

• It is never smaller than the smallest ND
  representation
• There is no known algorithm for finding it.
• In contrast, there is a method for finding the
  smallest ND representation

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Quine-McCluskey type exact ND SOP relation
minimization

• Given an ND relation, e.g. the complete
  flexibility, its i-set is the set of input
  minterms that can produce output value i.
• For each i-set, generate all its primes, Pi
• Form covering table with
• one column for each pj in Pi for all i
• one row for each minterm in the input space
• Solve minimum covering problem
• Primes chosen from Pk is the cover for kth i-set.

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Comparing Changes in Behaviors
Table 1. Comparing two computationally viable
theories
Operation NSC-behavior SS-behavior
elimination non-compliance compliance
node minimization compliance non-compliance
decomposition compliance non-compliance
merging cant change cant increase
node flexibility more less
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Experimental Setup

• These ideas have been implemented in a system,
  MVSIS
• The SS behavior has been used throughout.
• it is the easiest to use computationally
• global behavior can be expressed locally at each
  node as a BDD of the PI
• In the future we will experiment with using NSC
  behavior

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Experimental ObservationsSS Behavior

• Conformity is rarely lost but it does happen.
  This usually happens during node minimization.
• If we use an ND relation at the minimized node,
  then conformity is not guaranteed (only
  deterministic SOP guarantees conformity using SS)
• Often conformity is automatically regained by
  minimizing the next node.
• If the CF at the next node is well defined, this
  means that the network can be brought back to
  conformity.
• If it is not well defined, we leave the node
  relation alone and move to the next node.
• We have never experienced a final network that
  does not conform to the external specification.

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Goals of the Future Work

• Develop efficient Boolean optimization algorithms
  working on ND networks
• Explore common computational core of these
  algorithms

core
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Future Work

• We believe NSC behavior will be superior.
• need to solve computation efficiency problems
• elimination in reverse topological order means
  that intermediate variables have to be used
  (rather than only PI)
• which means that it is easier to
  maintain conformity.
• implies that NSC-CF contains more flexibility
  than SS-CF
• however, elimination can cause non-conformity

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The End
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Partial Encoding (Value Reducing)

• R2 is a wire, a cube, or a given function
• e.g. a library element (technology mapping)
• Transformation is accepted if
• ?log2 v2? ?log2 v1? ? ?log2 v? or
• v2 v1? v

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Merging

• Inverse of partial encoding

merge v1 and v2
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Decomposition (extraction) and Encoding

1. Select bound set XB
2. Imagine that a block B1 has v1 values. (v1 is the
   product of values in XB)
3. Compute the CF of block B1
4. Encode B1 using value-reducing encoding to get B2
5. Inputs in XC are shared, which leads to
   non-disjoint decomposition