Combinational and Sequential Flexibility in Logic Networks

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Combinational and Sequential Flexibility in Logic
Networks
290N The Unknown Component Problem Lecture 3
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Overview

• Logic networks
• Flexibility
• Complete vs. compatible flexibility
• Combinational case
• Satisfiability dont-cares
• Observability dont-cares
• External dont-cares
• Complete combinational flexibility
• Computational procedures
• Implementation

• Sequential case
• Input dont-care sequences
• Output dont-care sequences
• External specification
• Complete sequential flexibility

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Logic Network

• Logic network is a direct acyclic graph

Primary outputs (POs)
Internal nodes
Primary inputs (PIs)
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Fanin/Fanout of a Node

• Node has only one output.
• Node can have any number of inputs (fanins) and
  can be an input to any number of nodes (fanouts)

FO1
FO2
FO3
Fanouts
N
Node
FI2
FI3
FI1
Fanins
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Transitive Fanin/Fanout of a Node
Transitive fanout (TFO)
Node
Transitive fanin (TFI)
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Flexibility at a Node
External specification
Logic Network

• A flexibility at a node is a relation between the
  nodes inputs and outputs, such that any
  well-defined sub-relation used at the node leads
  to a logic network that conforms to the external
  specification.

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Complete vs Compatible Flexibility

• Definition. The complete flexibility (CF) is the
  maximum flexibility possible at a node, assuming
  that other nodes are fixed.
• Definition. A compatible flexibility (CF) is a
  flexibility at a node, assuming that other nodes
  can change, to some extent.
• Typically, compatible flexibilities are assigned
  to all of the nodes in one pass over the network.
  Then, each node is minimized independently.
• On the other hand, complete flexibility at a
  node, once computed, should be used before moving
  on to other nodes.

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Flexibility of Nodes in the Network

• Internal flexibility (dont-cares)
• Satisfiability dont-cares
• Some input combinations that never occur at a
  node
• Observability dont-cares
• Under some input combinations, the value produced
  at the output of the node does not matter
• External flexibility (dont-cares)
• Some input combinations never occur (unused
  codes, unreachable states)
• Complete flexibility (dont-cares)
• The sum of the three above

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Satisfiability Dont-Cares

• (x,y)(1,0) is a dont-care for node F

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Observability Dont-Cares

• (a,c)(1,1) is a dont-care for node F

z1
z2
F
a
b
c
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Global and Local Flexiblity
Local Space
Global Space
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Computing CF - global step
Original network Rspec( X, Z ) (can be also
given as an external specification)
Perturbed network R( X, yi, Z )
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Computing CF - local step
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Computing CF - local step
The same computation applies for multiple-output
nodes, i.e. where
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Sequential Flexibility for FSM
Mo

• Output dont-care sequences of Mx exist because
  Mo does not distinguish some string
• observability dont-cares

Mx

• Input dont-care sequences of Mx exist because Mi
  does not produce some strings
• satisfiability dont-cares

Mi
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FSM Networks

• Problem Given a network of finite state
  machines, compute the Complete Sequential
  Flexibility at a node

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Problem Formulation
Specification S (i,o) Context F
(i,v,u,o) Unknown X (u,v)
Problem Given S and F, find the Most General
Solution (MGS) of
Solution
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Definitions

• Most General Solution (MSG) is the automaton
  solution of the language equation F ? X ? S, such
  that any other automaton solution is contained in
  it.
• Complete Sequential Flexibility (CSF) is the
  maximum set of FSM behaviors (represented by a
  pseudo-non-deterministic FSM), such that
  implementing any sub-behavior of CSF, and
  replacing the sub-network by the implemented
  part, does not violate the specification of the
  total network.

• CSF is maximum sub-behavior of MGS, which is
  prefix-closed and u-progressive
• for unknown to be an FSM, it must be progressive
  in its inputs

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Algorithm to Compute
Algorithm CompleteSequentialFlexibility Input
FSM F, FSM (or automaton) S Output pseudo-non-
deterministic FSM X begin 01 X Complete ( S,
non-accepting ) 02 X Complement ( X ) 03 X
Support (X, (i,v,u,o)) lift 03 X Product
( F, X ) 04 X Support ( X, (u,v) ) -
restrict 05 X Determinize Complement( X
) 06 X PrefixCloseuProgressive( X,
u ) return X end