Inductively Finding a Reachable State Space Over-Approximation

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Inductively Finding a Reachable State Space
Over-Approximation

• EE 290a Project Presentation
• Mike Case

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Sequential Optimization

• One optimization approach

State space
Reachable state space
Can be used as dont cares

• Requires state reachability analysis
• Prohibitively expensive
• Can be approximated

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Van Eijks Method

• Uses induction rather than reachability analysis
• Fast but incomplete
• Finds sequentially equivalent nodes in a network
• Nodes are identical in every reachable state
• States where equivalences hold is an
  over-approximation of the reachable states

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Van Eijks Inductive Hypothesis

• Base Case
• A set of node equivalences holds for the initial
  state
• Inductive Hypothesis
• If equivalences hold in one state then they hold
  in every 1-reachable state as well

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Van Eijk Weaknesses

• Originally for equivalence checking
• Doesnt find many sequential equivalences in
  optimization

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Generalizing Van Eijk

• Find implications rather than exact equivalences

A B A ? B
0 0 1
0 1 1
1 0 0
1 1 1

• Implications subsume equivalences
• (A ? B) ? (B ? A) ? (A B)

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Implication Inductive Hypothesis

• Base Case
• A set of node implication holds for the initial
  state
• Inductive Hypothesis
• If implications hold in one state then they hold
  in every 1-reachable state as well
• Exactly like Van Eijk!

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State Reachability

• Induction gaurantees
• In every reachable state, implications hold

State space
States where implications hold
Reachable state space
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Sequential-Only Implications

• Combinational implications
• True for every state
• Tell us nothing
• Sequential implications
• True for every reachable state
• Gives reachable state space approximation

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Implementation Overview

• Implemented in MVSIS
• Used and-inverter graphs and SAT

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Base Case
1?
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InductiveStep
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Results - Performance
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Results State Space DCs
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Results - Synthesis