A Causality Interface for Deadlock Analysis in Dataflow

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A Causality Interface for Deadlock Analysis in
Dataflow

• Ye Zhou and Edward A. Lee
• University of California, Berkeley

EMSOFT 2006 Seoul, Korea Oct 22-25, 2006
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Outline

• Introduction
• Causality Interfaces
• Composition of Causality Interfaces
• Discussion
• Conclusion

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Introduction
connector
actor
output port
input port

• Actor receives tokens from input ports and reacts
  to these tokens by producing tokens on the output
  ports.
• What flows in the wires (connectors) are
  sequences of tokens.

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Introduction (Contd)

• Any actor network can be treated as a feedback
  system.
• We assume all actors are (Scott) continuous and
  use the least fixed point semantics as the
  behavior of a dataflow network.
• Question Will the network deadlock?

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Related Work

• Lee, Messerschmitt, 1987 Focuses on Synchronous
  Dataflow (SDF)
• Buck, 1993 Focuses on Boolean Dataflow
• Wadge, 1981 Cycle Sum Test
• Matthews, 1995 Partial Metrics
• Our approach interfaces

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Causality Interfaces

• A special family of behavioral interfaces.
• Capture the causality properties of an actor,
  which reflects the data dependency of an output
  port on an input port.
• A mathematic structure that helps to determine
  whether an actor network is live under certain
  model of computation.

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Outline

• Introduction
• Causality Interfaces
• General Definition
• Causality Interfaces for Dataflow
• Composition of Causality Interfaces
• Discussion
• Conclusion

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General Definition

• A causality interface for an actor a with input
  ports Pi and output ports Po is a function
• where D is a partially ordered set with elements
  called dependencies.

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How to compose dependencies?

• Serial connection
• Parallel connection
• We need two operators, one for serial connections
  ( ), one for parallel connections ( ).

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Dependency Algebra Axioms

• Dependency set D is a partially ordered set with
  two binary operators (for parallel) and
  (for serial) that satisfy the following axioms
• Associativity
• Commutativity and Idempotence (for only)

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Dependency Algebra Axioms (Contd)

• Ordering Axiom

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Dependency Algebra Examples

• We have previously given dependency algebras for
  the following models of computation.
• Discrete-event (DE) models
• Synchronous/Reactive (SR) models

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Causality Interfaces for Dataflow

• The dependency set D for dataflow models is a set
  of functions
• computes the greatest lower bound of two
  functions is function composition.

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Interpretation

• represents the causal
  relationship between the first n tokens at the
  input port pi and the first d(n) tokens at the
  output port po.
• For example, consider an actor with one input
  port pi and one output port po. Given n tokens at
  pi, there will be d(n) tokens at po.
• In general, an actor may have different
  interfaces d for each possible input sequence,
  but simple actors have just one.

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Outline

• Introduction
• Causality Interfaces
• Composition of Causality Interfaces
• Discussion
• Conclusion

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Feedforward Compositions

• Use for serial compositions and for
  parallel compositions.
• Example

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Feedback Compositions

• The gain of a cyclic path c (p1, p2, , pn, p1)
  is
• Productivity order

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Feedback Compositions (Contd)
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Liveness Condition
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Example Adaptive Filtering
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Outline

• Introduction
• Causality Interfaces
• Composition of Causality Interfaces
• Discussion
• Conclusion

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Decidability

• Buck, 1993 Deadlock is generally undecidable
  for dataflow models.
• Causality interfaces for some dataflow actors
  (e.g., boolean select and switch) depend on input
  sequences, so is in general
  undecidable.

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Causality Interfaces for Synchronous Dataflow
(SDF)

• Causality interfaces for SDF
• where N is the consumption rate, M is the
  production rate, and I is the number of initial
  tokens at the output.
• THEOREM 3 Deadlock is decidable for synchronous
  dataflow models with a finite number of actors.

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Complexity of the Analysis

• Question Do we need to check for all
  cyclic paths?
• (p1, , pn, p1) and (pi, , pn, p1, , pi) are
  two different cyclic paths of the same cycle.
• A simple cycle is a cycle that does not contain
  any other cycles.
• Answer it is sufficient to check one cyclic path
  of each simple cycle.

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Outline

• Introduction
• Causality Interfaces for Dataflow
• Composition of Causality Interfaces
• Discussion
• Conclusion

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Conclusion
Using the same algebraic structure previously
applied to DE and SR models, we give

• A causality interface theory for dataflow.
• An algebraic procedure to analyze liveness in
  dataflow networks.
• Liveness is decidable for synchronous dataflow.
• Causality analysis only needs to be performed for
  one cyclic path of each simple cycle.

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• Questions?