A Denotational Semantics For Dataflow with Firing

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A Denotational Semantics For Dataflow with Firing
Paper Discussion for

• Edward A. Lee
• Jike Chong
• Wei Zheng

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Kahn Process Network Definition

• Functional Nodes
• Unbounded FIFO
• Reads block
• Writes dont block
• Signals are streams

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Data-flow network Definition

• A collection of functional nodes
• Connected and communicate over unbounded FIFO
  queues
• Nodes are commonly called actors
• The bits of information that are communicated
  over the queues are commonly called tokens
• Source ee249 slides

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Question from this paper

• What is a denotational semantic?
• Why is Kahn Process Networks important?
• Why do we need firing?

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Question from this paper

• Denotational Semantic
• Two approach of semantics
• Denotational
• Developed by Scott and Strachey
• Meaning of the language is in terms of relation
• Operational
• Turing machine
• Meaning of the language is in terms of actions
  taken by some abstract machine
• Closer to implementation

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Question from this paper

• Why is Kahn Process Networks important?
• General
• Solid semantics
• Properties
• Functional Node
• Unbounded FIFO
• Blocking read
• No total order (concept of partial order)
• No over specification
• May be difficult to determine order in the real
  world

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Question from this paper

• Why do we need firing?
• Basic element of dataflow networks
• Provides notion of tokens passed from inputs to
  outputs
• Enables scheduling of input to output event
• Closer to real world scenarios
• Kahn Process Network Firing
• data-flow network

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Formal Semantic Model Kahn Process Network

• Provides a formal semantic model to construct
  rigorous proofs for Process Networks properties
• Need firing concepts based on the interactions of
  input sequences and the Process Network
  functional nodes

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Paper Outline

• Kahn Process Network property
• Prefix order
• Complete partial order
• Monotonic function
• Continuous function
• Composition of processes
• Determinacy of processes
• Least fixed point semantics
• Higher Order Function Example

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Paper Outline

• Kahn process network with Firing
• Dataflow actors
• Dataflow processes
• Extend Monotonic property to dataflow processes
• Extend Continuous property to dataflow processes
• Continuity of dataflow process
• Commutative firings and compositionality

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Prefix Order

• Prefix order
• is a prefix of ,
• Partially ordered set - poset
• , ,, -- is not a prefix of
  
• Empty signal
• l the empty sequence
• Chain totally ordered set
• , ,,,,

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Complete Partial Order

• Upper bound
• where every element is a prefix
• LUB least upper bound
• LUB(, ,,,,) ,, Union of all
  sets
• Complete Partial Order
• A poset with a bottom s.t. every chain in the set
  has a LUB

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Paper Outline

• Kahn Process Network property
• Prefix order
• Complete partial order
• Monotonic function
• Continuous function
• Composition of processes
• Determinacy of processes
• Least fixed point semantics
• Higher Order Function Example

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Monotonic Function

• untimed notion of causality

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Continuous Function

• Response to an infinite input sequence is the
  limit of its response to the finite approximation
  of this input

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Paper Outline

• Kahn Process Network property
• Prefix order
• Complete partial order
• Monotonic function
• Continuous function
• Composition of processes
• Determinacy of processes
• Least fixed point semantics
• Higher Order Function Example

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Composition of Processes
And a collection
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Determinate Composition

• A composition is determinate if given the input
  sequences, all other sequences are determined.

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Paper Outline

• Kahn Process Network property
• Prefix order
• Complete partial order
• Monotonic function
• Continuous function
• Composition of processes
• Determinacy of processes
• Least fixed point semantics
• Higher Order Function Example

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Least Fixed Point

• Composition does not deal with feedback
• Feedback loops may not have deterministic
  behavior
• Introduce the Least fixed point

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Least Fixed Point

• A method to gain deterministic behavior in
  process networks with feedback topologies
• Start with the empty sequence
• Apply the (monotonic) function
• Apply the function again to the result
• Repeat forever

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Higher Order Function

• Definition
• Functions that take functions as arguments and
  return functions
• CPO over functions
• Use a similar technique to study dataflow with
  firing
• (sm-) ? Is a CPO
• Proof
• All chains in the set have LUB
• Bottom element

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Higher Order Function

• Functionals
• example

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Higher Order Function Example

• Sieve of Eratosthenese
• Output prime numbers by recursive filtering
• mapping functionsonto functions

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Paper Outline

• Kahn Process Network property
• Kahn process network with Firing
• Dataflow actors
• Dataflow processes
• Extend Monotonic property to dataflow processes
• Extend Continuous property to dataflow processes
• Continuity of dataflow process
• Commutative firings and compositionality

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Dataflow with firings

• Based on continuous functionals on POSET of
  functions
• Dennis Dataflow
• Kahn process networks where processes are made up
  of a sequence of atomic computations called
  firings
• Firing can be described as functions
• Firing rule

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Dataflow actors
S needs to be finite
Concatenation of the two tuples of sequences
,
JOIN is defined to be LUB of the two tuples. If
the join exists, They are said to be joinable
Dataflow actor
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Dataflow processes

• Question
• Function F exists?
• Function F unique?
• Any assumption?

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Dataflow processes

• F exists, and unique in the pointwise prefix
  order

• We can prove that f is monotonic and continuous
• least fixed point F such that f(F) F, this F
  satisfied the definition in previous slides
• Find a constructive procedure for finding F
• F is OK to be the semantics of the dataflow
  process

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Paper Outline

• Kahn process network with Firing
• Dataflow actors
• Dataflow processes
• Extend Monotonic property to dataflow processes
• Extend Continuous property to dataflow processes
• Continuity of dataflow process
• Commutative firings and compositionality

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Continuity of dataflow process

• F is the least fixed point of the continuous
  functional f
• F(s) is the LUB of the chain given by (19)

• why do we care about the continuity?
• guarantee the determinacy

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Commutative firings and Compositionality
How to solve this problem? Replace rule 4 with
followings
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Redefine the functional
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Conclusion

• Bridge the gap
• Sequences of firings define a continuous Kahn
  process as the Least Fixed Point of an
  appropriately constructed functional

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References

• 1 A Denotational Semantics for Dataflow with
  Firing, Edward A. Lee, Technical Memorandum
  UCB/ERL M97/3, Electronics Research Laboratory,
  Berkeley
• 2 Models of Computation for Embedded System
  Design, L. Lavagno, A. Sangiovanni-Vincentelli
  and E. Sentovich, 1998 NATO ASI Proceedings on
  System Synthesis, Il Ciocco (Italy) 1998
• 3 Dataflow Process Networks, Edward A. Lee and
  Thomas M. Parks, Proceedings of the IEEE, vol.
  83, no. 5, pp. 773-801 May, 1995
• 4 Lecture notes from ee249/ee290n.