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## PowerPoint Slideshow about 'A Denotational Semantics For Dataflow with Firing' - Samuel

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Paper Outline

Paper Discussion for### A Denotational Semantics For Dataflow with Firing

Edward A. Lee

Jike Chong

Wei Zheng

Kahn Process Network Definition

- Functional Nodes
- Unbounded FIFO
- Reads block
- Writes don’t block
- Signals are streams

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

Question from this paper:

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

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

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

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

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

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

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

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

{{$}, {$,#},{$,#,%}}

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

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

Monotonic Function

- untimed notion of causality

Continuous Function

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

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

Composition of Processes

And a collection

Determinate Composition

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

- 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

Least Fixed Point

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

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

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

Higher Order Function

- Functionals
- example

Higher Order Function Example

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

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

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

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

Dataflow processes

- Question:
- Function F exists?
- Function F unique?
- Any assumption?

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

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

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

Commutative firings and Compositionality

How to solve this problem? Replace rule 4 with followings:

Conclusion

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

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-Vincentelliand 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.

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