Lusteral Uniform and Modular Composition of Data-flow & - - PowerPoint PPT Presentation

Synchron 2008 – CPL Aussois

Lusteral

Uniform and Modular Composition of Data-flow & Control-flow in the Lazy λ-Calculus

• M. Mendler, University of Bamberg

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joint work with Marc Pouzet

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Constructive Data Flow

• M. Mendler, The Otto-Friedrich University of Bamberg

Lusterel-2

Introduction

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Data Flow Programming

• M. Mendler, University of Bamberg

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Signal, Lustre, SCADE V4, LabView, Simulink, ...

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Control Flow Programming

• M. Mendler, University of Bamberg

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wA

A/arm

wB

B/arm

WaitAandB

done

/ AB

dA dB

ABSync Reset signal arm

idle cnt 1

arm Detection Inhib

Timer

2 T / disarm

eot

disarm

2

signal disarm

Esterel, SyncCharts, Argos, Stateflow, Statemate, ...

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Starting the 3rd Millennium ...

ReLuC, Lucide Synchrone, SCADE 6, 7, 8, Synoptic, ...

... we are looking at hybrid engines with tight integration

• f data flow and control flow:

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Starting the 3rd Millennium ...

... we need to understand the semantical mechanics of such a hybrid (4 education, certification, verification,...)

• M. Mendler, The Otto-Friedrich University of Bamberg

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Recent work (2005): J.-L. Colaço, B. Pagano, M. Pouzet give semantics of Scade V6 by

• syntactic compilation of
• SF DF using states-as-activation-clocks with explicit

absence values Open Question: Can SF and DF be unified into a

• compositional semantic model
• on equal footing, without „absence“ values ?

equational reasoning higher-order polymorphic declarative extensible

Lusterel Language Design Workbench

strong typing smooth morphing between data and control-flow

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The Lusteral Experiment

• Lustre-style DF + Esterel-style SF under the same (co-

recursive, functional) reactive process model

• shallow embedding in Haskell, aka lazy λ-calculus
• separation control ≠ data (no absence values)
• simple set of well-understood functional combinators
• M. Mendler, University of Bamberg

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We aim to integrate

Lustre DF Esterel SF

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Kahn Principle

• M. Mendler, University of Bamberg

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• At any time, a computing station is either computing or

waiting for information on one of its input lines

• Each computation station follows a sequential program

Working Hypothesis:

• Stream processing functions of Haskell are Kahn

processes: Lazy rewriting is deterministic and sequential !

• Kahn processes can be coded in Haskell. Kahn processes

exhibit parallelism of expression not of execution!

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Constructive Data Flow

• M. Mendler, The Otto-Friedrich University of Bamberg

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The Role of Laziness

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Orthogonality in Time and Space

• M. Mendler, University of Bamberg

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Data Flow

• M. Mendler, University of Bamberg

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data flow

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Data Flow

• M. Mendler, University of Bamberg

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data flow Q: How do we treat the cyclic DF dependencies ? A: Fixpoints on prefix-ordering, lazy recursion on streams!

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State Flow

• M. Mendler, University of Bamberg

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state flow

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State Flow

• M. Mendler, University of Bamberg

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state flow Q: How do we treat the cyclic SF dependencies ? A: Scott information-ordering, lazy recursion on state

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Fixed-point on Partial States

• M. Mendler, University of Bamberg

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xs = ys = zs = xs = f (3 fby (fst zs), snd zs) zs f h g

fby

3 ys xs ys = h (snd xs) zs = g (ys, fst xs) 3 5 6 2

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Fixed-point on Partial States

• M. Mendler, University of Bamberg

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xs = ys = zs = xs = f (3 fby (fst zs), snd zs) zs f h g

fby

3 ys xs ys = h (snd xs) zs = g (ys, fst xs) 3 9 5 6 1 2 9 4

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Fixed-point on Partial States

• M. Mendler, University of Bamberg

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4 1 9 3 5 2 5 2 1 xs = ys = zs = xs = f (3 fby (fst zs), snd zs) zs f h g

fby

3 ys xs ys = h (snd xs) zs = g (ys, fst xs) 7 7 9 5 1 3 5 6 2

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Where Eagerness is not Productive ...

• M. Mendler, University of Bamberg

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xs = f (3 fby (fst zs), snd zs) ys = h (snd xs) zs = g (ys, fst xs) fst (f(3:fst zs), snd(g(h(snd(f(3 fby (fst zs), snd zs))),fst xs) … loop ! fst xs fst (f(3 fby (fst zs), snd zs)) fst (f(3:(fst zs), snd zs)) fst (f(3:(fst zs), snd(g(ys,fst xs)))) fst (f(3:(fst zs), snd(g(h(snd xs),fst xs)) zs f h g

fby

3 ys xs Eager Evaluation

lazy on streams lazy on state (data)

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... Lazy Function Evaluation may be.

• M. Mendler, University of Bamberg

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xs = f (3 fby (fst zs), snd zs) ys = h (snd xs) zs = g (ys, fst xs) fst xs fst(f(3 fby (fst zs), snd zs)) fst((f1 × f2)(3 fby (fst zs), snd zs)) fst((λw.(f1(fst w), f2(snd w))(3 fby (fst zs), snd zs)) fst(f1(fst(3 fby (fst zs), snd zs)), f2(snd(3 fby (fst zs), snd zs))) f1(3:(fst zs)) f1(3) : f1(fst zs) f1(fst(3 fby (fst zs), snd zs)) f h g

fby

3 ys xs zs f1(3 fby (fst zs)) ... f1(3) : H(f1(3)) : H(H(f1(3))) : H(H(H(f1(3)))) : ... where H[v] = f1(g1(h(f2(g2(v)))) Lazy Evaluation

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Constructive Data Flow

• M. Mendler, The Otto-Friedrich University of Bamberg

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SF ≠ Clocked DF

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Clock Schedules are Implicit

• M. Mendler, University of Bamberg

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when

xs cs zs xs = x0 x1 ⊥ x2 ⊥ x3 x4 ⊥ x5 x6 ... cs = T F ⊥ F ⊥ T F ⊥ T T ... zs = x0 ⊥ ⊥ ⊥ ⊥ x3 ⊥ ⊥ x5 x6 ... cs?c xs?x zs!x tick c=F c=T tick absence ⊥ is not communicated !

loop read c from cs; if c then read x from xs; write x to zs; else pause; pause; end loop loop read c from cs; if c then read x from xs; write x to zs; else pause; pause; end loop

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Absence = Identity Reaction on ‘State Bus‘

• M. Mendler, University of Bamberg

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when

xs cs1 zs xs = x0 x1 ⊥ x2 ⊥ x3 x4 ⊥ x5 x6 ... cs1= T F ⊥ F ⊥ T F ⊥ T T ... zs = x0 ⊥ ⊥ ⊥ ⊥ x3 ⊥ ⊥ x5 x6 ... ys = y0 y1 ⊥ y2 ⊥ y3 y4 ⊥ y5 y6 ... cs2= F T ⊥ T ⊥ F T ⊥ F F ... zs = ⊥ y1 ⊥ y2 ⊥ ⊥ y4 ⊥ ⊥ ⊥ ...

when

ys cs2 zs

zs = x0 y1 ⊥ y2 ⊥ x3 y4 ⊥ x5 x6 ...

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Absence = Identity Reaction on ‘State Bus‘

• M. Mendler, University of Bamberg

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when

xs cs1 zs zs = x0 ⊥ ⊥ ⊥ ⊥ x3 ⊥ ⊥ x5 x6 ... zs = ⊥ y1 ⊥ y2 ⊥ ⊥ y4 ⊥ ⊥ ⊥ ... ``state bus´´ zs

when

ys cs2 zs state bus values = response functions: parallel emits = function composition: The last value dominates the bus

zs = x0 y1 ⊥ y2 ⊥ x3 y4 ⊥ x5 x6 ...

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Absence = Identity Reaction on ‘State Bus‘

• M. Mendler, University of Bamberg

Synchron 2008 – CPL Aussois 31

when

xs cs1 zs zs = x0 ⊥ ⊥ ⊥ ⊥ x3 ⊥ ⊥ x5 x6 ... zs = ⊥ y1 ⊥ y2 ⊥ ⊥ y4 ⊥ ⊥ ⊥ ... ``state bus´´ zs

when

ys cs2 zs

non-termination Ω ≠ identity ⊥ ≠ absence ∗ x ◦ y same as x default y (Signal) but deeply coded !

SF DF DF

zs = x0 y1 ⊥ y2 ⊥ x3 y4 ⊥ x5 x6 ...

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Absence = Identity Reaction on ‘State Bus‘

• M. Mendler, University of Bamberg

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zs = x0 ⊥ ⊥ ⊥ ⊥ x3 ⊥ ⊥ x5 x6 ... zs = ⊥ y1 ⊥ y2 ⊥ ⊥ y4 ⊥ ⊥ ⊥ ...

• M. Fourman [1989]:

Response functions (cpo) to model bi-directional instantaneous communication. non-termination Ω ≠ identity ⊥ ≠ absence ∗ x ◦ y same as x default y (Signal) but deeply coded !

SF DF DF

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Constructive Data Flow

• M. Mendler, The Otto-Friedrich University of Bamberg

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Synchronous Processes

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SF/DF Reaction Relations

• M. Mendler, University of Bamberg

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R::b

DF::SDF (a,b) c

E::a v::c

Data Flow

R::b

SF::SSM a b

E::a

State Flow

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Constructive Data Flow

• M. Mendler, The Otto-Friedrich University of Bamberg

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SF/DF Process Language

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State Flow

• M. Mendler, The Otto-Friedrich University of Bamberg

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state flow variable signal emission weak preemption parallel binding DF to signal local signal iteration wait branching signal input

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Data Flow

• M. Mendler, The Otto-Friedrich University of Bamberg

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close-off state flow static lifting of value operators up-sampling down-sampling delay initialisation initialised delay data flow variable feed-back, recursive data flow

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• M. Mendler, The Otto-Friedrich University of Bamberg

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Mode Automaton Example

SEC TDIV

STOP

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Macro States & Mode Automaton

START s := 0 last_s Leftb/

H

Leftb/ Rightb/ CONTROL last_s := pre s MEM CHRONO s := if False cnt < pre cnt then (last_s + 1) ´mod´ 60 else last_s cnt := 0 (pre cnt + 1) mod 100

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MEM

s

STOP

last_s s

START

s

SEC

cnt

TDIV

time

Data Flow

CONTROL

last_s

CHRONO

s

user action

system

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MEM

s last_s

Reaction interface „E;R“ MEM from Data Flow MEM as State Flow

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DF Interface STOP as Data Flow

STOP

s last_s

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SF Reaction interface „E;R“ STOP from Data Flow STOP as State Flow

STOP

s last_s

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START

s

SEC

cnt

TDIV

last_s

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CTRL

s env last_s STOP START CONTROL H

Leftb/ Leftb/ Rightb/

sy sx

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CTRL

s env last_s STOP START CONTROL H

Leftb/ Leftb/ Rightb/

sy sx

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CTRL

s env last_s

CONTROL as Data Flow

STOP START CONTROL H

Leftb/ Leftb/ Rightb/

sy sx

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MEM

CHRONO

CONTROL

s last_s s s

CHRONO

s

user action

CHRONO as Data Flow

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tick zs!s ys?act xs zs s ys isRbp act / s←0 i s L b p a c t / d ←

startb s d stopb s

zs!s tick ys?act isLbp act / / d←d+1 d=100 / s←s+1´mod´60 d← 0 1 2 3 2 3 1

chrono

/ s←0 d

Haskell implementation

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Constructive Data Flow

• M. Mendler, The Otto-Friedrich University of Bamberg

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ESTEREL is non-sequential !

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Concurrent AND, OR

• M. Mendler, The Otto-Friedrich University of Bamberg

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Esterel Control Flow is concurrent (non-sequential) ! Concurrent AND Æ

xs ys zs present x else pause ; present y then emit z present x else pause ; present y then emit z x y z present x then present y then emit z; present x then present y then emit z; x z y

z is absent if at least one of x or y is absent

present x then emit z || present y then emit z present x then emit z || present y then emit z x y z

Concurrent OR z is present if at least one of x or y is present

xs ys zs

Ç

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Constructive Data Flow

• M. Mendler, The Otto-Friedrich University of Bamberg

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Conclusion

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Summary

Kahn‘s original model of SF-DF can be uniformely coded in lazy λ-calculus without clocks and explicit „absent“ values Kahn‘s prefix ordering is not sufficient for SF causality, need a form of Scott-ordering on state Kahn SF is weaker than (full) Esterel SF, the latter is non-sequential with true concurrency of execution Lusteral implements sequential „token ring“ broadcast of signals not concurrent „ethernet“ (like Esterel)

Related Work

• Nowak, Beauvais, Talpin [1999]: Signal in Coq
• Elliott [1997], Hudak [2000]: Functional reactive

programming/animations (FRAN, FRP) in Haskell

• Kieburtz [1998]: Reactive Functional Programming
• Uustalu [2005]: DF with comonads and monads
• Pouzet, Hamón, Boulmé [≈ 2001]: Lucid

Synchrone in Coq

• Mandel [2004]: Reactive ML