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

Lusteral Uniform and Modular Composition of Data-flow & Control-flow in the Lazy λ -Calculus joint work with Marc Pouzet M. Mendler, University of Bamberg Synchron 2008 – CPL Aussois Synchron 2008 – CPL Aussois 1

Constructive Data Flow Introduction M. Mendler, The Otto-Friedrich University of Bamberg Synchron 2008 – CPL Aussois Lusterel-2

Data Flow Programming Signal, Lustre, SCADE V4, LabView, Simulink, ... M. Mendler, University of Bamberg Synchron 2008 – CPL Aussois Synchron 2008 – CPL Aussois 3

Control Flow Programming ABSync Detection 1 WaitAandB idle arm wA wB Reset Timer A/arm B /arm 2 T / disarm cnt 2 eot dA dB disarm Inhib / AB done signal arm signal disarm Esterel, SyncCharts, Argos, Stateflow, Statemate, ... M. Mendler, University of Bamberg Synchron 2008 – CPL Aussois Synchron 2008 – CPL Aussois 4

Starting the 3rd Millennium ... � ... we are looking at hybrid engines with tight integration of data flow and control flow: ReLuC, Lucide Synchrone, SCADE 6, 7, 8, Synoptic, ... Synchron 2008 – CPL Aussois

Starting the 3rd Millennium ... � ... we need to understand the semantical mechanics of such a hybrid (4 education, certification, verification,...) � 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 ? M. Mendler, The Otto-Friedrich University of Bamberg Synchron 2008 – CPL Aussois Lusterel-6

The Lusteral Experiment We aim to integrate • 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 Lusterel Language Design Workbench polymorphic equational reasoning higher-order extensible Lustre DF Esterel SF strong typing declarative smooth morphing between data and control-flow M. Mendler, University of Bamberg Synchron 2008 – CPL Aussois Synchron 2008 – CPL Aussois 7

Kahn Principle • 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! M. Mendler, University of Bamberg Synchron 2008 – CPL Aussois Synchron 2008 – CPL Aussois 8

Constructive Data Flow The Role of Laziness M. Mendler, The Otto-Friedrich University of Bamberg Synchron 2008 – CPL Aussois Lusterel-9

Orthogonality in Time and Space M. Mendler, University of Bamberg Synchron 2008 – CPL Aussois Synchron 2008 – CPL Aussois 10

Data Flow data flow M. Mendler, University of Bamberg Synchron 2008 – CPL Aussois Synchron 2008 – CPL Aussois 11

Data Flow Q: How do we treat the cyclic DF dependencies ? data flow A: Fixpoints on prefix-ordering, lazy recursion on streams! M. Mendler, University of Bamberg Synchron 2008 – CPL Aussois Synchron 2008 – CPL Aussois 12

State Flow state flow M. Mendler, University of Bamberg Synchron 2008 – CPL Aussois Synchron 2008 – CPL Aussois 13

State Flow Q: How do we treat the cyclic SF dependencies ? state flow A: Scott information-ordering, lazy recursion on state M. Mendler, University of Bamberg Synchron 2008 – CPL Aussois Synchron 2008 – CPL Aussois 14

Fixed-point on Partial States xs = f (3 fby (fst zs), snd zs) ys = h (snd xs) zs = g (ys, fst xs) fby 3 xs f xs = 5 3 h ys = 6 ys 2 zs = g 0 zs M. Mendler, University of Bamberg Synchron 2008 – CPL Aussois Synchron 2008 – CPL Aussois 16

Fixed-point on Partial States xs = f (3 fby (fst zs), snd zs) ys = h (snd xs) zs = g (ys, fst xs) fby 3 4 xs f xs = 5 0 3 h ys = 6 1 ys 2 9 zs = g 0 9 zs M. Mendler, University of Bamberg Synchron 2008 – CPL Aussois Synchron 2008 – CPL Aussois 17

Fixed-point on Partial States xs = f (3 fby (fst zs), snd zs) ys = h (snd xs) zs = g (ys, fst xs) fby 3 4 3 5 xs f xs = 5 0 7 7 3 h ys = 6 1 5 2 ys 2 9 2 1 zs = g 0 9 5 1 zs M. Mendler, University of Bamberg Synchron 2008 – CPL Aussois Synchron 2008 – CPL Aussois 18

Where Eagerness is not Productive ... xs = f (3 fby (fst zs), snd zs) fby xs f ys = h (snd xs) 3 zs = g (ys, fst xs) h Eager Evaluation ys fst xs � fst (f(3 fby (fst zs), snd zs)) g 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)) � fst (f(3:fst zs), snd(g(h(snd(f(3 fby (fst zs), snd zs))),fst xs) � … loop ! M. Mendler, University of Bamberg Synchron 2008 – CPL Aussois Synchron 2008 – CPL Aussois 19

... Lazy Function Evaluation may be. xs = f (3 fby (fst zs), snd zs) fby xs f ys = h (snd xs) 3 zs = g (ys, fst xs) h Lazy Evaluation ys fst xs � fst(f(3 fby (fst zs), snd zs)) g zs � fst((f 1 × f 2 )(3 fby (fst zs), snd zs)) � fst(( λ w.(f 1 (fst w), f 2 (snd w))(3 fby (fst zs), snd zs)) � fst(f 1 (fst(3 fby (fst zs), snd zs)), f 2 (snd(3 fby (fst zs), snd zs))) � f 1 (fst(3 fby (fst zs), snd zs)) lazy on state (data) � f 1 (3 fby (fst zs)) lazy on streams � f 1 (3:(fst zs)) where H[v] = f 1 (g 1 (h(f 2 (g 2 (v)))) � f 1 (3) : f 1 (fst zs) � ... f 1 (3) : H(f 1 (3)) : H(H(f 1 (3))) : H(H(H(f 1 (3)))) : ... M. Mendler, University of Bamberg Synchron 2008 – CPL Aussois Synchron 2008 – CPL Aussois 20

Constructive Data Flow SF ≠ Clocked DF M. Mendler, The Otto-Friedrich University of Bamberg Synchron 2008 – CPL Aussois Lusterel-21

Clock Schedules are Implicit xs = x 0 x 1 ⊥ x 2 ⊥ x 3 x 4 ⊥ x 5 x 6 ... xs when zs cs cs = T F ⊥ F ⊥ T F ⊥ T T ... zs = x 0 ⊥ ⊥ ⊥ ⊥ x 3 ⊥ ⊥ x 5 x 6 ... loop loop read c from cs; read c from cs; cs?c c=T if c then if c then c=F read x from xs; xs?x read x from xs; tick write x to zs; write x to zs; tick else pause; else pause; zs!x pause; pause; end loop absence ⊥ is not communicated ! end loop M. Mendler, University of Bamberg Synchron 2008 – CPL Aussois Synchron 2008 – CPL Aussois 28

Absence = Identity Reaction on ‘State Bus‘ xs = x 0 x 1 ⊥ x 2 ⊥ x 3 x 4 ⊥ x 5 x 6 ... xs when cs 1 zs cs 1 = T F ⊥ F ⊥ T F ⊥ T T ... zs = x 0 ⊥ ⊥ ⊥ ⊥ x 3 ⊥ ⊥ x 5 x 6 ... zs = ⊥ y 1 ⊥ y 2 ⊥ ⊥ y 4 ⊥ ⊥ ⊥ ... ys when cs 2 zs cs 2 = F T ⊥ T ⊥ F T ⊥ F F ... ys = y 0 y 1 ⊥ y 2 ⊥ y 3 y 4 ⊥ y 5 y 6 ... M. Mendler, University of Bamberg Synchron 2008 – CPL Aussois Synchron 2008 – CPL Aussois 29

Absence = Identity Reaction on ‘State Bus‘ xs when cs 1 zs zs = x 0 ⊥ ⊥ ⊥ ⊥ x 3 ⊥ ⊥ x 5 x 6 ... ``state bus´´ zs = ⊥ y 1 ⊥ y 2 ⊥ ⊥ y 4 ⊥ ⊥ ⊥ ... ys when cs 2 zs zs = x 0 y 1 ⊥ y 2 ⊥ x 3 y 4 ⊥ x 5 x 6 ... zs state bus values = response functions: parallel emits = function composition: The last value dominates the bus M. Mendler, University of Bamberg Synchron 2008 – CPL Aussois Synchron 2008 – CPL Aussois 30

Absence = Identity Reaction on ‘State Bus‘ xs when cs 1 zs zs = x 0 ⊥ ⊥ ⊥ ⊥ x 3 ⊥ ⊥ x 5 x 6 ... ``state bus´´ zs = ⊥ y 1 ⊥ y 2 ⊥ ⊥ y 4 ⊥ ⊥ ⊥ ... ys when cs 2 zs zs = x 0 y 1 ⊥ y 2 ⊥ x 3 y 4 ⊥ x 5 x 6 ... zs x ◦ y same as x default y (Signal) but deeply coded ! non-termination Ω ≠ identity ⊥ ≠ absence ∗ DF SF DF M. Mendler, University of Bamberg Synchron 2008 – CPL Aussois Synchron 2008 – CPL Aussois 31

Absence = Identity Reaction on ‘State Bus‘ M. Fourman [1989]: Response functions (cpo) zs = x 0 ⊥ ⊥ ⊥ ⊥ x 3 ⊥ ⊥ x 5 x 6 ... to model bi-directional instantaneous zs = ⊥ y 1 ⊥ y 2 ⊥ ⊥ y 4 ⊥ ⊥ ⊥ ... communication. zs = x 0 y 1 ⊥ y 2 ⊥ x 3 y 4 ⊥ x 5 x 6 ... x ◦ y same as x default y (Signal) but deeply coded ! non-termination Ω ≠ identity ⊥ ≠ absence ∗ DF SF DF M. Mendler, University of Bamberg Synchron 2008 – CPL Aussois Synchron 2008 – CPL Aussois 32

Constructive Data Flow Synchronous Processes M. Mendler, The Otto-Friedrich University of Bamberg Synchron 2008 – CPL Aussois Lusterel-33

SF/DF Reaction Relations State Flow SF::SSM a b E::a R::b Data Flow DF::SDF (a,b) c E::a R::b v::c M. Mendler, University of Bamberg Synchron 2008 – CPL Aussois Synchron 2008 – CPL Aussois 37

Constructive Data Flow SF/DF Process Language M. Mendler, The Otto-Friedrich University of Bamberg Synchron 2008 – CPL Aussois Lusterel-39

State Flow state flow variable signal input signal emission wait branching weak preemption parallel binding DF to signal local signal iteration M. Mendler, The Otto-Friedrich University of Bamberg Synchron 2008 – CPL Aussois Lusterel-40