The Synchrony Hypothesis or The Importance of Being Constructive - - PDF document

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The Synchrony Hypothesis or The Importance of Being Constructive - - PDF document

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The Synchrony Hypothesis or The Importance of Being Constructive inspired by Tom Shiple, Grard Berry M. Mendler, The University of Bamberg Synchron 2006 @ LAlpe dHuez-1 Whats this talk about ? to characterise in a

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  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-1

The Synchrony Hypothesis

  • r

The Importance of Being Constructive

inspired by Tom Shiple, Gérard Berry

  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-2

What‘s this talk about ?

  • to characterise in a mathematically precise way

the class of systems known, informally, as „constructive“ systems

  • to present correspondence theorems linking

denotational, operational and axiomatic semantics

  • to highlight the fact that there are different notions
  • f „causal“ systems depending on the MoCC

(model of coordination and communication)

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  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-3

Synchronous Abstraction

... why constructiveness matters

  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-4

Synchrony Hypothesis

Environment view: Reactions are

  • atomic
  • deterministic
  • bounded

System view: Reactions may be

  • non-atomic
  • non-deterministic
  • unbounded

LAGS

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  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-5

Soups of Soups

f1 f2 f3 f4 Glue Glue Soup Director Executive Scheduler Delay Model … Activation Condition Co-ordination Orchestration Control Contracts Component Protocols … Ptolemy ForSyDe BIP SystemC 42 …

  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-6

Soups of Soups

f2 f1 f3 f4 Glue

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  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-7

Operational Semantics for MyGlue

f2 f1 f3 f4 subset of execution traces F is My_Glue-combinational if ∀ i ∈ m. ∃ time bound Di and response value αi such that for all hi stabilises to αι at time Di.

Definition

My_Glue

  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-8

Denotational Semantics for MyGlue

f2 f1 f3 f4 monadic domain extension F is My_Glue-causal if

Definition

My_Glue

fixed point

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  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-9

soundness and completeness

The Full Abstraction Game

My Synchrony Hypothesis

?

Operational Semantics (F combinational) Denotational Semantics (F causal) Axiomatic Semantics

algebraic decision procedure m

  • d

e l t h e

  • r

y proof theory

  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-10

soundness and completeness

Constructive Semantics

Berry‘s Synchrony Hypothesis

?

Constructive Delay Model Ternary Simulation Constructive Logic

algebraic decision procedure m

  • d

e l t h e

  • r

y proof theory

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  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-11

What‘s Constructive Logic ?

Classical Logic ☻ Excluded Middle: e.g., ☻` (P = NP) ∨ (P ≠ NP) ☻` φ ∨ ¬ φ for all φ Double Negation: ☻` ¬¬ φ ≡ φ Constructive Logic ☺: Disjunction Property: Existential Property: If ☺` φ ∨ ψ then ☺` φ or ☺` ψ If ☺` ∃ x. φ(x) then there is a (closed) term t such that ☺` φ(t) Constructive proofs have computational meaning

  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-12

What does it buy us ?

Constructive reaction is always deterministic and bounded ! ` s stabilises to 0 ∨ s stabilises to 1 ⇒ ` s stabilises to 0 or ` s stabilises to 1 ` ∃ t. s stabilises at time t ⇒ for some delay bound D, ` s stabilises at time D

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  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-13

Some Folks‘ lore

  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-14

Some Folks‘ lore says ...

  • ... denotational semantics: Ternary Algebra is a

logic of constructive truth

  • ... axiomatic semantics: Intuitionistic Boolean

Logic has constructive provability

  • ... operational semantics: Inertial Delays are the

right real-time interpretation of Ternary Simulation

Really?...

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  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-15

1 Denotational Folklore

  • Ternary Algebra is like a logic of truth values

unknown, undefined, non-determinism,

  • scillation, deadlock, metastability,

unstable, transient, don‘t care, ... „discrete Scott domain“

x

... avoids dangerous classical equalities ☺:

not quite

... avoids equalities altogether ☹ ! Ternary logic has no theorems at all,

  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-16

2 Axiomatic Folklore

  • Constructiveness is not quite about provability in

Intuitionistic Boolean Logic !

intuitionistically, is equivalent to ⇒ for any formula ⇒ Intuitionism alone doesn‘t help, we must axiomatise delays (scheduling), too !

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  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-17

3 Operational Folklore

  • Inertial Delays are not quite the right operational

interpretation of Ternary Simulation !

  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-18

3 Operational Folklore

  • Inertial Delays are not quite the right operational

interpretation of Ternary Simulation !

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  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-19

3 Operational Folklore

  • Inertial Delays are not quite the right operational

interpretation of Ternary Simulation !

The ternary fixed point implies that s2 is non-constructive

  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-20

3 Operational Folklore

  • Inertial Delays are not quite the right operational

interpretation of Ternary Simulation !

0•1*0 0•00* 0•0*1 0•11 1•0*0* 1•0*1 1•11* 1•10 total state

stable states All possible inertial delay system trajectories according to General Multiple Winner model (GMW, (Brzozowski & Seger) converge !

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  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-21

Operational Semantics: Non-Inertial Delays

A constructive communication model for Boolean networks

  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-22

Up-bounded Non-Inertial (UN-) Delay

D=2

D

input changes too fast (1) Up-bounded Propagation: If the input remains stable for longer than D time, then the output stabilises to new value. (2) Non-inertial: If input changes, output totally uncontrolled until new value has propagated through.

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  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-23

Up-bounded Non-Inertial (UN-) Delay

D=3

  • scillation

Non-inertial (UN-) delays permit oscillation as predicted by ternary simulation.

  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-24

Execution of UN-Delay Networks

input nodes state nodes feedback vertex set Network excitation function Non-Inertial Network Behaviour trajectories h such that

  • h has all input signals constant at value a
  • h is right-continuous and non-Zeno
  • h is consistent with network excitation function S

and UN-delay scheduling constant input Let be the set of

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  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-25

Axiomatic Semantics: UN-Logic

A constructive „Boolean“ specification language for UN-delay networks

  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-26

Syntax and Semantics

Boolean expression in network variables delay parameter

Syntax

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  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-27

Basic Properties

Abbreviations UN-Logic contains Boolean Algebra logical equivalences

  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-28

Basic Properties

UN-Logic is constructive Monotonicity

always true

  • nly if s constant

e.g.,

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  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-29

Basic Properties

  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-30

Inertial Assignment

Let stand for

Abbreviation Proposition

logical equivalences

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  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-31

UN Network Specifications

N

  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-32

UN Network Specifications

Note: Equational Substitution in general is not sound unless !

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  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-33

UN Network Specifications

N

reduced substitution normal form

  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-34

Semantical Adequacy

input nodes state nodes feedback vertex set Network excitation function

Semantical Adequacy Theorem

constant input The non-inertial network behaviour UN-exec(N, a) coincides with the models of the formula

ΨN,a ≡d

f ^ si∈S

si :=Di Si ∧

^ xi∈X

xi = ai

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  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-35

UN-Calculus

  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-36

Theorems

Soundness Completeness* Constructiveness

What do these theorems buy us ? ...

Φ ` Θ ⇒ Φ | = Θ. Φ | = Θ ⇒ Φ ` Θ.

If Φ ` Θ, then there exists a single θ ∈ Θ such that Φ ` θ.

*under some weak assumptions

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  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-37

Reactivity of UN-Networks

no non-determinism no metastability bounded reaction

  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-38

Quantifier swap !

Reactivity of UN-Networks

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  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-39

Reactivity of UN-Networks

adequacy completeness constructiveness soundness

  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-40

Reactivity of UN-Networks

Corollary

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  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-41

Denotational Semantics: Timed Ternary Simulation

A sound and complete algebraic decision procedure for UN-Logic

  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-49

Classes of Synchronous Causality

f1 f2 f3 f4

glue

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  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-50

Soup of Delays

Example

  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-51

Soup of Delays

Definition

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  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-52

Some Synchronous Causality Classes

UI-DI = UI*-DI = UN-DI = UN*-DI UN-SI = UN*-SI UI*-SI UI-SI Constructive Systems

  • M. Mendler, The University of Bamberg

Synchron 2006 @ L‘Alpe d‘Huez-53

Summary

Theorem The following statements are equivalent: A system (decomposition) S is constructive, i.e., provably stable in UN-Logic The ternary simulation of S in the chosen state variables generates Boolean solutions S stabilises in bounded time to a unique steady state under non-inertial delay assumptions