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

The Synchrony Hypothesis or The Importance of Being Constructive inspired by Tom Shiple, Gérard Berry M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-1 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 of „causal“ systems depending on the MoCC (model of coordination and communication) M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-2 1

Synchronous Abstraction ... why constructiveness matters M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-3 Synchrony Hypothesis Environment view: Reactions are • atomic • deterministic • bounded System view: Reactions may be • non-atomic LAGS • non-deterministic • unbounded M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-4 2

Soups of Soups Ptolemy Glue f 3 ForSyDe BIP f 4 f 1 SystemC f 2 42 … Soup Activation Condition Glue Director Co-ordination Executive Orchestration Scheduler Control Contracts Delay Model Component Protocols … … M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-5 Soups of Soups Glue f 3 f 4 f 1 f 2 M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-6 3

Operational Semantics for MyGlue subset of execution traces Definition F is My_Glue-combinational if ∀ i ∈ m. ∃ time bound D i and response value α i such that for all h i stabilises to α ι at time D i . My_Glue f 3 f 1 f 4 f 2 M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-7 Denotational Semantics for MyGlue monadic domain extension fixed point Definition F is My_Glue-causal if My_Glue f 3 f 4 f 1 f 2 M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-8 4

The Full Abstraction Game Operational Semantics (F combinational) soundness and completeness m o d e l t h Axiomatic Semantics e o r y My Synchrony ? Hypothesis proof theory algebraic decision procedure Denotational Semantics (F causal) M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-9 Constructive Semantics Constructive Delay Model soundness and completeness m o d e l t Constructive Logic h e Berry‘s o r y ? Synchrony proof theory Hypothesis algebraic decision procedure Ternary Simulation M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-10 5

What‘s Constructive Logic ? Classical Logic ☻ ☻ ` φ ∨ ¬ φ for all φ Excluded Middle: e.g., ☻ ` (P = NP) ∨ (P ≠ NP) Double Negation: ☻ ` ¬¬ φ ≡ φ Constructive Logic ☺ : If ☺ ` φ ∨ ψ then ☺ ` φ or ☺ ` ψ Disjunction Property: If ☺ ` ∃ x. φ (x) then there is a Existential Property: (closed) term t such that ☺ ` φ (t) Constructive proofs have computational meaning M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-11 What does it buy us ? ` 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 Constructive reaction is always deterministic and bounded ! M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-12 6

Some Folks‘ lore M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-13 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?... M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-14 7

1 Denotational Folklore not quite • Ternary Algebra is like a logic of truth values „discrete Scott domain“ unknown, undefined, non-determinism, oscillation, deadlock, metastability, unstable, transient, don‘t care, ... ... avoids dangerous classical equalities ☺ : x ... avoids equalities altogether ☹ ! Ternary logic has no theorems at all, M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-15 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 ! M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-16 8

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-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 9

3 Operational Folklore • Inertial Delays are not quite the right operational interpretation of Ternary Simulation ! The ternary fixed point implies that s 2 is non-constructive 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 ! total state 0 • 1*0 1 • 0*0* 0 • 00* 1 • 0*1 All possible inertial delay system trajectories according to General Multiple 1 • 11* 0 • 0*1 Winner model (GMW, (Brzozowski & Seger) converge ! 1 • 10 0 • 11 stable states M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-20 10

Operational Semantics: Non-Inertial Delays A constructive communication model for Boolean networks M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-21 Up-bounded Non-Inertial (UN-) Delay input changes too fast D=2 D (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. M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-22 11

Up-bounded Non-Inertial (UN-) Delay oscillation D=3 Non-inertial (UN-) delays permit oscillation as predicted by ternary simulation. M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-23 Execution of UN-Delay Networks state nodes feedback vertex set input nodes constant input Network excitation function Non-Inertial Network Behaviour Let be the set of 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 M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-24 12

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-25 Syntax and Semantics Syntax Boolean expression in network variables delay parameter M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-26 13

Basic Properties Abbreviations UN-Logic contains Boolean Algebra logical equivalences M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-27 Basic Properties Monotonicity UN-Logic is constructive e.g., always true only if s constant M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-28 14

Basic Properties M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-29 Inertial Assignment Abbreviation Let stand for Proposition logical equivalences M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-30 15

UN Network Specifications N M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-31 UN Network Specifications Note: Equational Substitution in general is not sound unless ! M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-32 16

UN Network Specifications N reduced substitution normal form M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-33 Semantical Adequacy state nodes feedback vertex set input nodes constant input Network excitation function Semantical Adequacy Theorem The non-inertial network behaviour UN-exec( N, a ) coincides with the models of the formula ^ ^ Ψ N,a ≡ d s i := D i S i ∧ x i = a i f s i ∈ S x i ∈ X M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-34 17

UN-Calculus M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-35 Theorems Soundness Φ ` Θ ⇒ Φ | = Θ. Completeness* Φ | = Θ ⇒ Φ ` Θ. Constructiveness If Φ ` Θ, then there exists a single θ ∈ Θ such that Φ ` θ . What do these theorems buy us ? ... *under some weak assumptions M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-36 18

Reactivity of UN-Networks bounded reaction no non-determinism no metastability M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-37 Reactivity of UN-Networks Quantifier swap ! M. Mendler, The University of Bamberg Synchron 2006 @ L‘Alpe d‘Huez-38 19