A Complete Approximation Theory for Weighted Transition Systems

Transcription

1 A Complete Approximation Theory for Weighted Transition Systems December 1, 2015 Peter Christoffersen Mikkel Hansen Mathias R. Pedersen Radu Mardare Kim G. Larsen Department of Computer Science Aalborg University Denmark

2 Agenda 1 Introduction Logic Axiomatization Canonical model construction Weak completeness Conclusion

3 Motivation 2 Today microchips are used nearly everywhere we look. Cyber-physical systems The idea of combining computation and the physical world. Use sensors and input devices for humans to affect the computation. Motors, actuators and other mechanics can alter and affect the world.

4 Motivation 2 Today microchips are used nearly everywhere we look. Cyber-physical systems The idea of combining computation and the physical world. Use sensors and input devices for humans to affect the computation. Motors, actuators and other mechanics can alter and affect the world. When dealing with real-world processes you often rely on resources such as: Energy, money, distances etc.

5 Motivation Resource modeling 3 Weighted Transition Systems (WTS) can encode this quantitative behaviour, though in a strictly precise fashion. WTS example: Robot vacuum cleaner Clean? Yes. Room is Cleaned. The room takes 20 units, e.g. time or energy, to clean. Y 20 0 C

6 Motivation Resource modeling 3 Weighted Transition Systems (WTS) can encode this quantitative behaviour, though in a strictly precise fashion. WTS example: Robot vacuum cleaner Clean? Yes. Room is Cleaned. The room takes 20 units, e.g. time or energy, to clean. Y 20 0 C What if the room had a very varying degree of dirtiness?

7 Motivation Resource modeling 4 Cyber-physical systems Sensors and inputs from the world affects computations, likewise mechanical output affects the world. The settings these systems operate in are often unpredictable, and the inputs are always with some imprecision. Problems Tolerance of sensors. Unpredictable environment. We can only reason about what is encoded in the model.

8 Motivation Resource modeling 5 Solution Let the model account for the imprecision so we can reason about it. We extend the notion of WTS with bounds x, y on transitions. This captures the imprecision in the modeling domain by denoting a whole range of values. WTS example: Robot vacuum cleaner Clean? Yes. Room is Cleaned. The room takes 5 to 20 units, e.g. time or energy, to clean. Y 5, 20 0 C

9 Contribution 6 An extension of Weighted Transition Systems with bounds, as well as a suitable notion of bisimulation. Logic to reason with bounds that has the Hennessy-Milner property. Weak-complete axiomatization of the logic.

10 Bounds 7 Bounds A bound B R 2 0 is either the empty set or a tuple x, y where x y. Denote the set of all bounds by B. A B iff B A and A + B + B A A + B + A B = min{a, B }, max{a +, B + } B B + A A + + A B = max{a, B }, min{a +, B + } B A B + A + +

11 Generalized Weighted Transition Systems 8 A Generalized Weighted Transition System (GTS) is a tuple G = (S, θ, l), where Transition function θ : S (2 S B) is a transition function satisfying the following conditions: θ (s) ( ) =, ( ) θ (s) S i = θ (s) (S i ), and i i ( ) ( ) θ (s) S i = θ (s) S i = i i i θ (s) (S i ). (I) (II) (III)

12 GTS: Transition function Property II 9 ( ) θ (s) S i = i i θ (s) (S i ) θ (s) ({t 1 } {t 3 } {t 4 }) = min{1, 3, 6}, max{1, 4, 6} = 1, 6 1, 1 t 1 s 3, 4 6, 6 t 4 t 1 s 1, 6 t 4 t3 t3 t 2 t 2

13 Logic Syntax 10 Syntax L : ϕ, ψ ::= p ϕ ϕ ψ L r ϕ M r ϕ where r Q 0 and p AP. Semantics G, s = L r ϕ iff can reach a state satisfying ϕ with weight at least r G, s = M r ϕ iff can reach a state satisfying ϕ with weight at most r

14 Logic Syntax 10 Syntax L : ϕ, ψ ::= p ϕ ϕ ψ L r ϕ M r ϕ where r Q 0 and p AP. Semantics G, s = L r ϕ iff θ (s) ( ϕ ) and θ (s) ( ϕ ) r G, s = M r ϕ iff θ (s) ( ϕ ) and θ + (s) ( ϕ ) r where ϕ is the set of all GTS states with the property ϕ, i.e. ϕ = {s (S, θ, l) G : s S and G, s = ϕ}

15 Logic Example 11 Example s 2, 4 s ϕ s +

16 Logic Example 12 Example s 2 2, 4 s ϕ G, s = L 2 ϕ s +

17 Logic Example 13 Example s 2, 4 s ϕ G, s = L 2 ϕ G, s = M 4 ϕ s 4

18 Logic Example 14 Example s G, s = L 2 ϕ 2, 4 s ϕ 3 G, s = M 4 ϕ G, s = L 3 ϕ s +

19 Logic Example 15 Example s G, s = L 2 ϕ 2, 4 G, s = M 4 ϕ s ϕ 3 G, s = L 3 ϕ s + G, s = M 3 ϕ

20 Logic Derived operators 16 In addition to the operators defined by the syntax, we have the following derived operators Derived operators ϕ ψ def = ϕ ϕ def = def = ( ϕ ψ) ϕ ψ def = ϕ ψ

21 Logic Derived operators 16 In addition to the operators defined by the syntax, we have the following derived operators Derived operators ϕ ψ def = ϕ ϕ def = def = ( ϕ ψ) ϕ ψ def = ϕ ψ We can encode and with their usual semantics, semantics ϕ ϕ def = L 0 ϕ def = ϕ = L 0 ϕ

22 Bisimulation 17 Bisimulation Given GTS G = (S, θ, l), an equivalence relation R on S is a bisimulation relation iff srt implies l(s) = l(t) and θ(s)(t ) = θ(t)(t ) for all equivalence classes T S/R. Bisimulation invariance (Hennessy-Milner property) s t iff [ ϕ L : G, s = ϕ G, t = ϕ].

23 Filters 18 Filter A non-empty subset F of L is called a filter iff / F, ϕ F and ϕ ψ implies ψ F, and ϕ F and ψ F implies ϕ ψ F. Ultrafilter A filter F is called an ultrafilter iff for every ϕ L either ϕ F or ϕ F, but not both.

24 Axioms 19 (A1): L 0 (A2): L r+s ϕ L r ϕ, s > 0 (A2 ): M r ϕ M r+s ϕ, s > 0 (A3): L r ϕ L s ψ L min{r,s} (ϕ ψ) (A3 ): M r ϕ M s ψ M max{r,s} (ϕ ψ) (A4): ((L r ϕ) (L s ψ)) ( L 0 (ϕ ψ) L max{r,s} (ϕ ψ) ) (A4 ): ((M r ϕ) (M s ψ)) ( L 0 (ϕ ψ) M min{r,s} (ϕ ψ) ) (A5): ((L 0 ϕ) ( L r ϕ) (L 0 ψ) ( L s ψ)) L max{r,s} (ϕ ψ) (A5 ): ((L 0 ϕ) ( M r ϕ) (L 0 ψ) ( M s ψ)) M min{r,s} (ϕ ψ) (A6): L r (ϕ ψ) L r ϕ L r ψ (A6 ): M r (ϕ ψ) M r ϕ M r ψ (A7): L 0 ψ (L r ϕ L r (ϕ ψ)) (A7 ): L 0 ψ (M r ϕ M r (ϕ ψ)) (A8): L r+s ϕ M r ϕ, s > 0 (A9): M r ϕ L 0 ϕ

25 Axioms A2 and A2 20 (A2) L r+s ϕ L r ϕ, s > 0 (A2 ) M t ϕ M t+q ϕ, q > 0 + r + s t ϕ

26 Axioms A3 21 (A3) L r ϕ L s ψ L min{r,s} (ϕ ψ) r s + ϕ ϕ ψ ψ

27 Axioms A3 22 (A3 ) M r ϕ M s ψ M max{r,s} (ϕ ψ) r s + ϕ ϕ ψ ψ

28 Axioms A8 23 (A8) L r+s ϕ M r ϕ, s > 0 + r + s r ϕ

29 Axioms A9 24 (A9) M r ϕ L 0 ϕ 0 r + ϕ

30 Axioms 25 (R1): {L s ϕ s < r} L r ϕ (R1 ): {M s ϕ s > r} M r ϕ (R2): ϕ ψ = ((L r ψ) (L 0 ϕ)) L r ϕ (R2 ): ϕ ψ = ((M s ψ) (L 0 ϕ)) M s ϕ (R3): ϕ ψ = L 0 ϕ L 0 ψ (R4): { M r ϕ r Q 0 } L 0 ϕ (R5): (R5 ): {ϕ i i N} ϕ ϕ i+1 ϕ i ϕ ϕ i i N, s > 0 { L r ϕ i i N} L r+s ϕ {ϕ i i N} ϕ ϕ i+1 ϕ i ϕ ϕ i i N, s > 0 { M r+s ϕ i i N} M r ϕ (R6): {L r+s ϕ ϕ F } { L r ψ F ψ} (R6 ): {M r+s ϕ ϕ F } { M r ψ F ψ}

31 Axioms R2 and R2 26 (R2) ϕ ψ = ((L r ψ) (L 0 ϕ)) L r ϕ (R2 ) ϕ ψ = ((M s ψ) (L 0 ϕ)) M s ϕ r s + ψ ϕ

32 Axioms R5 27 (R5) {ϕ i i N} ϕ ϕ i+1 ϕ i ϕ ϕ i i N, s > 0 { L r ϕ i i N} L r+s ϕ + r ϕ 1 ϕ 2 ϕ 3 ϕ

33 Axioms R5 28 (R5 ) {ϕ i i N} ϕ ϕ i+1 ϕ i ϕ ϕ i i N, s > 0 { M r+s ϕ i i N} M r ϕ + r ϕ 1 ϕ 2 ϕ 3 ϕ

34 Axioms Soundness 29 Lemma (Soundness) ϕ implies = ϕ.

35 Canonical model construction 30 GTS with ultrafilters as states. Transition function must satisfy conditions I-III. θl : U [L B] θf : U [F { } B] θu : U [2 U B] Labeling function l U : U 2 AP. lu(u) = {p AP p u}

36 Formulae 31 Transition function to formulae { θ L (u)(ϕ) = sup{r L r ϕ u}, inf{s M s ϕ u} if L 0 ϕ / u otherwise. The function θ L assigns a bound to each transition from an ultrafilter to a formula. Lemma L 0 ϕ u implies sup{r L r ϕ u} inf{s M s ϕ u}. This means that the definition for θ L does not give ill-formed bounds.

37 Filters 32 Transition function to filters θ F (u)(f) = ϕ F θ L (u)(ϕ), Φ = { { } if Φ = {ϕ L ϕ ψ for all ψ Φ} otherwise.

38 Ultrafilters 33 f 2 U F θ F (u) f θ F (u) B f is an isomorphism between 2 U and F given by f (U) = u. u U

39 Ultrafilters 34 Transition function to sets of ultrafilters θ U (u)(u) = θ F (u)(f (U)). Theorem The canonical model G U = (U, θ U, l U ) is a GTS.

40 Truth Lemma 35 Truth lemma For consistent ϕ L, G U, u = ϕ iff ϕ u.

41 Weak completeness 36 Weak completeness = ϕ implies ϕ. Proof = ϕ implies ϕ iff ϕ implies = ϕ iff the consistency of ϕ implies the existences of a model for ϕ and this is true because of Lindenbaum s lemma and the truth lemma.

42 Conclusion 37 Contribution New modelling formalism and logic with bounds to encode imprecisions. Logic has the Hennessy-Milner property. Weak-complete axiomization.

43 Conclusion 37 Contribution New modelling formalism and logic with bounds to encode imprecisions. Logic has the Hennessy-Milner property. Weak-complete axiomization. Future work Strong completeness. Dependent axioms. Remove axioms with uncountably many instances. Relationship between WTS and GTS.

44 Thank you 38 Thank you!