Reasoning About Bounds In Weighted Transition Systems

Transcription

1 Reasoning About Bounds In Weighted Transition Systems QuantLA 2017 September 18, 2017 Mikkel Hansen, Kim Guldstrand Larsen, Radu Mardare, Mathias Ruggaard Pedersen and Bingtian Xue {mhan, kgl, mardare, mrp, Department of Computer Science Aalborg University Denmark Slides adapted from Mikkel Hansen

2 Introduction

3 Introduction Background 1 Cyber-Physical Systems operate in environments that are inherently unpredictable and imprecise. inc./dec. temp. start temp. reading

4 Introduction Motivation 2 Observation When modeling physical phenomena, their quantitative nature is most often captured by means of observations and measurements. The accuracy of these measurements is highly dependent on the methods/equipment used to obtain them as well as the environment in which they are obtained.

5 Introduction Motivation 2 Observation When modeling physical phenomena, their quantitative nature is most often captured by means of observations and measurements. The accuracy of these measurements is highly dependent on the methods/equipment used to obtain them as well as the environment in which they are obtained. x ± 1 Example Consider measuring some physical phenomena with a piece of equipment that is known to be accurate within one unit.

6 Introduction Motivation 3 Example x ± 1 Consider measuring some physical phenomena with a piece of equipment that is known to be accurate within one unit. We may obtain a reading of x and thus conclude that the actual value is somewhere in the interval [x 1; x + 1].

7 Introduction Motivation 3 Example x ± 1 Consider measuring some physical phenomena with a piece of equipment that is known to be accurate within one unit. We may obtain a reading of x and thus conclude that the actual value is somewhere in the interval [x 1; x + 1]. Problem What quantity should we use in our model?

8 Introduction Motivation 4 Problem What quantity should we use in our model? Solution 1: Naive Approach Use the concrete value x regardless of the possible inaccuracy of the measurement. {a} 5 2 {b} {b}

9 Introduction Motivation 5 Solution 2: Abstractions Realize the inherent uncertainty in the quantitative measures and try to abstract it away in the models. In our example, a natural solution could be to use the interval [x 1; x + 1] as a quantitative measure. {a} [4; 6] [1; 3] {a} {b} The model then describes how the system can or may be implemented.

10 Introduction Motivation 6 Our Approach We abstract away the intermediate transitions in [x 1; x + 1], 4 {a} {a} {b}

11 Introduction Motivation 6 Our Approach We abstract away the intermediate transitions in [x 1; x + 1], 4 {a} {a} {b} and instead reason only about the lower/upper bounds on transition weights.

12 Introduction Contribution 7 Our Contribution Reasoning about min and max transitions, including bisimulation.

13 Introduction Contribution 7 Our Contribution Reasoning about min and max transitions, including bisimulation. Modal logic that is invariant under bisimulation.

14 Introduction Contribution 7 Our Contribution Reasoning about min and max transitions, including bisimulation. Modal logic that is invariant under bisimulation. Finite model property and satisfiability checking algorithm.

15 Introduction Contribution 7 Our Contribution Reasoning about min and max transitions, including bisimulation. Modal logic that is invariant under bisimulation. Finite model property and satisfiability checking algorithm. Complete axiomatization for the logic.

16 Models

17 Model Weighted Transition Systems 8 Definition: Weighted Transition System A weighted transition system (WTS) is a tuple M = (S,, l), where S is a non-empty set of states, S R 0 S is the transition relation, and l : S 2 AP is a labeling function mapping to each state a set of atomic propositions. {waiting} 1 s s s 3 {cleaning} {charging}

18 Model Example: Robot Vacuum Cleaner 9 Example: A model of a robot vacuum cleaner {waiting} 1 s s s 3 {cleaning} {charging} 3 states: waiting, cleaning and charging. The time it takes to move between states is given by the transition weights.

19 Model Image Sets 10 Definition: Image Set For arbitrary WTS M = (S,, l) the function θ : S (2 S 2 R 0 ) is defined for any state s S and any set of states T S as { } θ = x R 0 t T such that s x t We refer to θ(s)(t ) as the image from s to T or simply an image set. θ (s) (T ) = inf θ (s) (T ) and θ + (s) (T ) = sup θ (s) (T )

20 Model Example: Robot Vacuum Cleaner 11 Example: A model of a robot vacuum cleaner {waiting} 1 s s s 3 {cleaning} {charging} θ (s 1 ) ({s 2 }) = {0} θ (s 2 ) ({s 1 }) = {5, 10, 15} θ (s 1 ) ({s 2 }) = 0 θ (s 2 ) ({s 1 }) = 5 θ + (s 1 ) ({s 2 }) = 0 θ + (s 2 ) ({s 1 }) = 15

21 Model Bisimulation 12 Definition: Weighted Bisimilarity W Given a WTS M = (S,, l), a binary relation R on S is called a weighted bisimulation relation if and only if srt implies (Atomic Harmony) l(s) = l(t), (Zig) s x s implies t (Zag) t x t such that s Rt, and x t implies s x s such that s Rt. s 1 {a} 1 10 {b} s 2 {b} s 3 {a} {b} t t 2 R = (s 1, t 1 ), (t 1, s 1 ), (s 2, t 2 ), (t 2, s 2 ), (s 3, t 2 ), (t 2, s 3 ), (s 2, s 3 ), (s 3, s 2 ) is a bisimulation relation

22 Model Bisimulation 13 Definition: Generalized Weighted Bisimilarity Given a WTS M = (S,, l), an equivalence relation R on S is called a generalized weighted bisimulation iff srt implies (Atomic Harmony) l(s) = l(t), (Lower Bound) θ (s) (T ) = θ (t) (T ), and (Upper Bound) θ + (s) (T ) = θ + (t) (T ) for any R-equivalence class T S/R. {a} s t {a} {b} s t {b} θ (s) ({s, t }) = θ (t) ({s, t }) and θ + (s) ({s, t }) = θ + (t) ({s, t }) so s t but s W t

23 Model Bisimulation 14 Theorem: Relation between and W Generalized weighted bisimilarity is coarser than weighted bisimilarity, i.e. W

24 Logic

25 Logic Syntax and Semantics 15 Definition: Syntax of L L : ϕ, ψ ::= p ϕ ϕ ψ L r ϕ M r ϕ Definition: L r and M r Semantics M, s = L r ϕ iff θ (s) ( ϕ ) r M, s = M r ϕ iff θ + (s) ( ϕ ) r where ϕ = {t S M, t = ϕ}.

26 Logic Syntax and Semantics 16 Definition: L r and M r Semantics M, s = L r ϕ iff θ (s) ( ϕ ) r M, s = M r ϕ iff θ + (s) ( ϕ ) r where ϕ = {t S M, t = ϕ}. θ (s) ( ϕ ) θ (s) ( ϕ ) r θ θ + θ θ + r (a) M, s = L r ϕ (b) M, s = M r ϕ

27 Logic Syntax and Semantics 17 It is possible that L r ϕ and M r ϕ are both satisfied at the same time. θ (s) ( ϕ ) r θ θ + This also holds for their negation, L r ϕ and M r ϕ. θ (s) ( ϕ ) r θ θ +

28 Logic Bisimulation Invariance 18 Theorem: Bisimulation Invariance For any image-finite WTS M = (S,, l) and states s, t S it holds that s t iff [ ϕ L. M, s = ϕ iff M, t = ϕ]

29 Metatheory

30 Metatheory Axioms 19 (A1): L 0 (A2): L r+q ϕ L r ϕ if q > 0 (A2 ): M r ϕ M r+q ϕ if q > 0 (A3): L r ϕ L q ψ L min{r,q} (ϕ ψ) (A3 ): M r ϕ M q ψ M max{r,q} (ϕ ψ) (A4): L r (ϕ ψ) L r ϕ L r ψ (A5): L 0 ψ (L r ϕ L r (ϕ ψ)) (A5 ): L 0 ψ (M r ϕ M r (ϕ ψ)) (A6): L r+q ϕ M r ϕ if q > 0 (A7): M r ϕ L 0 ϕ (R1): ϕ ψ = ((L r ψ) (L 0 ϕ)) L r ϕ (R1 ): ϕ ψ = ((M r ψ) (L 0 ϕ)) M r ϕ (R2): ϕ ψ = L 0 ϕ L 0 ψ

31 Metatheory Axioms 19 (A1): L 0 (A2): L r+q ϕ L r ϕ if q > 0 (A2 ): M r ϕ M r+q ϕ if q > 0 (A3): L r ϕ L q ψ L min{r,q} (ϕ ψ) (A3 ): M r ϕ M q ψ M max{r,q} (ϕ ψ) (A4): L r (ϕ ψ) L r ϕ L r ψ (A5): L 0 ψ (L r ϕ L r (ϕ ψ)) (A5 ): L 0 ψ (M r ϕ M r (ϕ ψ)) (A6): L r+q ϕ M r ϕ if q > 0 (A7): M r ϕ L 0 ϕ (R1): ϕ ψ = ((L r ψ) (L 0 ϕ)) L r ϕ (R1 ): ϕ ψ = ((M r ψ) (L 0 ϕ)) M r ϕ (R2): ϕ ψ = L 0 ϕ L 0 ψ

32 Metatheory 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

33 Metatheory A3 21 (A3) L r ϕ L q ψ L min{r,q} (ϕ ψ) r q + ϕ ϕ ψ ψ

34 Metatheory A3 22 (A3 ) M r ϕ M q ψ M max{r,q} (ϕ ψ) r q + ϕ ϕ ψ ψ

35 Metatheory A6 23 (A8) L r+q ϕ M r ϕ, q > 0 + r + q r ϕ

36 Metatheory R1 and R1 24 (R1) ϕ ψ = ((L r ψ) (L 0 ϕ)) L r ϕ (R1 ) ϕ ψ = ((M s ψ) (L 0 ϕ)) M s ϕ r s + ψ ϕ

37 Metatheory Soundness 25 Lemma: Soundness ϕ implies = ϕ

38 Metatheory Finite Model Construction 26 Starting from a consistent formula ϕ L, we do the following based on ϕ 1 Construct a finite subset L[ϕ] of L 2 Construct a model M ϕ where: States are the maximal consistent sets (ultrafilters) of L[ϕ] Transitions are given by the formulae in these sets, i.e. L r ϕ u implies u x ϕ where x r. Then, M ϕ is a model for ϕ.

39 Metatheory Finite Model Property 27 Lemma: Truth Lemma If ϕ L is a consistent formula, then for formulae all ψ L[ϕ] and ultrafilters u L[ϕ] we have M ϕ, u = ψ iff ψ u Lemma: Finite Model Property For any consistent formula ϕ L, there exists a finite WTS M = (S,, l) and a state s S such that M, s = ϕ

40 Metatheory Completeness 28 Theorem: Completeness For any formula ϕ L it holds that = ϕ implies ϕ

41 Conclusions

42 Concluding Remarks 29 We have proposed abstracting away individual transition weights in a WTS by introducing the notion of an image set. We have proposed a modal logic that reasons about the minimum and maximum weights on transitions and shown that it has the finite model property. We have presented a procedure for determining the satisfiability of formulae in our logic (not covered in this talk). We have proposed an axiomatization for our logic and shown that is is complete.

43 Open Problems

44 Open Problems Strong Completeness 30 There is a stronger notion of completeness, often called strong completeness, which asserts that Φ = ϕ implies Φ ϕ for any set of formulae Φ L. Completeness is a special case of strong completeness where Φ =. In the case of compact logics, strong completeness follows directly from completeness. However, our logic is non-compact. Φ = {L q ϕ q < r} { L r ϕ}

45 Open Problems Temporal Specification 31 Many interesting properties of CPSs such as liveness and deadlock freeness cannot be expressed in modal logic. We would like to explore temporal specification languages such as CTL or LTL.

46 Thank you