Relaxing Goodness is Still Good
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1 Relaxing Goodness is Still Good Gordon Pace 1 Gerardo Schneider 2 1 Dept. of Computer Science and AI University of Malta 2 Dept. of Informatics University of Oslo ICTAC 08 September 1-3, Istanbul, Turkey Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
2 Reachability Analysis of GSPDIs Hybrid System: combines discrete and continuous dynamics Examples: thermostat, robot, chemical reaction Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
3 Reachability Analysis of GSPDIs Hybrid System: combines discrete and continuous dynamics Examples: thermostat, robot, chemical reaction e 3 R 4 e 2 R 2 e 4 e 5 R 5 R 3 R 7 x f? R 1 e 1 e 8 R 6 x 0 e 6 e 7 R 8 Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
4 Outline 1 Polygonal Hybrid Systems (SPDIs) 2 Generalized Polygonal Hybrid Systems (GSPDIs) 3 Reachability Analysis of GSPDIs Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
5 Outline 1 Polygonal Hybrid Systems (SPDIs) 2 Generalized Polygonal Hybrid Systems (GSPDIs) 3 Reachability Analysis of GSPDIs Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
6 Polygonal Hybrid Systems (SPDIs) Preliminaries A constant differential inclusion (angle between vectors a and b): ẋ b a x b a R x Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
7 Polygonal Hybrid Systems (SPDIs) A finite partition of (a subset of) the plane into convex polygonal sets (regions) Dynamics given by the angle determined by two vectors: ẋ b a e 3 R 4 e 2 R 2 e 4 R 5 R 3 R 1 e 1 e 5 R 7 e 8 R 6 e 6 e 7 R 8 Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
8 Polygonal Hybrid Systems (SPDIs) A finite partition of (a subset of) the plane into convex polygonal sets (regions) Dynamics given by the angle determined by two vectors: ẋ b a R 4 e 3 x f e 2 R 2 e 4 R 3 e 1 R 5 R 1 e 5 R 7 e 8 R 6 x 0 e 6 e 7 R 8 Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
9 Polygonal Hybrid Systems (SPDIs) Goodness Goodness Assumption The dynamics of an SPDI only allows trajectories traversing any edge only in one direction Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
10 Polygonal Hybrid Systems (SPDIs) Goodness Goodness Assumption The dynamics of an SPDI only allows trajectories traversing any edge only in one direction e 4 e 3 e 5 e 5 b P b P e e 2 e 2 6 a a e 1 e 1 Good region e 6 e 4 Bad region e3 Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
11 Polygonal Hybrid Systems (SPDIs) Goodness Goodness Assumption The dynamics of an SPDI only allows trajectories traversing any edge only in one direction entry only e 4 exit only e 3 exit only inout e 4 e 5 P e P 5e6 b a e e 2 6 e 1 e 1 Good region b a entry only Bad region e 2 Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
12 Polygonal Hybrid Systems (SPDIs) Goodness Goodness Assumption The dynamics of an SPDI only allows trajectories traversing any edge only in one direction entry only e 4 exit only e 3 exit only inout e 4 e 5 P e P 5e6 b a e e 2 6 e 1 e 1 Good region b a entry only Bad region e 2 Theorem Under the goodness assumption, reachability for SPDIs is decidable Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
13 Outline 1 Polygonal Hybrid Systems (SPDIs) 2 Generalized Polygonal Hybrid Systems (GSPDIs) 3 Reachability Analysis of GSPDIs Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
14 GSPDI: Generalized SPDI Definition An SPDI without the goodness assumption is called a GSPDI e 3 R 4 e 2 e 4 R 2 e 1 R 5 R 3 R 7 R 1 e 5 e 8 R 6 e 6 e 7 R 8 Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
15 Why Goodness is Good e e R R 2 e 4 e e R R 2 e 1 1 e 4 1 e e R e 1 R 3 R e 4 x 3 x f R 3 f xf R 5 R 1 R 5 R 1 R 5 R 1 R 7 R 7 R 7 R 2 e 1 e 5 e 8 e 5 e 8 e 5 e 8 R 6 x 0 e 6 e 7 R 8 R 6 x 0 e 6 e 7 R 8 R 6 x 0 e 6 e 7 R e 6 e 7 e 8 (e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 ) 5 e 9 (e 6 e 7 e 8 e 1 e 2 e 3 e 4 e 5 ) 5 e 6 e 7 e 8 e 9 4 e 6 e 7 e 8 (e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 ) e 9 r 1 s 1 r 2 Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
16 Why Goodness is Good Lemma (Truncated Affine Multivalued Functions (TAMF)) Successors are positive TAMFs: Succ(x) = F({x} S) J where F(x) = [a 1 x + b 1, a 2 x + b 2 ] with a 1, a 2 > 0 Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
17 Why Goodness is Good Lemma (Truncated Affine Multivalued Functions (TAMF)) Successors are positive TAMFs: Succ(x) = F({x} S) J where F(x) = [a 1 x + b 1, a 2 x + b 2 ] with a 1, a 2 > 0 Theorem An edge-signature σ = e 1... e p can always be abstracted into types of signatures of the form σ A = r 1 s 1... r n s nr n+1, where r i is a sequence of pairwise different edges and all s i are disjoint simple cycle. There are finitely many type of signatures. Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
18 Why Goodness is Good Lemma (Truncated Affine Multivalued Functions (TAMF)) Successors are positive TAMFs: Succ(x) = F({x} S) J where F(x) = [a 1 x + b 1, a 2 x + b 2 ] with a 1, a 2 > 0 Theorem An edge-signature σ = e 1... e p can always be abstracted into types of signatures of the form σ A = r 1 s 1... r n s nr n+1, where r i is a sequence of pairwise different edges and all s i are disjoint simple cycle. There are finitely many type of signatures. Many proofs (decidability, soundess, completeness) depend on the goodness assumption Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
19 Problems when Relaxing Goodness Successors are not longer guaranteed to be positive TAMFs Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
20 Problems when Relaxing Goodness Successors are not longer guaranteed to be positive TAMFs Proper inout edges and bounces: e e I 5 I 4 I 1 e I 4 e e e I 3 I 3 I 1 I 0 I 2 I 2 e I 0 e Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
21 Problems when Relaxing Goodness Finiteness argument for types of signature is broken for GSPDIs b c Type of signature: d (abcd) (dcba) (abcd) a a d Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
22 Problems when Relaxing Goodness Finiteness argument for types of signature is broken for GSPDIs c b Type of signature: d (abcd) (dcba) (abcd) a a d Decidability of Reachability for GSPDIs Reduce GSPDI reachability to SPDI reachability (not possible: [SAC 08]) A decidability proof extending that of SPDI Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
23 Outline 1 Polygonal Hybrid Systems (SPDIs) 2 Generalized Polygonal Hybrid Systems (GSPDIs) 3 Reachability Analysis of GSPDIs Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
24 Decidability for GSPDI 1 Prove it is enough to consider trajectories without self-crossing 2 Redefine successors and prove them to be positive TAMFs 3 Duplicate inout edges (e and e 1 ) s.t. each edge is traversed in one direction only 4 Prove that signatures not containing bounces (sub-sequences ee 1 ) behave as for SPDIs 5 Show that we can treat signatures containing bounces 6 Prove soundness and termination Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
25 1. It is enough to consider trajectories without self-crossing Idem as for SPDIs Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
26 2. Successors are positive TAMFs Lemma In GSPDIs, successors can always be written as positive TAMFs. Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
27 2. Successors are positive TAMFs Lemma In GSPDIs, successors can always be written as positive TAMFs. We apply a transformation to successors of regions containing inout edges e e b b a a e e Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
28 3. Duplicate inout edges (e and e 1 ) Out Out a b R 2 In In Out Out a b R 2 In In Out Out a b R 2 In In Out In In? R 1 b a In Out Out Out In In e 1 1 e 1 R 1 b a In Out Out Out In In R 1 b a In Out Out In Out In Out In Out (a) (b) (c) Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
29 4. Signatures without bounces behaves as for SPDIs Lemma For any two edges e 0 and e 1, Succ e0 e 1 is always a positive TAMF, whenever e 1 e 1 0. Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
30 5. We can treat signatures containing bounces Flip Lemma Let Flip[l, u] = [1 u, 1 l]. Then Succ ee 1 = Flip Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
31 5. We can treat signatures containing bounces Flip Lemma Let Flip[l, u] = [1 u, 1 l]. Then Succ ee 1 = Flip Lemma Composing Flip with an inverted TAMF gives a positive TAMF and an inverted TAMF if we compose it with a positive TAMF. Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
32 5. We can treat signatures containing bounces Signatures with even number of bounces Corollary Any signature with an even number of bounces has its behaviour characterised by a positive TAMF. Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
33 5. We can treat signatures containing bounces Signatures with even number of bounces Corollary Any signature with an even number of bounces has its behaviour characterised by a positive TAMF. Lemma Given a simple cycle σ containing an even number of bounces, its iterated behaviour can be calculated as for SPDIs. Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
34 5. We can treat signatures containing bounces Signatures with odd number of bounces Lemma Given a simple cycle s with an odd number of bounces, we can calculate the limit of its iterated behaviour without iterating. Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
35 6. Soundness, Completeness and Termination Algorithm 1 Pre-processing: Redefine successors Transform H: e and e 1 as different edges 2 Generate the finite set of types of signatures Σ = {σ 0,..., σ n } Simple cycles are all distinct 3 Apply Reach σi (x 0, x f ) for each σ i Σ 4 Reach(H, x 0, x f ) = Yes iff for some σ i Σ, Reach σi (x 0, x f ) = Yes. Theorem Reach(H, x 0, x f ) is a sound and complete algorithm calculating GSPDI reachability. The algorithm always terminates. Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
36 Final Remarks This presentation A (constructive) proof that reachability is decidable for GSPDIs Reuse of reachability algorithm for SPDI Acceleration of simple cycles Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
37 Final Remarks This presentation A (constructive) proof that reachability is decidable for GSPDIs Reuse of reachability algorithm for SPDI Acceleration of simple cycles Current Work Implementation of the algorithm (Hallstein A. Hansen) Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
38 Final Remarks This presentation A (constructive) proof that reachability is decidable for GSPDIs Reuse of reachability algorithm for SPDI Acceleration of simple cycles Current Work Implementation of the algorithm (Hallstein A. Hansen) Future Work Application to analysis of nonlinear differential equations Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
39 Motivation Use of SPDIs for approximating nonlinear differential equations Example Pendulum with friction coefficient k, mass M, pendulum length R and gravitational constant g. Behaviour: ẋ = y and ẏ = ky g sin(x) MR 2 R Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
40 Motivation Use of SPDIs for approximating nonlinear differential equations Example Pendulum with friction coefficient k, mass M, pendulum length R and gravitational constant g. Behaviour: ẋ = y and ẏ = ky g sin(x) MR 2 R Triangulation of the plane: Huge number of regions Need to reduce the complexity... without too much overhead Relax Goodness: GSPDI Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
41 Motivation Use of SPDIs for approximating nonlinear differential equations Gerardo Schneider (UiO) Relaxing Goodness is Still Good ICTAC / 27
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