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Reachability Analysis of Generalized Polygonal Hybrid Systems (GSPDIs)

Gerardo Schneider

Department of Informatics University of Oslo

SAC-SV’08 March 16–20, 2008 - Fortaleza

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 1 / 44

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Reachability Analysis of GSPDIs

Hybrid System: combines discrete and continuous dynamics Examples: thermostat, robot, chemical reaction

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 3 / 44

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Reachability Analysis of GSPDIs

Hybrid System: combines discrete and continuous dynamics Examples: thermostat, robot, chemical reaction

?

e5 R4 e2 e6 x0 e3 e8 R8 e7 e1 R2 R5 R3 R7 xf R6 R1 e4

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 3 / 44

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Outline

1

Polygonal Hybrid Systems (SPDIs) and Motivation

2

Generalized Polygonal Hybrid Systems (GSPDIs)

3

Reachability Analysis of GSPDIs

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 4 / 44

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Outline

1

Polygonal Hybrid Systems (SPDIs) and Motivation

2

Generalized Polygonal Hybrid Systems (GSPDIs)

3

Reachability Analysis of GSPDIs

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 5 / 44

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Polygonal Hybrid Systems (SPDIs)

Preliminaries

A constant differential inclusion (angle between vectors a and b): ˙ x ∈ ∠b

a

x’ x b a R

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 7 / 44

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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: ˙ x ∈ ∠b

a

R7 R6 R1 e5 e4 R4 e2 e6 e3 e8 R8 e7 e1 R2 R5 R3

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 9 / 44

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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: ˙ x ∈ ∠b

a

R6 R4 e4 e5 R1 xf R7 R3 R5 R2 e1 e7 R8 e8 e3 x0 e6 e2

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 9 / 44

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Polygonal Hybrid Systems (SPDIs)

Goodness

Goodness Assumption

The dynamics of an SPDI only allows trajectories traversing any edge

• nly in one direction

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 11 / 44

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Polygonal Hybrid Systems (SPDIs)

Goodness

Goodness Assumption

The dynamics of an SPDI only allows trajectories traversing any edge

• nly in one direction

Good region Bad region

e5 e2 e3 e6 e1 e2 e5 e6 e4 e3 e4 P P e1

a b a b

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 11 / 44

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Polygonal Hybrid Systems (SPDIs)

Goodness

Goodness Assumption

The dynamics of an SPDI only allows trajectories traversing any edge

• nly in one direction

exit−only

Good region Bad region

entry−only exit−only inout entry−only

P e5 e4 e4 e3 e2 e1 e2 e3 e6 e1 e5 e6 P

b a b a

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 13 / 44

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Polygonal Hybrid Systems (SPDIs)

Goodness

Goodness Assumption

The dynamics of an SPDI only allows trajectories traversing any edge

• nly in one direction

exit−only

Good region Bad region

entry−only exit−only inout entry−only

P e5 e4 e4 e3 e2 e1 e2 e3 e6 e1 e5 e6 P

b a b a Theorem

Under the goodness assumption, reachability for SPDIs is decidable

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 13 / 44

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Motivation

Use of SPDIs for approximating non-linear differential equations

Example

Pendulum with friction coefficient k, mass M, pendulum length R and gravitational constant g. Behaviour: ˙ x = y and ˙ y = − ky

MR2 − g sin(x) R

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 15 / 44

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Motivation

Use of SPDIs for approximating non-linear differential equations

Example

Pendulum with friction coefficient k, mass M, pendulum length R and gravitational constant g. Behaviour: ˙ x = y and ˙ y = − ky

MR2 − g sin(x) R

Triangulation of the plane: Huge number of regions Need to reduce the complexity ... without too much overhead

Relax Goodness: GSPDI

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 15 / 44

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Motivation

Use of SPDIs for approximating non-linear differential equations

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 17 / 44

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Outline

1

Polygonal Hybrid Systems (SPDIs) and Motivation

2

Generalized Polygonal Hybrid Systems (GSPDIs)

3

Reachability Analysis of GSPDIs

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 18 / 44

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GSPDI: Generalized SPDI

Definition

An SPDI without the goodness assumption is called a GSPDI

R2 R6 R1 e5 e4 R4 e2 e6 e3 e8 R8 e7 e1 R5 R3 R7

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 20 / 44

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Why Goodness is Good

1 4 3 2 1

r1 s∗

1 r2

r e6e7e8(e1e2e3e4e5e6e7e8)∗e9 e6e7e8e1e2e3

xf xf xf R7 R3 R5 R2 e1 e7 R8 e8 e3 x0 e6 e2 R4 e4 e5 R1 R6 R7 R3 R5 R2 e1

e6e7e8e1e2e3 (e6e7e8e1e2e3e4e5)5e6e7e8e9

e7 R8 e8 e3 x0 e6 e2 R4 e4 e5 R1 R6 R7 R3 R5 R2 e1 e7 R8 e8 e3 x0 e6

e6e7e8(e1e2e3e4e5e6e7e8)5e9 e6e7e8e1e2e3

e2 R4 e4 e5 R1 R6 Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 22 / 44

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Why Goodness is Good

Theorem

An edge-signature σ = e1 . . . ep can always be abstracted into types of signatures of the form σA = r1s∗

1 . . . rn s∗ nrn+1, where ri is a sequence

• f pairwise different edges and all si are disjoint simple cycle.

There are finitely many type of signatures.

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 24 / 44

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Why Goodness is Good

Theorem

An edge-signature σ = e1 . . . ep can always be abstracted into types of signatures of the form σA = r1s∗

1 . . . rn s∗ nrn+1, where ri is a sequence

• f pairwise different edges and all si are disjoint simple cycle.

There are finitely many type of signatures. Many proofs (decidability, soundess, completeness) depend on the goodness assumption

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 24 / 44

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Problems when Relaxing Goodness

Finiteness argument for types of signature is broken for GSPDIs

d (abcd)∗ (dcba)∗ (abcd)∗ a

c a d b Type of signature:

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 26 / 44

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Problems when Relaxing Goodness

Finiteness argument for types of signature is broken for GSPDIs

d (abcd)∗ (dcba)∗ (abcd)∗ a

c a d b Type of signature:

Challenge: Reachability analysis of GSPDIs

Reduce GSPDI reachability to SPDI reachability; or Provide a completely new decidability proof for GSPDI.

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 26 / 44

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Outline

1

Polygonal Hybrid Systems (SPDIs) and Motivation

2

Generalized Polygonal Hybrid Systems (GSPDIs)

3

Reachability Analysis of GSPDIs

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 27 / 44

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Getting a Decision Algorithm for GSPDIs Based on that of SPDIs

1

It is enough to consider trajectories without self-crossing

2

It is possible to eliminate all inout edges, preserving reachability

3

It is possible to eliminate all sliding edges, preserving reachability

4

Re-state and prove some results on SPDI reachability useful to GPSDI reachability analysis

5

Prove soundness and termination

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 29 / 44

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Getting a Decision Algorithm for GSPDIs Based on that of SPDIs

1

It is enough to consider trajectories without self-crossing

2

It is possible to eliminate all inout edges, preserving reachability

3

It is possible to eliminate all sliding edges, preserving reachability

4

Re-state and prove some results on SPDI reachability useful to GPSDI reachability analysis

5

Prove soundness and termination

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 31 / 44

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Getting a Decision Algorithm for GSPDIs Based on that of SPDIs

1

It is enough to consider trajectories without self-crossing

2

It is possible to eliminate all inout edges, preserving reachability

3

It is possible to eliminate all sliding edges, preserving reachability

4

Re-state and prove some results on SPDI reachability useful to GPSDI reachability analysis

5

Prove soundness and termination No decision algorithm for reachability of GSPDIs... We will give a semi-test algorithm!

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 31 / 44

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It is not possible to eliminate inout edges

x2 e4 a b e3 e2 x1 e1 x0 xf e6 e5 P

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 33 / 44

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It is not possible to eliminate inout edges

x2 e4 a b e3 e2 x1 e1 x0 xf e6 e5 P

Theorem

There is no structure-preserving reduction from the GSPDI reachability problem to the SPDI reachability problem.

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 33 / 44

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A Semi-Test Algorithm for GSPDIs

Hred = {H1, . . . , Hn}: all possible underlying SPDIs, after fixing all the inout edges of H as entry-only or exit-only

Algorithm

1

Detect all the inout edges;

2

Generate the set of SPDIs Hred = {H1, . . . , Hn};

3

Apply the reachability algorithm for SPDIs to each Hi (1 ≤ i ≤ n), ReachSPDI(Hi, x0, xf).

4

If there exists at least one SPDI Hi ∈ Hred such that ReachSPDI(Hi, x0, xf) = Yes then Reach(H, x0, xf) = Yes,

• therwise we do not know.

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 35 / 44

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A Semi-Test Algorithm for GSPDIs

• 1. It is enough to consider trajectories without self-crossing

Idem as for SPDIs

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 37 / 44

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A Semi-Test Algorithm for GSPDIs

• 3. It is possible to eliminate all sliding edges, preserving reachability

Theorem

If there exists a sliding trajectory segment from points x0 ∈ e0 to xf ∈ ef then there always exists a non-sliding trajectory segment between them.

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 39 / 44

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A Semi-Test Algorithm for GSPDIs

• 3. It is possible to eliminate all sliding edges, preserving reachability

Theorem

If there exists a sliding trajectory segment from points x0 ∈ e0 to xf ∈ ef then there always exists a non-sliding trajectory segment between them.

(a’) e

(b’) (b)

e xf xf e x0

(a)

x0 xf xf e

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 39 / 44

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A Semi-Test Algorithm for GSPDIs

• 4. Re-state and prove some results on SPDI reachability useful to GPSDI reachability

analysis

1

Redefine the edge-to-edge successor function

2

Rephrase topologically results using contiguity between entry-only and exit-only edges

3

Re-prove soundess of some algorithms

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 41 / 44

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A Semi-Test Algorithm for GSPDIs

• 5. Soundness and termination

Theorem

Given a GSPDI H, Reach(H, x0, xf) = Yes if ReachSPDI(Hi, x0, xf) = Yes for some Hi ∈ Hred. On the other hand, Reach(H, x0, xf) is inconclusive if for all Hi ∈ Hred, ReachSPDI(Hi, x0, xf) = No. The algorithm always terminate.

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 43 / 44

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Final Remarks

News at submission to SAC-SV

A semi-test for reachability analysis of GSPDIs (this presentation)

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 44 / 44

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Final Remarks

News at submission to SAC-SV

A semi-test for reachability analysis of GSPDIs (this presentation)

Later news after acceptance to SAC-SV

Reachability for GSPDIs is decidable (submitted, not published yet)

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 44 / 44

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Final Remarks

News at submission to SAC-SV

A semi-test for reachability analysis of GSPDIs (this presentation)

Later news after acceptance to SAC-SV

Reachability for GSPDIs is decidable (submitted, not published yet)

Latest news at SAC-SV

Implementation of reachability algorithm and application (current work)

Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 44 / 44