IMITATOR II A Tool for Solving the Good Parameters Problem in - PowerPoint PPT Presentation

INFINITY ’2010 Singapore 21st September 2010 IMITATOR II A Tool for Solving the Good Parameters Problem in Timed Automata ´ Etienne Andr´ e Laboratoire Sp´ ecification et V´ erification LSV, ENS de Cachan & CNRS, France Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 1 / 23

Introduction Context The Good Parameters Problem Context: Verification of timed systems ◮ Use of timing parameters (unknown constants) ◮ Model of Parametric Timed Automata (PTA) Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 2 / 23

Introduction Context The Good Parameters Problem Context: Verification of timed systems ◮ Use of timing parameters (unknown constants) ◮ Model of Parametric Timed Automata (PTA) The good parameters problem: [Frehse et al., 2008] ◮ “Given a bounded parameter domain V 0 , find a dense set of points (timing parameters) of good behavior in V 0 (ideally the largest one)” V 0 Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 2 / 23

Introduction Context The Good Parameters Problem Context: Verification of timed systems ◮ Use of timing parameters (unknown constants) ◮ Model of Parametric Timed Automata (PTA) The good parameters problem: [Frehse et al., 2008] ◮ “Given a bounded parameter domain V 0 , find a dense set of points (timing parameters) of good behavior in V 0 (ideally the largest one)” K 0 V 0 Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 2 / 23

Introduction Context Classical approaches Verification of the property on a set of discrete points ◮ Drawback: would need an infinite number of verifications to obtain a dense set of points Computation of all the reachable states of a PTA, and intersection with the set of bad states [Alur et al., 1995] ◮ Drawback: too costly in practice Approach based on CEGAR [Clarke et al., 2000, Frehse et al., 2008] ◮ Drawback: underapproximation Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 3 / 23

Introduction Context Classical approaches Verification of the property on a set of discrete points ◮ Drawback: would need an infinite number of verifications to obtain a dense set of points Computation of all the reachable states of a PTA, and intersection with the set of bad states [Alur et al., 1995] ◮ Drawback: too costly in practice Approach based on CEGAR [Clarke et al., 2000, Frehse et al., 2008] ◮ Drawback: underapproximation New approach implemented in Imitator II ◮ Method of behavioral cartography Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 3 / 23

Introduction Preliminaries Good and Bad Traces Trace over a PTA: finite alternating sequence of locations and actions (time-abstract run) A trace is said to be good if it verifies a given property ◮ Example of property φ : “ b always occurs before c ” ◮ Example of good trace w.r.t. φ a e d a b f c ◮ Example of bad trace w.r.t. φ a e d a f c b Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 4 / 23

Outline Outline The Inverse Method Algorithm 1 The Behavioral Cartography Algorithm 2 Implementation and Case Studies 3 Final Remarks 4 Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 5 / 23

The Inverse Method Algorithm Outline The Inverse Method Algorithm 1 The Behavioral Cartography Algorithm 2 Implementation and Case Studies 3 Final Remarks 4 Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 6 / 23

The Inverse Method Algorithm The Inverse Method (1/2) PTA A Imitator II Constraint K 0 on the parameters Inverse Method Reference point π 0 Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 7 / 23

The Inverse Method Algorithm The Inverse Method (2/2) Input ◮ A PTA A ◮ A reference valuation π 0 of all the parameters of A · π 0 Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 8 / 23

The Inverse Method Algorithm The Inverse Method (2/2) Input ◮ A PTA A ◮ A reference valuation π 0 of all the parameters of A Output: tile K 0 ◮ Convex constraint on the parameters such that ⋆ π 0 | = K 0 ⋆ For all point π | = K 0 , A under π has the same trace set as for π 0 [Andr´ e et al., 2009] K 0 · π 0 Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 8 / 23

The Inverse Method Algorithm Application to the Root Contention Protocol rc slow min 220 Root contention protocol of the IEEE 210 1394 (“FireWire”) High Performance 200 Serial Bus [Hune et al., 2002] 190 180 Input: IEEE reference valuation rc slow min = 159 ns 170 · delay = 30 ns 160 π 0 150 140 130 120 110 100 90 80 delay 00 10 20 30 40 50 60 70 80 90 100 Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 9 / 23

The Inverse Method Algorithm Application to the Root Contention Protocol rc slow min 220 Root contention protocol of the IEEE 210 1394 (“FireWire”) High Performance 200 Serial Bus [Hune et al., 2002] K 0 190 180 Input: IEEE reference valuation rc slow min = 159 ns 170 · delay = 30 ns 160 π 0 Output: 150 K 0 : 2 delay < 76 140 ∧ 2 delay + 85 < rc slow min 130 120 110 100 90 80 delay 00 10 20 30 40 50 60 70 80 90 100 Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 9 / 23

The Inverse Method Algorithm Application to the Root Contention Protocol rc slow min 220 Root contention protocol of the IEEE 210 1394 (“FireWire”) High Performance 200 Serial Bus [Hune et al., 2002] K 0 190 180 Input: IEEE reference valuation rc slow min = 159 ns 170 · delay = 30 ns 160 π 0 Output: 150 K 0 : 2 delay < 76 140 ∧ 2 delay + 85 < rc slow min 130 Prop 3 : The minimum probability that 120 a leader is elected after three rounds 110 or less is greater or equal to 0 . 75 100 ◮ For all π | = K 0 , Prop 3 is 90 satisfied 80 delay 00 10 20 30 40 50 60 70 80 90 100 Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 9 / 23

The Behavioral Cartography Algorithm Outline The Inverse Method Algorithm 1 The Behavioral Cartography Algorithm 2 Implementation and Case Studies 3 Final Remarks 4 Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 10 / 23

The Behavioral Cartography Algorithm The Behavioral Cartography Algorithm Goal: Find the maximal set of points corresponding to a good behavior Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 11 / 23

The Behavioral Cartography Algorithm The Behavioral Cartography Algorithm Goal: Find the maximal set of points corresponding to a good behavior Method: Iterate the inverse method for all the integer points of a given rectangle V 0 Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 11 / 23

The Behavioral Cartography Algorithm The Behavioral Cartography Algorithm Goal: Find the maximal set of points corresponding to a good behavior Method: Iterate the inverse method for all the integer points of a given rectangle V 0 Output: set of tiles for all the integer points of V 0 ◮ � behavioral cartography of the parameter space [Andr´ e and Fribourg, 2010] PTA A Cartography Cover Algorithm Rectangle V 0 Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 11 / 23

The Behavioral Cartography Algorithm The Root Contention Protocol: Cartography rc slow min 220 210 200 190 We consider the following V 0 : rc slow min ∈ [140; 200] and 180 delay ∈ [1; 50] 170 160 150 140 130 120 110 100 90 80 delay 00 10 20 30 40 50 60 70 80 90 100 Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 12 / 23

The Behavioral Cartography Algorithm The Root Contention Protocol: Cartography rc slow min 220 210 200 190 We consider the following V 0 : rc slow min ∈ [140; 200] and 180 1 delay ∈ [1; 50] 170 160 150 140 130 120 110 100 90 80 delay 00 10 20 30 40 50 60 70 80 90 100 Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 12 / 23

The Behavioral Cartography Algorithm The Root Contention Protocol: Cartography rc slow min 220 210 200 190 We consider the following V 0 : rc slow min ∈ [140; 200] and 180 1 delay ∈ [1; 50] 170 160 150 2 140 130 120 110 100 90 80 delay 00 10 20 30 40 50 60 70 80 90 100 Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 12 / 23

The Behavioral Cartography Algorithm The Root Contention Protocol: Cartography rc slow min 220 210 200 190 We consider the following V 0 : rc slow min ∈ [140; 200] and 180 1 delay ∈ [1; 50] 170 160 150 2 140 3 130 120 110 100 90 80 delay 00 10 20 30 40 50 60 70 80 90 100 Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 12 / 23

The Behavioral Cartography Algorithm The Root Contention Protocol: Cartography rc slow min 220 210 200 190 We consider the following V 0 : rc slow min ∈ [140; 200] and 180 1 delay ∈ [1; 50] 170 160 150 4 2 140 3 130 120 110 100 90 80 delay 00 10 20 30 40 50 60 70 80 90 100 Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 12 / 23

The Behavioral Cartography Algorithm The Root Contention Protocol: Cartography rc slow min 220 210 200 190 We consider the following V 0 : rc slow min ∈ [140; 200] and 180 1 delay ∈ [1; 50] 170 160 5 150 4 2 140 3 130 120 110 100 90 80 delay 00 10 20 30 40 50 60 70 80 90 100 Etienne ANDR´ ´ E (LSV) IMITATOR II 21st September 2010 12 / 23