Decidability of the Reachability for a Family of Linear Vector - PowerPoint PPT Presentation

Background Reachable Sets of LDSs Decision Procedure for T e Isolating Real Roots of PEFs Evaluation Discussions and Conclusions Decidability of the Reachability for a Family of Linear Vector Fields Ting Gan 1 , Mingshuai Chen 2 , Liyun Dai 1 , Bican Xia 1 , and Naijun Zhan 2 1 LMAM & School of Mathematical Sciences, Peking University 2 State Key Lab. of Computer Science, Institute of Software, Chinese Academy of Sciences Shanghai, October 2015 Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Shanghai, ATVA 2015 1 / 37

Background Reachable Sets of LDSs Decision Procedure for T e Isolating Real Roots of PEFs Evaluation Discussions and Conclusions Outline 1 Background Computing Reachable Sets of Linear Dynamics Systems (LDSs) with Inputs 2 Decision Procedure for T e 3 4 Isolating Real Roots of PEFs Evaluation Results 5 Discussions and Conclusions 6 Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Shanghai, ATVA 2015 2 / 37

x t = Temperature in the attic, x t = Temperature in the living area, x t = Temperature in the basement, t = Time in hours. x x x x x x x x x x x x x x T with the initial set X x x x x x x . Is it possible for the temperature x getting over than F (unsafe) ? UNBOUNDED. Background Reachable Sets of LDSs Decision Procedure for T e Isolating Real Roots of PEFs Evaluation Discussions and Conclusions Example : Home Heating Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Shanghai, ATVA 2015 3 / 37

x t = Temperature in the attic, x t = Temperature in the living area, x t = Temperature in the basement, t = Time in hours. x x x x x x x x x x x x x x T with the initial set X x x x x x x . Is it possible for the temperature x getting over than F (unsafe) ? UNBOUNDED. Background Reachable Sets of LDSs Decision Procedure for T e Isolating Real Roots of PEFs Evaluation Discussions and Conclusions Example : Home Heating Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Shanghai, ATVA 2015 3 / 37

x x x x x x x x x x x x x x T with the initial set X x x x x x x . Is it possible for the temperature x getting over than F (unsafe) ? UNBOUNDED. Background Reachable Sets of LDSs Decision Procedure for T e Isolating Real Roots of PEFs Evaluation Discussions and Conclusions Example : Home Heating x 3 ( t ) = Temperature in the attic, x 2 ( t ) = Temperature in the living area, x 1 ( t ) = Temperature in the basement, t = Time in hours. Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Shanghai, ATVA 2015 3 / 37

T with the initial set X x x x x x x . Is it possible for the temperature x getting over than F (unsafe) ? UNBOUNDED. Background Reachable Sets of LDSs Decision Procedure for T e Isolating Real Roots of PEFs Evaluation Discussions and Conclusions Example : Home Heating x 3 ( t ) = Temperature in the attic, x 2 ( t ) = Temperature in the living area, x 1 ( t ) = Temperature in the basement, t = Time in hours. x 1 = 1 2 (45 − x 1 ) + 1 ˙ 2 ( x 2 − x 1 ) , x 2 = 1 2 ( x 1 − x 2 ) + 1 4 (35 − x 2 ) + 1 4 ( x 3 − x 2 ) + 20 , ˙ x 3 = 1 4 ( x 2 − x 3 ) + 3 4 (35 − x 3 ) , ˙ Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Shanghai, ATVA 2015 3 / 37

Is it possible for the temperature x getting over than F (unsafe) ? UNBOUNDED. Background Reachable Sets of LDSs Decision Procedure for T e Isolating Real Roots of PEFs Evaluation Discussions and Conclusions Example : Home Heating x 3 ( t ) = Temperature in the attic, x 2 ( t ) = Temperature in the living area, x 1 ( t ) = Temperature in the basement, t = Time in hours. x 1 = 1 2 (45 − x 1 ) + 1 ˙ 2 ( x 2 − x 1 ) , x 2 = 1 2 ( x 1 − x 2 ) + 1 4 (35 − x 2 ) + 1 4 ( x 3 − x 2 ) + 20 , ˙ x 3 = 1 4 ( x 2 − x 3 ) + 3 4 (35 − x 3 ) , ˙ with the initial set X = { ( x 1 , x 2 , x 3 ) T | 1 − ( x 1 − 45) 2 − ( x 2 − 35) 2 − ( x 3 − 35) 2 > 0 } . Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Shanghai, ATVA 2015 3 / 37

UNBOUNDED. Background Reachable Sets of LDSs Decision Procedure for T e Isolating Real Roots of PEFs Evaluation Discussions and Conclusions Example : Home Heating x 3 ( t ) = Temperature in the attic, x 2 ( t ) = Temperature in the living area, x 1 ( t ) = Temperature in the basement, t = Time in hours. x 1 = 1 2 (45 − x 1 ) + 1 ˙ 2 ( x 2 − x 1 ) , x 2 = 1 2 ( x 1 − x 2 ) + 1 4 (35 − x 2 ) + 1 4 ( x 3 − x 2 ) + 20 , ˙ x 3 = 1 4 ( x 2 − x 3 ) + 3 4 (35 − x 3 ) , ˙ with the initial set X = { ( x 1 , x 2 , x 3 ) T | 1 − ( x 1 − 45) 2 − ( x 2 − 35) 2 − ( x 3 − 35) 2 > 0 } . Is it possible for the temperature x 2 getting over than 70 ◦ F (unsafe) ? Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Shanghai, ATVA 2015 3 / 37

Background Reachable Sets of LDSs Decision Procedure for T e Isolating Real Roots of PEFs Evaluation Discussions and Conclusions Example : Home Heating x 3 ( t ) = Temperature in the attic, x 2 ( t ) = Temperature in the living area, x 1 ( t ) = Temperature in the basement, t = Time in hours. x 1 = 1 2 (45 − x 1 ) + 1 ˙ 2 ( x 2 − x 1 ) , x 2 = 1 2 ( x 1 − x 2 ) + 1 4 (35 − x 2 ) + 1 4 ( x 3 − x 2 ) + 20 , ˙ x 3 = 1 4 ( x 2 − x 3 ) + 3 4 (35 − x 3 ) , ˙ with the initial set X = { ( x 1 , x 2 , x 3 ) T | 1 − ( x 1 − 45) 2 − ( x 2 − 35) 2 − ( x 3 − 35) 2 > 0 } . Is it possible for the temperature x 2 getting over than 70 ◦ F (unsafe) ? UNBOUNDED. Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Shanghai, ATVA 2015 3 / 37

Background Reachable Sets of LDSs Decision Procedure for T e Isolating Real Roots of PEFs Evaluation Discussions and Conclusions Hybrid Systems Hybrid systems exhibit combinations of discrete jumps and continuous evolution, many of which are Safety-critical. Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Shanghai, ATVA 2015 4 / 37

Background Reachable Sets of LDSs Decision Procedure for T e Isolating Real Roots of PEFs Evaluation Discussions and Conclusions Safety Verification Using Reachable Sets  System is safe, if no trajectory enters the unsafe set. 1. The figure is taken from [M. Althoff, 2010]. Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Shanghai, ATVA 2015 5 / 37

Quantifier Elimination : T Example T x y x xy b x ay b a b = = Background Reachable Sets of LDSs Decision Procedure for T e Isolating Real Roots of PEFs Evaluation Discussions and Conclusions Tarski Algebra and Quantifier Elimination Tarski Algebra ( T ( R ) )= real numbers with arithmetic and ordering. Example ϕ := ∀ x ∃ y : x 2 + xy + b > 0 ∧ x + ay 2 + b ≤ 0 Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Shanghai, ATVA 2015 6 / 37

Background Reachable Sets of LDSs Decision Procedure for T e Isolating Real Roots of PEFs Evaluation Discussions and Conclusions Tarski Algebra and Quantifier Elimination Tarski Algebra ( T ( R ) )= real numbers with arithmetic and ordering. Example ϕ := ∀ x ∃ y : x 2 + xy + b > 0 ∧ x + ay 2 + b ≤ 0 Quantifier Elimination : T ( R ) | = ϕ ← → ϕ ′ Example = ∀ x ∃ y ( x 2 + xy + b > 0 ∧ x + ay 2 + b ≤ 0) T ( R ) | → a < 0 ∧ b > 0 ← � �� � � �� � ϕ ′ ϕ Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Shanghai, ATVA 2015 6 / 37

Background Reachable Sets of LDSs Decision Procedure for T e Isolating Real Roots of PEFs Evaluation Discussions and Conclusions Quantifier Elimination Survey of QE Algorithms Tarski's algorithm [Tarski 51] : the first one, but its complexity is nonelementary, impratical, simplified by Seidenberg [Seidenberg 54]. Collins' algorithm [Collins 76] : based on cylindrical algebraic decomposition (CAD), double exponential in the number of variables, improved by Hoon Hong [Hoon Hong 92] by combining with SAT engine partial cylindrical algebraic decomposition (PCAD), implemented in many computer algebra tools, e.g., QEPCAD,REDLOG, . . . . Ben-Or, Kozen and Reif's algorithm [Ben-Or, Kozen & Reif 86] : double exponential in the number of variables using sequential computation, single exponential using parallel computation, based on Sturm sequence and Sturm Theorem. More efficient algorithms [Grigor'ev & Vorobjov 88, Grigor'ev 88], [Renegar 89], [Heintz, Roy & Solerno 89], [Basu,Pollack & Roy 96] : mainly based on BKR's approach, double exponential in the number of quantifier alternation, no implementation yet. Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Shanghai, ATVA 2015 7 / 37

TC is still open. In 2008, Strzebonski showed the decidability of e , the extension of TA with polynomial exponential functions (PEFs) : m i t f t x f i t x e i Background Reachable Sets of LDSs Decision Procedure for T e Isolating Real Roots of PEFs Evaluation Discussions and Conclusions Tarski's Conjecture (TC) Whether the extension of TA with exponentiation is decidable ? Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Shanghai, ATVA 2015 8 / 37