Gentle Invitation to“Real Quantifier Elimination”

Hoon Honghong@ncsu.edu

Real Quantifier Elimination?Input:

Output:

2 1 0x x∀ + >

True

Input:

Output:

2 3 1 0x x x∀ + + >

False

Input:

Output:

2 1 0x x bx∀ + + >

2 2b− < <

Input:

Output:

2

2

0

0

x y x xy b

x ay b

∀ ∃ + + >∧

+ + ≤

0 0a b< ∧ >

Real Quantifier Elimination?

Input:

Output:

Real Quantifier Elimination?

Input:

Output:

Expression made ofAlgebraic expressions

∀ ∃

∧ ∨ ¬ ⇒ ⇐ ⇔

= ≠ > < ≥ ≤

Expressionequivalentwithout ∀ ∃

Real Quantifier Elimination?

Why work on it?Foundation of Mathematics

• Hilbert’s Program (1900)

Applications in Science and Engineering• Stability analysis of PDE and Finite differences • Robust control system design• Reachability analysis• Parametric optimization• Hybrid system analysis• Parameter estimation• Robot motion planning• Computer vision• Dynamic geometric constraint solving• Education software for real analysis• ….

“Ability”◦ Stablizability◦ Reachability◦ Satisfiability

“Robustness” ◦ Robust control◦ Tolerant system◦ Stability

→∃

→∀

Why applicable?

App: Nonlinear Control of Aircraft

α

β

M. Jirstrand J. Symbolic Computation (1997)

Where should the actuator B be located so that the armsatisfies “stability” and “positive realness (PR)” ?

A B C

A c.g.C

f1(x,y)

f2(x,y)x

y

H. Anai, S. Hara (2000) Fujitsu

App: Design of HDD Swing-arm

App: Optimal Numerical AlgorithmM. Erascu, H. Hong, Reliable Computing (2013)

App: Stability of Numerical PDEH. Hong and M. Safey-Eldin, J. of Symbolic Computation (2012)

App: Stability AnalysisMulti-stable model of Cell DifferentiationH, Hong, X. Tang, B. Xia (2013)

App: Hopf BifurcationGene regulatory networkCircadian clock of green alga

T. Sturm, et al (2013)

Why work on it? (Recap)

Arise as a fundamental question in the logical foundation of mathematics

Numerous problems from science and engineering can be reduced to QE problems.

◦ ~1930: Tarski

222⋰𝑛𝑛

◦ ~1975: Collins

22𝑛𝑛

◦ ~1990: Canny, Grigorev, Renegar, Roy, Basu,…

2𝑛𝑛 for existential inputs

◦ ----- : Arnon, McCallum, Hong, Brown, Strezbonski,…

Efficient algorithms for moderate inputs

◦ ----- : Weispfenning, Sturm, Dolzman, Hong, Safey-Eldin, Anai, Xia, Chen, Kosta, . . .

Efficient algorithms for special inputs

History/State of the Art

Software Packages

General Inputs• QEPCAD (in C)• Resolve (in Mathematica)• Regular chain (in Maple)• Discoverer (in Maple)

Special Inputs• Redlog (in REDUCE)• SyNRAC (in Maple)• …

How does it work?

Deep theories from (Real) Algebraic geometryCommutative algebra Real/complex analysis

Efficient operations fromComputational algebraComputational geometry List processing

Taste of a mathematical theory…

2

2

2

2

1 0 41 0 4

( , )1 2 1

1 2 1

xx

res f gx x

x x

−−

=− −

− −

( )

2 2

2 2

( , ) 4( , ) 2 1

f x y y xg x y y x y x

= + −

= − + +

( ) ?h x =

4 3 24 4 15 16 12x x x x+ − − +=0.538, 1.801x =

Challenges!

Improve the efficiency of QETackle nontrivial engineering problems using QE

Exploit the inherent structure of the inputs Collaboration between

Mathematicians (QE) Engineers

◦ Computational Quantifier Elimination Computer Journal Edited by Hong◦ Collins’ 65th Birthday Conference RISC monograph Edited by Johnson and Caviness◦ Application of Quantifier Elimination Journal of Symbolic Computation Edited by Hong◦ Algorithms in Real Algebraic Geometry Springer Authored by Basu, Pollack, Roy◦ Numerous individual articles Journal of Symbolic Computation

Some References

Come and Join!