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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− < <
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Input:
Output:
2
2
0
0
x y x xy b
x ay b
∀ ∃ + + >∧
+ + ≤
0 0a b< ∧ >
Real Quantifier Elimination?
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Input:
Output:
Real Quantifier Elimination?
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Input:
Output:
Expression made ofAlgebraic expressions
∀ ∃
∧ ∨ ¬ ⇒ ⇐ ⇔
= ≠ > < ≥ ≤
Expressionequivalentwithout ∀ ∃
Real Quantifier Elimination?
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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• ….
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“Ability”◦ Stablizability◦ Reachability◦ Satisfiability
“Robustness” ◦ Robust control◦ Tolerant system◦ Stability
→∃
→∀
Why applicable?
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App: Nonlinear Control of Aircraft
α
β
M. Jirstrand J. Symbolic Computation (1997)
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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
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App: Optimal Numerical AlgorithmM. Erascu, H. Hong, Reliable Computing (2013)
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App: Stability of Numerical PDEH. Hong and M. Safey-Eldin, J. of Symbolic Computation (2012)
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App: Stability AnalysisMulti-stable model of Cell DifferentiationH, Hong, X. Tang, B. Xia (2013)
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App: Hopf BifurcationGene regulatory networkCircadian clock of green alga
T. Sturm, et al (2013)
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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.
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◦ ~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
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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)• …
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How does it work?
Deep theories from (Real) Algebraic geometryCommutative algebra Real/complex analysis
Efficient operations fromComputational algebraComputational geometry List processing
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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 =
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Challenges!
Improve the efficiency of QETackle nontrivial engineering problems using QE
Exploit the inherent structure of the inputs Collaboration between
Mathematicians (QE) Engineers
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◦ 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
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Come and Join!