LIBPOLY: A LIBRARY FOR REASONING ABOUT POLYNOMIALS Dejan Jovanovi - PowerPoint PPT Presentation

LIBPOLY: A LIBRARY FOR REASONING ABOUT POLYNOMIALS Dejan Jovanović Bruno Dutertre SRI International SMT Workshop 2017

OUTLINE INTRODUCTION LIBPOLY Working with Polynomials Constructing a Sign Table Cylindrical Algebraic Decomposition CONCLUSION

OUTLINE INTRODUCTION LIBPOLY Working with Polynomials Constructing a Sign Table Cylindrical Algebraic Decomposition CONCLUSION

NON-LINEAR REASONING T x 20 1 T z T y MANY APPLICATIONS T x T z T y Example from Narkawicz, Muõz, Formal Verification of Conflict Detection Algorithms for Arbitrary Trajectories, 2012 1 ( t ) = 3 . 2484 + 270 . 7t + 433 . 12t 2 − 324 . 83999t 3 20 1 ( t ) = 15 . 1592 + 108 . 28t + 121 . 2736t 2 − 649 . 67999t 3 0 � 20 1 ( t ) = 38980 . 8 + 5414t − 21656t 2 + 32484t 3 2 ( t ) = 1 . 0828 − 135 . 35t + 234 . 9676t 2 2 + 3248 . 4t 3 40 000 2 ( t ) = 18 . 40759 − 230 . 6364t − 121 . 2736t 2 − 649 . 67999t 3 2 ( t ) = 40280 . 15999 − 10828t + 24061 . 9816t 2 − 32484t 3 39 500 39 000 0 D = 5 H = 1000 0 ≤ t ≤ 20 40 2 ( t )) 2 + ( T y 2 ( t )) 2 ≤ D 2 | T z 1 ( t ) − T z 2 ( t ) | ≤ H ( T x 1 ( t ) − T x 1 ( t ) − T y 60

NON-LINEAR REASONING 1 T x T z T y T x MANY APPLICATIONS T z Example from Narkawicz, Muõz, Formal Verification of Conflict Detection Algorithms for Arbitrary Trajectories, 2012 20 RUN SMT SOLVER 319 T y 1 ( t ) = 3 . 2484 + 270 . 7t + 433 . 12t 2 − 324 . 83999t 3 20 1 ( t ) = 15 . 1592 + 108 . 28t + 121 . 2736t 2 − 649 . 67999t 3 0 � 20 1 ( t ) = 38980 . 8 + 5414t − 21656t 2 + 32484t 3 2 ( t ) = 1 . 0828 − 135 . 35t + 234 . 9676t 2 2 + 3248 . 4t 3 40 000 2 ( t ) = 18 . 40759 − 230 . 6364t − 121 . 2736t 2 − 649 . 67999t 3 t �→ 16384 ≈ 0 . 019470215 2 ( t ) = 40280 . 15999 − 10828t + 24061 . 9816t 2 − 32484t 3 39 500 39 000 0 D = 5 H = 1000 0 ≤ t ≤ 20 40 2 ( t )) 2 + ( T y 2 ( t )) 2 ≤ D 2 | T z 1 ( t ) − T z 2 ( t ) | ≤ H ( T x 1 ( t ) − T x 1 ( t ) − T y 60

NON-LINEAR REASONING SMT TECHNIQUES Popular techniques in SMT (QF_NRA): Cylindrical Algebraic Decomposition (CAD): complete method, currently state-of-the-art; requires advanced polynomial operations. ◮ Interval reasoning: RASAT ◮ Linear reasoning + model-based refinement: CVC4 ◮ DPLL(T) + VTS: VERIT ◮ DPLL(T) + CAD: SMTRAT, VERIT ◮ MCSAT + CAD: Z3, YICES2

NON-LINEAR REASONING SMT TECHNIQUES Popular techniques in SMT (QF_NRA): Cylindrical Algebraic Decomposition (CAD): ◮ Interval reasoning: RASAT ◮ Linear reasoning + model-based refinement: CVC4 ◮ DPLL(T) + VTS: VERIT ◮ DPLL(T) + CAD: SMTRAT, VERIT ◮ MCSAT + CAD: Z3, YICES2 ◮ complete method, currently state-of-the-art; ◮ requires advanced polynomial operations.

NON-LINEAR REASONING SMT SOLVERS (2016) 1e+06 100000 10000 cumulative time (s) 1000 100 10 1 z3 yices2 0.1 smtrat rasat-0.4 0.01 rasat-0.3 cvc4 0.001 0 2000 4000 6000 8000 10000 benchmarks solved

SMT SOLVERS (2017) NON-LINEAR REASONING 1e+06 100000 10000 cumulative time (s) 1000 100 10 1 yices2 0.1 z3 verit+rasat+redlog 0.01 cvc4 smtrat 0.001 0 2000 4000 6000 8000 10000 benchmarks solved

NON-LINEAR REASONING CAD-BASED REASONING 2. Basic operations: 3. Solving and model representation: 4. Projection and symbolic explanations: 1. Representation of polynomials. ◮ variables, variable ordering; ◮ arithmetic (addition, multiplication, …); ◮ GCD computation; ◮ some factorization. ◮ Sturm sequences; ◮ interval reasoning; ◮ root isolation (multivariate); ◮ resultants; ◮ computation with algebraic numbers. ◮ principal subresultant coefgicients.

NON-LINEAR REASONING CAD-BASED REASONING Use LIBPOLY Implement yourself Borrow and adapt code Use a computer algebra system Use an existing library HOW TO GET THESE? 4. Projection and symbolic explanations: . 2. Basic operations: 3. Solving and model representation: 1. Representation of polynomials. ◮ variables, variable ordering; ◮ arithmetic (addition, multiplication, …); ◮ GCD computation; ◮ some factorization. ◮ Sturm sequences; ◮ interval reasoning; ◮ root isolation (multivariate); ◮ resultants; ◮ computation with algebraic numbers. ◮ principal subresultant coefgicients.

NON-LINEAR REASONING CAD-BASED REASONING Use LIBPOLY Implement yourself Borrow and adapt code Use a computer algebra system HOW TO GET THESE? 4. Projection and symbolic explanations: . 3. Solving and model representation: 2. Basic operations: 1. Representation of polynomials. ◮ variables, variable ordering; ◮ arithmetic (addition, multiplication, …); ◮ GCD computation; ◮ some factorization. ◮ Use an existing library ◮ Sturm sequences; ◮ interval reasoning; ◮ root isolation (multivariate); ◮ resultants; ◮ computation with algebraic numbers. ◮ principal subresultant coefgicients.

NON-LINEAR REASONING CAD-BASED REASONING Use LIBPOLY Implement yourself Borrow and adapt code Use a computer algebra system HOW TO GET THESE? 4. Projection and symbolic explanations: . 3. Solving and model representation: 2. Basic operations: 1. Representation of polynomials. ◮ variables, variable ordering; ◮ arithmetic (addition, multiplication, …); ◮ GCD computation; ◮ some factorization. ◮ Use an existing library � ◮ Sturm sequences; ◮ interval reasoning; ◮ root isolation (multivariate); ◮ resultants; ◮ computation with algebraic numbers. ◮ principal subresultant coefgicients.

NON-LINEAR REASONING CAD-BASED REASONING Use LIBPOLY Implement yourself Borrow and adapt code HOW TO GET THESE? 4. Projection and symbolic explanations: . 2. Basic operations: 3. Solving and model representation: 1. Representation of polynomials. ◮ variables, variable ordering; ◮ arithmetic (addition, multiplication, …); ◮ GCD computation; ◮ some factorization. ◮ Use an existing library � ◮ Sturm sequences; ◮ Use a computer algebra system ◮ interval reasoning; ◮ root isolation (multivariate); ◮ resultants; ◮ computation with algebraic numbers. ◮ principal subresultant coefgicients.

NON-LINEAR REASONING CAD-BASED REASONING Use LIBPOLY Implement yourself Borrow and adapt code HOW TO GET THESE? 4. Projection and symbolic explanations: . 2. Basic operations: 3. Solving and model representation: 1. Representation of polynomials. ◮ variables, variable ordering; ◮ arithmetic (addition, multiplication, …); ◮ GCD computation; ◮ some factorization. ◮ Use an existing library � ◮ Use a computer algebra system � ◮ Sturm sequences; ◮ interval reasoning; ◮ root isolation (multivariate); ◮ resultants; ◮ computation with algebraic numbers. ◮ principal subresultant coefgicients.

NON-LINEAR REASONING 3. Solving and model representation: Use LIBPOLY Implement yourself HOW TO GET THESE? 4. Projection and symbolic explanations: CAD-BASED REASONING . 2. Basic operations: 1. Representation of polynomials. ◮ variables, variable ordering; ◮ arithmetic (addition, multiplication, …); ◮ GCD computation; ◮ some factorization. ◮ Use an existing library � ◮ Use a computer algebra system � ◮ Sturm sequences; ◮ interval reasoning; ◮ Borrow and adapt code ◮ root isolation (multivariate); ◮ resultants; ◮ computation with algebraic numbers. ◮ principal subresultant coefgicients.

NON-LINEAR REASONING 3. Solving and model representation: Use LIBPOLY Implement yourself HOW TO GET THESE? 4. Projection and symbolic explanations: CAD-BASED REASONING . 2. Basic operations: 1. Representation of polynomials. ◮ variables, variable ordering; ◮ arithmetic (addition, multiplication, …); ◮ GCD computation; ◮ some factorization. ◮ Use an existing library � ◮ Use a computer algebra system � ◮ Sturm sequences; ◮ interval reasoning; ◮ Borrow and adapt code � ◮ root isolation (multivariate); ◮ resultants; ◮ computation with algebraic numbers. ◮ principal subresultant coefgicients.

NON-LINEAR REASONING 3. Solving and model representation: Use LIBPOLY HOW TO GET THESE? 4. Projection and symbolic explanations: CAD-BASED REASONING . 2. Basic operations: 1. Representation of polynomials. ◮ variables, variable ordering; ◮ arithmetic (addition, multiplication, …); ◮ GCD computation; ◮ some factorization. ◮ Use an existing library � ◮ Use a computer algebra system � ◮ Sturm sequences; ◮ interval reasoning; ◮ Borrow and adapt code � ◮ root isolation (multivariate); ◮ Implement yourself � ◮ resultants; ◮ computation with algebraic numbers. ◮ principal subresultant coefgicients.

NON-LINEAR REASONING HOW TO GET THESE? 2. Basic operations: 3. Solving and model representation: CAD-BASED REASONING 4. Projection and symbolic explanations: 1. Representation of polynomials. ◮ variables, variable ordering; ◮ arithmetic (addition, multiplication, …); ◮ GCD computation; ◮ some factorization. ◮ Use an existing library � ◮ Use a computer algebra system � ◮ Sturm sequences; ◮ interval reasoning; ◮ Borrow and adapt code � ◮ root isolation (multivariate); ◮ Implement yourself � ◮ resultants; ◮ computation with algebraic numbers. ◮ Use LIBPOLY � . ◮ principal subresultant coefgicients.

OUTLINE INTRODUCTION LIBPOLY Working with Polynomials Constructing a Sign Table Cylindrical Algebraic Decomposition CONCLUSION

LIBPOLY ◮ Open source: https://github.com/SRI-CSL/libpoly . ◮ Permissive License: LGLP ◮ Lightweight: Implemented in C, 15KLOC. ◮ Only depends on GMP. ◮ Basis for non-linear reasoning in YICES2.