Interpolant Synthesis for Quadratic Polynomial Inequalities and - PowerPoint PPT Presentation

Key ideas Generating interpolants for CQI Combination with EUF Evaluation results Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . Interpolant Synthesis for Quadratic Polynomial Inequalities and Combination with EUF Deepak Kapur Department of Computer Science, University of New Mexico Joint work with Ting Gan, Liyun Dai, Bican Xia, Naijun Zhan, and Mingshuai Chen Dagstuhl, September 2015 Deepak Kapur University of New Mexico Interpolant synthesis for CQI+EUF Dagstuhl, September 2015 1 / 29

NSOSC condition : generalized Motzkin's theorem applies SOS ( NSOSC not satisifed) : equalities from expressions in a sum of squares being equal to 0. similar to the linear case Generalization of Motzkin's transposition theorem Concave quadratic polynomials Positive constant replaced by sum of squares 2 Generating interpolants for Concave Quadratic Polynomial inequalities 3 Combination with uninterpreted function symbols ( EUF ) 4 Concluding remarks Key ideas Generating interpolants for CQI Combination with EUF Evaluation results Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . Outline 1 Key ideas Deepak Kapur University of New Mexico Interpolant synthesis for CQI+EUF Dagstuhl, September 2015 2 / 29

NSOSC condition : generalized Motzkin's theorem applies SOS ( NSOSC not satisifed) : equalities from expressions in a sum of squares being equal to 0. similar to the linear case Concave quadratic polynomials Positive constant replaced by sum of squares 2 Generating interpolants for Concave Quadratic Polynomial inequalities 3 Combination with uninterpreted function symbols ( EUF ) 4 Concluding remarks Key ideas Generating interpolants for CQI Combination with EUF Evaluation results Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . Outline 1 Key ideas Generalization of Motzkin's transposition theorem Deepak Kapur University of New Mexico Interpolant synthesis for CQI+EUF Dagstuhl, September 2015 2 / 29

NSOSC condition : generalized Motzkin's theorem applies SOS ( NSOSC not satisifed) : equalities from expressions in a sum of squares being equal to 0. similar to the linear case Positive constant replaced by sum of squares 2 Generating interpolants for Concave Quadratic Polynomial inequalities 3 Combination with uninterpreted function symbols ( EUF ) 4 Concluding remarks Key ideas Generating interpolants for CQI Combination with EUF Evaluation results Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . Outline 1 Key ideas Generalization of Motzkin's transposition theorem Concave quadratic polynomials Deepak Kapur University of New Mexico Interpolant synthesis for CQI+EUF Dagstuhl, September 2015 2 / 29

NSOSC condition : generalized Motzkin's theorem applies SOS ( NSOSC not satisifed) : equalities from expressions in a sum of squares being equal to 0. similar to the linear case 2 Generating interpolants for Concave Quadratic Polynomial inequalities 3 Combination with uninterpreted function symbols ( EUF ) 4 Concluding remarks Key ideas Generating interpolants for CQI Combination with EUF Evaluation results Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . Outline 1 Key ideas Generalization of Motzkin's transposition theorem Concave quadratic polynomials Positive constant replaced by sum of squares Deepak Kapur University of New Mexico Interpolant synthesis for CQI+EUF Dagstuhl, September 2015 2 / 29

similar to the linear case NSOSC condition : generalized Motzkin's theorem applies SOS ( NSOSC not satisifed) : equalities from expressions in a sum of squares being equal to 0. 3 Combination with uninterpreted function symbols ( EUF ) 4 Concluding remarks Key ideas Generating interpolants for CQI Combination with EUF Evaluation results Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . Outline 1 Key ideas Generalization of Motzkin's transposition theorem Concave quadratic polynomials Positive constant replaced by sum of squares 2 Generating interpolants for Concave Quadratic Polynomial inequalities Deepak Kapur University of New Mexico Interpolant synthesis for CQI+EUF Dagstuhl, September 2015 2 / 29

similar to the linear case SOS ( NSOSC not satisifed) : equalities from expressions in a sum of squares being equal to 0. 3 Combination with uninterpreted function symbols ( EUF ) 4 Concluding remarks Key ideas Generating interpolants for CQI Combination with EUF Evaluation results Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . Outline 1 Key ideas Generalization of Motzkin's transposition theorem Concave quadratic polynomials Positive constant replaced by sum of squares 2 Generating interpolants for Concave Quadratic Polynomial inequalities NSOSC condition : generalized Motzkin's theorem applies Deepak Kapur University of New Mexico Interpolant synthesis for CQI+EUF Dagstuhl, September 2015 2 / 29

similar to the linear case 3 Combination with uninterpreted function symbols ( EUF ) 4 Concluding remarks Key ideas Generating interpolants for CQI Combination with EUF Evaluation results Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . Outline 1 Key ideas Generalization of Motzkin's transposition theorem Concave quadratic polynomials Positive constant replaced by sum of squares 2 Generating interpolants for Concave Quadratic Polynomial inequalities NSOSC condition : generalized Motzkin's theorem applies SOS ( NSOSC not satisifed) : equalities from expressions in a sum of squares being equal to 0. Deepak Kapur University of New Mexico Interpolant synthesis for CQI+EUF Dagstuhl, September 2015 2 / 29

similar to the linear case 4 Concluding remarks Key ideas Generating interpolants for CQI Combination with EUF Evaluation results Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . Outline 1 Key ideas Generalization of Motzkin's transposition theorem Concave quadratic polynomials Positive constant replaced by sum of squares 2 Generating interpolants for Concave Quadratic Polynomial inequalities NSOSC condition : generalized Motzkin's theorem applies SOS ( NSOSC not satisifed) : equalities from expressions in a sum of squares being equal to 0. 3 Combination with uninterpreted function symbols ( EUF ) Deepak Kapur University of New Mexico Interpolant synthesis for CQI+EUF Dagstuhl, September 2015 2 / 29

4 Concluding remarks Key ideas Generating interpolants for CQI Combination with EUF Evaluation results Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . Outline 1 Key ideas Generalization of Motzkin's transposition theorem Concave quadratic polynomials Positive constant replaced by sum of squares 2 Generating interpolants for Concave Quadratic Polynomial inequalities NSOSC condition : generalized Motzkin's theorem applies SOS ( NSOSC not satisifed) : equalities from expressions in a sum of squares being equal to 0. 3 Combination with uninterpreted function symbols ( EUF ) similar to the linear case Deepak Kapur University of New Mexico Interpolant synthesis for CQI+EUF Dagstuhl, September 2015 2 / 29

Key ideas Generating interpolants for CQI Combination with EUF Evaluation results Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . Outline 1 Key ideas Generalization of Motzkin's transposition theorem Concave quadratic polynomials Positive constant replaced by sum of squares 2 Generating interpolants for Concave Quadratic Polynomial inequalities NSOSC condition : generalized Motzkin's theorem applies SOS ( NSOSC not satisifed) : equalities from expressions in a sum of squares being equal to 0. 3 Combination with uninterpreted function symbols ( EUF ) similar to the linear case 4 Concluding remarks Deepak Kapur University of New Mexico Interpolant synthesis for CQI+EUF Dagstuhl, September 2015 2 / 29

Key ideas Generating interpolants for CQI Combination with EUF Evaluation results Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview Overview of the idea Example (running example) Consider two formulas A and B with A ∧ B | = ⊥ , where A := − x 12 + 4 x 1 + x 2 − 4 ≥ 0 ∧ − x 1 − x 2 + 3 − y 2 > 0 , B := − 3 x 12 − x 22 + 1 ≥ 0 ∧ x 2 − z 2 ≥ 0 We aim to generate an interpolant I for A and B , on the common variables ( x 1 and x 2 ), such that A | = I and I ∧ B | = ⊥ . An intuitive description of a candidate interpolant is as the purple curve in the above right figure, which separates A and B in the panel of x 1 and x 2 . Deepak Kapur University of New Mexico Interpolant synthesis for CQI+EUF Dagstuhl, September 2015 3 / 29