interpolant synthesis for quadratic polynomial
play

Interpolant Synthesis for Quadratic Polynomial Inequalities and - PowerPoint PPT Presentation

Feb 20, 2023 •315 likes •990 views

Key ideas Generating interpolants for CQI Combination with EUF Evaluation results Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . Interpolant Synthesis for Quadratic Polynomial Inequalities and Combination with EUF

  1. 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

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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

Recommend

Reducing Interpolant Circuit Size by Ad-Hoc Logic Synthesis and SAT-Based Weakening Gianpiero

Reducing Interpolant Circuit Size by Ad-Hoc Logic Synthesis and SAT-Based Weakening Gianpiero

Reducing Interpolant Circuit Size by Ad-Hoc Logic Synthesis and SAT-Based Weakening Gianpiero Cabodi Paolo Camurati Paolo Pasini Marco Palena Danilo Vendraminetto Politecnico di Torino Torino, Italy 1 Outline l Motivations &

675 views • 51 slides
A Semidefinite Relaxation Scheme for Multivariate Quartic Polynomial Optimization With Quadratic

A Semidefinite Relaxation Scheme for Multivariate Quartic Polynomial Optimization With Quadratic

A Semidefinite Relaxation Scheme for Multivariate Quartic Polynomial Optimization With Quadratic Constraints Zhi-Quan Luo Department of Electrical and Computer Engineering University of Minnesota Shuzhong Zhang Department of Systems

629 views • 34 slides
Interpolation polynomial * Means finding an approximate value to a function as a polynomial of

Interpolation polynomial * Means finding an approximate value to a function as a polynomial of

Interpolation polynomial * Means finding an approximate value to a function as a polynomial of degree n using n+1 points. Methods of solution 1- Quadratic Lagrange x x x x 1 2 0 f x y y y 0 1 2 P

154 views • 13 slides
The quadratic formula You may recall the quadratic formula for roots of quadratic polynomials ax 2

The quadratic formula You may recall the quadratic formula for roots of quadratic polynomials ax 2

The quadratic formula You may recall the quadratic formula for roots of quadratic polynomials ax 2 + bx + c . It says that the solutions to this polynomial are b 2 4 ac b . 2 a The quadratic formula You may recall the quadratic

1.63k views • 104 slides
Automatic Synthesis of Fast and Certified Code for Polynomial Evaluation through the example of

Automatic Synthesis of Fast and Certified Code for Polynomial Evaluation through the example of

ANR MetaLibm kick-off meeting Lyon, 22 January, 2014 Automatic Synthesis of Fast and Certified Code for Polynomial Evaluation through the example of the CGPE tool Guillaume Revy quipe-projet DALI, Univ. Perpignan Via Domitia LIRMM, CNRS:

570 views • 30 slides
Learning convex bounds for linear quadratic control policy synthesis Jack Umenberger Thomas

Learning convex bounds for linear quadratic control policy synthesis Jack Umenberger Thomas

Learning convex bounds for linear quadratic control policy synthesis Jack Umenberger Thomas B. Schn Learning to control data control (observations of the (stabilize the upright dynamical system) equilibrium position) learning

743 views • 34 slides
Introduction The MP Problem Solving a system of m multivariate polynomial equations in n variables

Introduction The MP Problem Solving a system of m multivariate polynomial equations in n variables

Fast Exhaustive Search for Quadratic Systems in F 2 on FPGAs Charles Bouillaguet, Chen-Mou Cheng, Tung Chou, Ruben Niederhagen, Bo- Yin Yang August 15, 2013 Introduction The MP Problem Solving a system of m multivariate polynomial equations in

790 views • 32 slides
On A Quadratic Eigenproblem Arising In The Analysis of Delay Equations Heike Fabender AG

On A Quadratic Eigenproblem Arising In The Analysis of Delay Equations Heike Fabender AG

Introduction Polynomial Matrix Eigenproblem Spectral Symmetry Cayley Transformations Structured Linearization Conclusions On A Quadratic Eigenproblem Arising In The Analysis of Delay Equations Heike Fabender AG Numerik Institut

920 views • 70 slides
11. Quadratic forms and ellipsoids Quadratic forms Orthogonal decomposition Positive

11. Quadratic forms and ellipsoids Quadratic forms Orthogonal decomposition Positive

CS/ECE/ISyE 524 Introduction to Optimization Spring 201718 11. Quadratic forms and ellipsoids Quadratic forms Orthogonal decomposition Positive definite matrices Ellipsoids Laurent Lessard (www.laurentlessard.com) Quadratic

594 views • 21 slides
Polynomial Solutions of Recurrence Relations O. Shkaravska M. van Eekelen A. Tamalet Digital

Polynomial Solutions of Recurrence Relations O. Shkaravska M. van Eekelen A. Tamalet Digital

Motivation: recurrences in program analysis and math. Our Contribution: multi-step quadratic recurrences for 1-variable polynomials Polynomial Solutions of Recurrence Relations O. Shkaravska M. van Eekelen A. Tamalet Digital Security, ICIS

537 views • 18 slides
Key Terms Solve Quadratic Equations by Factoring Solve Quadratic Equations Using Square Roots

Key Terms Solve Quadratic Equations by Factoring Solve Quadratic Equations Using Square Roots

Slide 1 / 175 Slide 2 / 175 Algebra I Quadratics 2015-11-04 www.njctl.org Slide 3 / 175 Slide 4 / 175 Table of Contents Key Terms Click on the topic to go to that section Characteristics of Quadratic Equations Transforming Quadratic

567 views • 30 slides
7.3 Real Life Examples Example 3. The length of a

7.3 Real Life Examples Example 3. The length of a

SET 1 Chapter 7 Quadratic Equations Chapter 7: Quadratic Equations 1 7.1 Introduction A quadratic equation is one in which the highest power of the unknown quantity is 2. For

222 views • 4 slides
3.2 Graphing Quadratic Functions The equation of a quadratic relation may be written in several

3.2 Graphing Quadratic Functions The equation of a quadratic relation may be written in several

MPM2D1 U3L2 Graphing Quadratic Functions 3.2 Graphing Quadratic Functions The equation of a quadratic relation may be written in several forms: Factored Form: Standard Form: Vertex Form: Last class, we looked at graphing equations in

195 views • 7 slides
I : Theorem Great Galois ' when polynomial a theorem that down write today a expresses

I : Theorem Great Galois ' when polynomial a theorem that down write today a expresses

Applications of statement I : Theorem Great Galois ' when polynomial a theorem that down write today a expresses in something analogous to expressible roots that are has the gvntmhi formula : quadratic formula , cubic formula ,

471 views • 23 slides
Efficient Interpolant Generation in Satisfiability Modulo Linear Integer Arithmetic Alberto

Efficient Interpolant Generation in Satisfiability Modulo Linear Integer Arithmetic Alberto

Deduction at Scale Seminar 2011 Efficient Interpolant Generation in Satisfiability Modulo Linear Integer Arithmetic Alberto Griggio FBK-IRST, Trento joint work with Thi Thieu Hoa Le and Roberto Sebastiani, DISI - Univ. Trento Introduction

767 views • 39 slides
Polynomial Time corresponds to Solutions of Polynomial Ordinary Differential Equations of

Polynomial Time corresponds to Solutions of Polynomial Ordinary Differential Equations of

Polynomial Time corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length Olivier Bournez, Daniel Graa and Amaury Pouly July 13, 2015 Main result and consequences Theorem (Informal) PTIME = PIVP of

633 views • 47 slides
Non-linear Interpolant Generation and Its Application to Program Verification Naijun Zhan State

Non-linear Interpolant Generation and Its Application to Program Verification Naijun Zhan State

Problem Description Synthesizing Non-linear Interpolants Archimedean Condition To invariant generation To machine-lea Non-linear Interpolant Generation and Its Application to Program Verification Naijun Zhan State Key Laboratory of Computer

426 views • 40 slides
Spinor regular ternary quadratic lattices Anna Haensch Duquesne 1 University Joint work with Andy

Spinor regular ternary quadratic lattices Anna Haensch Duquesne 1 University Joint work with Andy

Spinor regular ternary quadratic lattices Anna Haensch Duquesne 1 University Joint work with Andy Earnest Computational Challenges in the Theory of Lattices ICERM 27 April 2018 1 doo-KANE 1 / 39 A rational polynomial f ( x 1 , ..., x n )

954 views • 94 slides
PARABOLA 1 I NTRODUCTION All along, we have been talking about quadratic equations, graphs of

PARABOLA 1 I NTRODUCTION All along, we have been talking about quadratic equations, graphs of

D AY 136 E QUATION OF A PARABOLA 1 I NTRODUCTION All along, we have been talking about quadratic equations, graphs of quadratic equations and so on. We would like now to move a notch higher and approach this quadratic equation as a form of

154 views • 13 slides
Interpolant Compaction In Unbounded Model Checking Danilo Vendraminetto PhD student at Formal

Interpolant Compaction In Unbounded Model Checking Danilo Vendraminetto PhD student at Formal

Alpine Verification Meeting 2013, FBK, Trento, Italy. Optimization Techniques For Craig Interpolant Compaction In Unbounded Model Checking Danilo Vendraminetto PhD student at Formal Methods Group, Politecnico di Torino, Torino, Italy. Before

880 views • 40 slides
Polynomial Hierarchy A polynomial-bounded version of Kleenes Arithmetic Hierarchy becomes

Polynomial Hierarchy A polynomial-bounded version of Kleenes Arithmetic Hierarchy becomes

Polynomial Hierarchy A polynomial-bounded version of Kleenes Arithmetic Hierarchy becomes trivial if P = NP . Karp, 1972 Computational Complexity, by Fu Yuxi Polynomial Hierarchy 1 / 44 Larry Stockmeyer and Albert Meyer introduced

584 views • 45 slides
Property of the interior polynomial from the HOMFLY polynomial

Property of the interior polynomial from the HOMFLY polynomial

Computing the HOMFLY polynomial using the combinatorics Properties of the interior polynomial Property of the interior polynomial from the HOMFLY polynomial 2017 12

554 views • 24 slides
Synthesis of Ranking Functions and Synthesis of Inductive Invariants and Synthesis of

Synthesis of Ranking Functions and Synthesis of Inductive Invariants and Synthesis of

Synthesis of Ranking Functions and Synthesis of Inductive Invariants and Synthesis of Recurrence Sets via Constraint Solving Andreas Podelski January 17, 2012 1 Program Verification and Constraints Reasoning about program

422 views • 28 slides
Bzier Curves CPSC 453 Fall 2018 Sonny Chan Todays Outline Quadratic Bzier curves

Bzier Curves CPSC 453 Fall 2018 Sonny Chan Todays Outline Quadratic Bzier curves

Bzier Curves CPSC 453 Fall 2018 Sonny Chan Todays Outline Quadratic Bzier curves de Casteljau formulation Bernstein polynomial form Bzier splines Cubic Bzier curves Drawing Bzier curves freeform shape?

687 views • 37 slides

More recommend