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Interpolant Synthesis for Quadratic Polynomial Inequalities and Combination with EUF

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9706)

Abstract

An algorithm for generating interpolants for formulas which are conjunctions of quadratic polynomial inequalities (both strict and nonstrict) is proposed. The algorithm is based on a key observation that quadratic polynomial inequalities can be linearized if they are concave. A generalization of Motzkin’s transposition theorem is proved, which is used to generate an interpolant between two mutually contradictory conjunctions of polynomial inequalities, using semi-definite programming in time complexity \(\mathcal {O}(n^3+nm)\), where n is the number of variables and m is the number of inequalities (This complexity analysis assumes that despite the numerical nature of approximate SDP algorithms, they are able to generate correct answers in a fixed number of calls.). Using the framework proposed in [22] for combining interpolants for a combination of quantifier-free theories which have their own interpolation algorithms, a combination algorithm is given for the combined theory of concave quadratic polynomial inequalities and the equality theory over uninterpreted functions (EUF).

Keywords

Program verification Interpolant Concave quadratic polynomial Motzkin’s theorem SOS Semi-definite programming 

Notes

Acknowledgement

The first three authors are supported partly by NSFC under grants 11290141, 11271034 and 61532019; the fourth and sixth authors are supported partly by “973 Program” under grant No. 2014CB340701, by NSFC under grant 91418204, by CDZ project CAP (GZ 1023), and by the CAS/SAFEA International Partnership Program for Creative Research Teams; the fifth author is supported partly by NSF under grant DMS-1217054 and by the CAS/SAFEA International Partnership Program for Creative Research Teams.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.LMAM & School of Mathematical SciencesPeking UniversityBeijingChina
  2. 2.State Key Laboratory of Computer ScienceInstitute of Software, Chinese Academy of SciencesBeijingChina
  3. 3.Department of Computer ScienceUniversity of New MexicoAlbuquerqueUSA
  4. 4.State Key Laboratory of Software EngineeringWuhan UniversityWuhanChina

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