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).
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Notes
- 1.
Under the assumption that SDP tool returns an approximate but correct answer in a fixed number of calls.
- 2.
The proposed algorithm and its way of handling of combined theories do not crucially depend upon using algorithms in [20]; however, adopting their approach makes proofs and presentation totally on CQI.
- 3.
After properly handling \(N_{\text {mix}}\) since Horn clauses have symbols both from \(\overline{\phi }\) and \(\overline{\psi }\).
- 4.
The tool and all case studies can be found at http://lcs.ios.ac.cn/~chenms/tools/InterCQI_v1.1.tar.bz2.
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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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Gan, T., Dai, L., Xia, B., Zhan, N., Kapur, D., Chen, M. (2016). Interpolant Synthesis for Quadratic Polynomial Inequalities and Combination with EUF . In: Olivetti, N., Tiwari, A. (eds) Automated Reasoning. IJCAR 2016. Lecture Notes in Computer Science(), vol 9706. Springer, Cham. https://doi.org/10.1007/978-3-319-40229-1_14
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