Proving UNSAT in SMT: The Case of Quantifier Free Non-Linear Real Arithmetic

We discuss the topic of unsatisfiability proofs in SMT, particularly with reference to quantifier free non-linear real arithmetic. We outline how the methods here do not admit trivial proofs and how past formalisation attempts are not sufficient. We note that the new breed of local search based algorithms for this domain may offer an easier path forward.

[1]  Christopher W. Brown.  Open Non-uniform Cylindrical Algebraic Decompositions , 2015, ISSAC.

[2]  Christopher W. Brown,et al.  Constructing a single cell in cylindrical algebraic decomposition , 2015, J. Symb. Comput..

[3]  Matthew England,et al.  New Opportunities for the Formal Proof of Computational Real Geometry? (Extended Abstract) , 2020, PAAR+SC²@IJCAI.

[4]  Leonardo Mendonça de Moura,et al.  A Model-Constructing Satisfiability Calculus , 2013, VMCAI.

[5]  Leonardo Mendonça de Moura,et al.  Solving non-linear arithmetic , 2012, ACCA.

[6]  Mizuhito Ogawa,et al.  Subtropical Satisfiability , 2017, FroCoS.

[7]  Assia Mahboubi,et al.  Formal proofs in real algebraic geometry: from ordered fields to quantifier elimination , 2012, Log. Methods Comput. Sci..

[8]  Matthew England,et al.  Using Machine Learning to Decide When to Precondition Cylindrical Algebraic Decomposition with Groebner Bases , 2016, 2016 18th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC).

[9]  Mizuhito Ogawa,et al.  raSAT: An SMT Solver for Polynomial Constraints , 2016, IJCAR.

[10]  Mizuhito Ogawa,et al.  Wrapping Computer Algebra is Surprisingly Successful for Non-Linear SMT , 2018, SC-Square@FLOC.

[11]  Alberto Griggio,et al.  Incremental Linearization for Satisfiability and Verification Modulo Nonlinear Arithmetic and Transcendental Functions , 2018, ACM Trans. Comput. Log..

[12]  Assia Mahboubi,et al.  Implementing the cylindrical algebraic decomposition within the Coq system , 2007, Mathematical Structures in Computer Science.

[13]  Assia Mahboubi,et al.  A formal quantifier elimination for algebraically closed fields , 2010, AISC'10/MKM'10/Calculemus'10.

[14]  Cesare Tinelli,et al.  LFSC for SMT proofs: Work in progress , 2012 .

[15]  B. F. Caviness,et al.  Quantifier Elimination and Cylindrical Algebraic Decomposition , 2004, Texts and Monographs in Symbolic Computation.

[16]  Daniel Kroening,et al.  SC2: Satisfiability Checking Meets Symbolic Computation - (Project Paper) , 2016, CICM.

[17]  George E. Collins,et al.  Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975 .

[18]  C. Barrett,et al.  Proofs in satisfiability modulo theories , 2014 .

[19]  Marijn J. H. Heule,et al.  Proceedings of SAT Competition 2017: Solver and Benchmark Descriptions , 2017 .

[20]  Matthew England,et al.  Cylindrical Algebraic Decomposition with Equational Constraints , 2019, J. Symb. Comput..

[21]  Volker Weispfenning,et al.  Quantifier Elimination for Real Algebra — the Quadratic Case and Beyond , 1997, Applicable Algebra in Engineering, Communication and Computing.

[22]  Pascal Fontaine,et al.  Scalable Fine-Grained Proofs for Formula Processing , 2017, CADE.

[23]  James H. Davenport,et al.  Deciding the Consistency of Non-Linear Real Arithmetic Constraints with a Conflict Driven Search Using Cylindrical Algebraic Coverings , 2020, ArXiv.

[24]  César A. Muñoz,et al.  Formally-Verified Decision Procedures for Univariate Polynomial Computation Based on Sturm’s and Tarski’s Theorems , 2015, Journal of Automated Reasoning.

[25]  Peter Lammich.  Efficient Verified (UN)SAT Certificate Checking , 2017, CADE.