Subtropical Satisfiability and off-the-shelf computer algebra as - PowerPoint PPT Presentation

Subtropical Satisfiability and off-the-shelf computer algebra as procedures for SMT Pascal Fontaine , Mizuhito Ogawa, Thomas Sturm, Van Khanh To, Xuan Tung Vu Univ. of Lorraine, CNRS, Inria, LORIA Japan Advanced Institute of Science and Technology (JAIST) University of Engineering and Technology, VNU, Hanoi, Vietnam MPI Informatics and Saarland University 17 May, 2018 SC 2 workshop, Trento 1/35

Outline Introduction Understanding the subtropical method Interlude: from SAT to SMT Combining with other techniques Conclusion 2/35

Outline Introduction Understanding the subtropical method Interlude: from SAT to SMT Combining with other techniques Conclusion 3/35

SMT + non linear arithmetics ◮ High demand for non linear arithmetic reasoning capability ◮ Theory of real closed fields: decidable (QE: CAD, virtual substitution,. . . ) ◮ Doubly exponential (existential fragment also high complexity) ◮ Complete decision procedure not always efficient enough ◮ Need for good heuristics Contribution: subtropical satisfiability Simple heuristic to quickly discharge many proof obligations (or failing quickly) ◮ Based on subtropical method: quickly find positive solution for f = 0 where f has hundreds of thousand of monomials, with dozen variables, degrees around 10 in each variable ◮ Here: find real solution for f 1 > 0 ∧ · · · ∧ f n > 0 4/35

Subtropical method: univariate case Consider x ≥ 0 , ◮ 1 − 2 x + x 3 > 0 5/35

Subtropical method: univariate case Consider x ≥ 0 , ◮ 1 − 2 x + x 3 > 0 , satisfiable: x = 0 1 0 1 2 − 1 − 2 y = 1 − 2 x + x 3 5/35

Subtropical method: univariate case Consider x ≥ 0 , ◮ 1 − 2 x + x 3 > 0 , satisfiable: x = 0 ◮ − 1 − 2 x + x 3 > 0 1 0 1 2 − 1 − 2 y = 1 − 2 x + x 3 5/35

Subtropical method: univariate case Consider x ≥ 0 , ◮ 1 − 2 x + x 3 > 0 , satisfiable: x = 0 ◮ − 1 − 2 x + x 3 > 0 , satisfiable: with sufficiently large x 1 1 0 0 1 2 1 2 − 1 − 1 − 2 − 2 y = 1 − 2 x + x 3 y = − 1 − 2 x + x 3 5/35

Subtropical method: univariate case Consider x ≥ 0 , ◮ 1 − 2 x + x 3 > 0 , satisfiable: x = 0 ◮ − 1 − 2 x + x 3 > 0 , satisfiable: with sufficiently large x ◮ 2 x − x 3 > 0 1 1 0 0 1 2 1 2 − 1 − 1 − 2 − 2 y = 1 − 2 x + x 3 y = − 1 − 2 x + x 3 5/35

Subtropical method: univariate case Consider x ≥ 0 , ◮ 1 − 2 x + x 3 > 0 , satisfiable: x = 0 ◮ − 1 − 2 x + x 3 > 0 , satisfiable: with sufficiently large x ◮ 2 x − x 3 > 0 , satisfiable: with sufficiently small x 1 1 1 0 0 0 1 2 1 2 1 2 − 1 − 1 − 1 − 2 − 2 − 2 y = 1 − 2 x + x 3 y = − 1 − 2 x + x 3 y = 2 x − x 3 5/35

Subtropical method: univariate case Consider x ≥ 0 , ◮ 1 − 2 x + x 3 > 0 , satisfiable: x = 0 ◮ − 1 − 2 x + x 3 > 0 , satisfiable: with sufficiently large x ◮ 2 x − x 3 > 0 , satisfiable: with sufficiently small x Find a model for f > 0 ? 1 1 1 0 0 0 1 2 1 2 1 2 − 1 − 1 − 1 − 2 − 2 − 2 y = 1 − 2 x + x 3 y = − 1 − 2 x + x 3 y = 2 x − x 3 5/35

Subtropical method: univariate case Consider x ≥ 0 , ◮ 1 − 2 x + x 3 > 0 , satisfiable: x = 0 ◮ − 1 − 2 x + x 3 > 0 , satisfiable: with sufficiently large x ◮ 2 x − x 3 > 0 , satisfiable: with sufficiently small x Find a model for f > 0 ? Check coefficient sign for lowest or highest degree monomial 1 1 1 0 0 0 1 2 1 2 1 2 − 1 − 1 − 1 − 2 − 2 − 2 y = 1 − 2 x + x 3 y = − 1 − 2 x + x 3 y = 2 x − x 3 5/35

Subtropical method: univariate case Consider x ≥ 0 , ◮ 1 − 2 x + x 3 > 0 , satisfiable: x = 0 ◮ − 1 − 2 x + x 3 > 0 , satisfiable: with sufficiently large x ◮ 2 x − x 3 > 0 , satisfiable: with sufficiently small x Find a model for f > 0 ? Check coefficient sign for lowest or highest degree monomial Incomplete: − 1 + 2 x − x 3 > 0 ? 1 1 1 0 0 0 1 2 1 2 1 2 − 1 − 1 − 1 − 2 − 2 − 2 y = 1 − 2 x + x 3 y = − 1 − 2 x + x 3 y = 2 x − x 3 5/35

Subtropical method: univariate case Consider x ≥ 0 , ◮ 1 − 2 x + x 3 > 0 , satisfiable: x = 0 ◮ − 1 − 2 x + x 3 > 0 , satisfiable: with sufficiently large x ◮ 2 x − x 3 > 0 , satisfiable: with sufficiently small x Find a model for f > 0 ? Check coefficient sign for lowest or highest degree monomial Incomplete: − 1 + 2 x − x 3 > 0 ? ( x = 0 . 8 ) 1 1 1 1 0 0 0 0 1 2 1 2 1 2 1 2 − 1 − 1 − 1 − 1 − 2 − 2 − 2 − 2 y = 1 − 2 x + x 3 y = − 1 − 2 x + x 3 y = 2 x − x 3 y = − 1 + 2 x − x 3 5/35

Subtropical method: univariate case Consider x ≥ 0 , ◮ 1 − 2 x + x 3 > 0 , satisfiable: x = 0 ◮ − 1 − 2 x + x 3 > 0 , satisfiable: with sufficiently large x ◮ 2 x − x 3 > 0 , satisfiable: with sufficiently small x Find a model for f > 0 ? Check coefficient sign for lowest or highest degree monomial Incomplete: − 1 + 2 x − x 3 > 0 ? ( x = 0 . 8 ) But certainly fast 1 1 1 1 0 0 0 0 1 2 1 2 1 2 1 2 − 1 − 1 − 1 − 1 − 2 − 2 − 2 − 2 y = 1 − 2 x + x 3 y = − 1 − 2 x + x 3 y = 2 x − x 3 y = − 1 + 2 x − x 3 5/35

Outline Introduction Understanding the subtropical method Interlude: from SAT to SMT Combining with other techniques Conclusion 6/35

Outline Introduction Understanding the subtropical method Interlude: from SAT to SMT Combining with other techniques Conclusion 7/35

Subtropical method: towards the multivariate case Polynomial − 2 x 5 1 + x 2 1 x 2 − 3 x 2 1 − x 3 2 + 2 x 2 2 can be ◮ negative, e.g. − 2 x 5 1 dominates if x 1 large enough w.r.t. x 2 ◮ positive, e.g. 2 x 2 2 dominates if x 2 small enough (not zero) and an even smaller x 1 . 8/35

Subtropical method: towards the multivariate case Polynomial − 2 x 5 1 + x 2 1 x 2 − 3 x 2 1 − x 3 2 + 2 x 2 2 can be ◮ negative, e.g. − 2 x 5 1 dominates if x 1 large enough w.r.t. x 2 ◮ positive, e.g. 2 x 2 2 dominates if x 2 small enough (not zero) and an even smaller x 1 . Handling the multivariate case: ◮ reduce to univariate, setting all variables but one to 0 ◮ consider monomial of highest/lowest total degree (if unique) ◮ ordering? lexicographic? 8/35

Subtropical method: towards the multivariate case Polynomial − 2 x 5 1 + x 2 1 x 2 − 3 x 2 1 − x 3 2 + 2 x 2 2 can be ◮ negative, e.g. − 2 x 5 1 dominates if x 1 large enough w.r.t. x 2 ◮ positive, e.g. 2 x 2 2 dominates if x 2 small enough (not zero) and an even smaller x 1 . Handling the multivariate case: ◮ reduce to univariate, setting all variables but one to 0 ◮ consider monomial of highest/lowest total degree (if unique) ◮ ordering? lexicographic? Contribution monotonic total preorders on the exponent vectors 8/35

Subtropical method: towards the multivariate case (2) Theorem Strictly max. monomials (w.r.t. monotonic total preorders on the exponent vectors) can dominate, for suitable (positive) values of variables. 9/35

Subtropical method: towards the multivariate case (2) Theorem Strictly max. monomials (w.r.t. monotonic total preorders on the exponent vectors) can dominate, for suitable (positive) values of variables. Equivalent to consider the vertices of the Newton polytope, i.e. the set of exponent vectors. 9/35

Subtropical method: towards the multivariate case (2) Theorem Strictly max. monomials (w.r.t. monotonic total preorders on the exponent vectors) can dominate, for suitable (positive) values of variables. Equivalent to consider the vertices of the Newton polytope, i.e. the set of exponent vectors. f = − 2 x 5 1 + x 2 1 x 2 − 3 x 2 1 − x 3 2 +2 x 2 2 9/35

Subtropical method: towards the multivariate case (2) Theorem Strictly max. monomials (w.r.t. monotonic total preorders on the exponent vectors) can dominate, for suitable (positive) values of variables. Equivalent to consider the vertices of the Newton polytope, i.e. the set of exponent vectors. f = − 2 x 5 1 + x 2 1 x 2 − 3 x 2 1 − x 3 2 +2 x 2 2 (0 , 3) (0 , 2) (2 , 1) (2 , 0) (5 , 0) 9/35

Subtropical method: towards the multivariate case (2) Theorem Strictly max. monomials (w.r.t. monotonic total preorders on the exponent vectors) can dominate, for suitable (positive) values of variables. Equivalent to consider the vertices of the Newton polytope, i.e. the set of exponent vectors. ◮ − 2 x 5 1 , − 3 x 2 1 , − x 3 2 and 2 x 2 2 f = − 2 x 5 1 + x 2 1 x 2 − 3 x 2 1 − x 3 2 +2 x 2 correspond to vertices 2 (0 , 3) (0 , 2) (2 , 1) (2 , 0) (5 , 0) 9/35

Subtropical method: towards the multivariate case (2) Theorem Strictly max. monomials (w.r.t. monotonic total preorders on the exponent vectors) can dominate, for suitable (positive) values of variables. Equivalent to consider the vertices of the Newton polytope, i.e. the set of exponent vectors. ◮ − 2 x 5 1 , − 3 x 2 1 , − x 3 2 and 2 x 2 2 f = − 2 x 5 1 + x 2 1 x 2 − 3 x 2 1 − x 3 2 +2 x 2 correspond to vertices 2 ◮ These monomials can dominate for (0 , 3) suitable values of variables (0 , 2) (2 , 1) (2 , 0) (5 , 0) 9/35