A Symbiosis of Interval Constraint Propagation and Cylindrical Algebraic Decomposition

Loup, Ulrich; Scheibler, Karsten; Corzilius, Florian; Ábrahám, Erika; Becker, Bernd

doi:10.1007/978-3-642-38574-2_13

Cited by 11 publications

(6 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Secondly, our own SMT-solver based on our SMT-RAT (SMT Real-Arithmetic Toolbox) library [19], which we continuously improve and extend. The current release offers, besides some non-arithmetic components, the CArL [13] library for arithmetic datatypes and basic computations with them, and solver modules for simplex, the cylindrical algebraic decomposition [36], the virtual substitution method [18], Gröbner bases [33], interval constraint propagation, branch and bound, and their strategic combination [19].…”

Section: Arithmetic Theories In Smt Solversmentioning

confidence: 99%

Building Bridges between Symbolic Computation and Satisfiability Checking

Ábrahám

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

View full text Add to dashboard Cite

The satisfiability problem is the problem of deciding whether a logical formula is satisfiable. For first-order arithmetic theories, in the early 20th century some novel solutions in form of decision procedures were developed in the area of mathematical logic. With the advent of powerful computer architectures, a new research line started to develop practically feasible implementations of such decision procedures. Since then, symbolic computation has grown to an extremely successful scientific area, supporting all kinds of scientific computing by efficient computer algebra systems.Independently, around 1960 a new technology called SAT solving started its career. Restricted to propositional logic, SAT solvers showed to be very efficient when employed by formal methods for verification. It did not take long till the power of SAT solving for Boolean problems had been extended to cover also different theories. Nowadays, fast SAT-modulo-theories (SMT) solvers are available also for arithmetic problems.Due to their different roots, symbolic computation and SMT solving tackle the satisfiability problem differently. We discuss differences and similarities in their approaches, highlight potentials of combining their strengths, and discuss the challenges that come with this task.

show abstract

Section: Arithmetic Theories In Smt Solversmentioning

confidence: 99%

Building Bridges between Symbolic Computation and Satisfiability Checking

Ábrahám

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

View full text Add to dashboard Cite

show abstract

“…For the SMT domain of non-linear real arithmetic (NRA), CAD and more generally computer algebra systems can play the role of such theory solvers 1 . SMT-RAT contains a tailored CAD implementation for use in SMT (Loup et al, 2013;Kremer and Ábrahám, 2019), while Z3 contains an algorithm by Jovanovic and de Moura (2012) which uses CAD theory without producing actual CADs. The latter inspired new developments in symbolic computation such as non-uniform CAD by Brown (2015).…”

Section: Cad and Scmentioning

confidence: 99%

“…There are high barriers to implementing CAD without the support of a computer algebra system, however, SMT solvers such as SMT-RAT by Loup et al (2013); Kremer and Ábrahám (2019) and Z3 by Jovanovic and de Moura (2012) show it is possible. Indeed, the developers of SMT-RAT are now beginning to expand their CAD module to include a variety of projection operators (Viehmann et al, 2017) and even EC functionality as described by Haehn et al (2018).…”

Section: Reasons For Optimismmentioning

confidence: 99%

Cylindrical algebraic decomposition with equational constraints

England

Bradford

Davenport

2020

Journal of Symbolic Computation

View full text Add to dashboard Cite

Cylindrical Algebraic Decomposition (CAD) has long been one of the most important algorithms within Symbolic Computation, as a tool to perform quantifier elimination in first order logic over the reals. More recently it is finding prominence in the Satisfiability Checking community as a tool to identify satisfying solutions of problems in nonlinear real arithmetic.The original algorithm produces decompositions according to the signs of polynomials, when what is usually required is a decomposition according to the truth of a formula containing those polynomials. One approach to achieve that coarser (but hopefully cheaper) decomposition is to reduce the polynomials identified in the CAD to reflect a logical structure which reduces the solution space dimension: the presence of Equational Constraints (ECs). This paper may act as a tutorial for the use of CAD with ECs: we describe all necessary background and the current state of the art. In particular, we present recent work on how McCallum's theory of reduced projection may be leveraged to make further savings in the lifting phase: both to the polynomials we lift with and the cells lifted over. We give a new complexity analysis to demonstrate that the double exponent in the worst case complexity bound for CAD reduces in line with the number of ECs. We show that the reduction can apply to both the number of polynomials produced and their degree.

show abstract

“…While alternatives exist [6,25,41], they have not seen much use [27], and CAD-based methods are the only sound and complete methods in practical use today. CAD-based methods used in modern SMT solvers include incremental CAD implementations [34,36] and cylindrical algebraic coverings [3], both of which are integrated in the traditional CDCL(T) framework for SMT [40].…”

Section: Introductionmentioning

confidence: 99%

Cooperating Techniques for Solving Nonlinear Real Arithmetic in the cvc5 SMT Solver (System Description)

Kremer

Reynolds

Barrett

et al. 2022

Lecture Notes in Computer Science

View full text Add to dashboard Cite

The SMT solver solves quantifier-free nonlinear real arithmetic problems by combining the cylindrical algebraic coverings method with incremental linearization in an abstraction-refinement loop. The result is a complete algebraic decision procedure that leverages efficient heuristics for refining candidate models. Furthermore, it can be used with quantifiers, integer variables, and in combination with other theories. We describe the overall framework, individual solving techniques, and a number of implementation details. We demonstrate its effectiveness with an evaluation on the SMT-LIB benchmarks.

show abstract

A Symbiosis of Interval Constraint Propagation and Cylindrical Algebraic Decomposition

Cited by 11 publications

References 13 publications

Building Bridges between Symbolic Computation and Satisfiability Checking

Building Bridges between Symbolic Computation and Satisfiability Checking

Cylindrical algebraic decomposition with equational constraints

Cooperating Techniques for Solving Nonlinear Real Arithmetic in the cvc5 SMT Solver (System Description)

Contact Info

Product

Resources

About