$$\mathsf {SC}^\mathsf{2} $$ : Satisfiability Checking Meets Symbolic Computation

Ábrahám, Erika; Abbott, John; Becker, Bernd; Bigatti, Anna Maria; Brain, Martin; Buchberger, Bruno; Cimatti, Alessandro; Davenport, James H.; England, Matthew; Fontaine, Pascal; Forrest, Stephen; Griggio, Alberto; Kroening, Daniel; Seiler, Werner M.; Sturm, Thomas

doi:10.1007/978-3-319-42547-4_3

Cited by 24 publications

(32 citation statements)

References 49 publications

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“…It is a "natural" example in that it comes from 2D geometry. It would be possible to build artificial examples that had, for example, rotational symmetry but no translational symmetry, but, in the spirit of [2], we have started with natural problems. From the basis of this limited analysis, we draw the following provisional conclusions.…”

Section: Resultsmentioning

confidence: 99%

“…This is a very powerful form of human reasoning. Harrison [6] asks, and substantially answers, the question of how, assuming the symmetry is stated, it can be incorporated into formal proof: here we ask an equivalent question for computation, notably in the context of Symbolic Computation and Satisfiability Checking [2]. The challenge turns out to be the detection, rather than the use, of the symmetry.…”

Section: (T T C) P(t F C) and P(f F C) Since P(f T C) Is Eqmentioning

confidence: 99%

“…At its most basic, a symmetry is some transformation of an object that leaves the object (or some aspect of the object) unchanged. [11] That quotation comes from a major survey of symmetry in purely Boolean satisfiability problems, but our setting is "Satisfiability Modulo Theories" over the real numbers, and the desire to enhance this with techniques from Computer Algebra, notably Cylindrical Algebraic Decomposition: see [1,2]. Hence we need to worry about symmetry in the underlying theory as well as in the Boolean formulation, and ask what alignment there is between symmetry in the underlying theory and in the Boolean satisfiability problem that encodes the problem.…”

Section: Introductionmentioning

confidence: 99%

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What Does “Without Loss of Generality” Mean, and How Do We Detect It

Davenport

2017

Math.Comput.Sci.

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When one goes from a geometrical statement to an algebraic statement, the immediate translation is to replace every point by a pair of coordinates, if in the plane (or more as required). A statement with N points is then a statement with 2N (or 3N or more) variables, and the complexity of tools like cylindrical algebraic decomposition is doubly exponential in the number of variables. Hence one says "without loss of generality, A is at (0,0)" and so on. How might one automate this, in particular the detection that a "without loss of generality" argument is possible, or turn it into a procedure (and possibly even a formal proof)?

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Section: Resultsmentioning

confidence: 99%

Section: (T T C) P(t F C) and P(f F C) Since P(f T C) Is Eqmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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What Does “Without Loss of Generality” Mean, and How Do We Detect It

Davenport

2017

Math.Comput.Sci.

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“…In [3] we reported on the activities of our SC 2 project to strengthen the connections between these areas. To mention the most important measures, as a platform for interdisciplinary communication and exchange, we initiated an annual SC 2 workshop [2].…”

Section: Resultsmentioning

confidence: 99%

“…Highly interesting techniques were implemented in SATmodulo-theories (SMT) solvers [5,17] for checking easier theories, but the development for quantifier-free non-linear real and integer arithmetic 1 is still in its infancy. The SC 2 (Satisfiability Checking and Symbolic Computation) initiative, described further in [3], aims at bridging these two communities, so that members are well informed about both fields, and thus able to combine the knowledge and techniques of both fields to develop new research and to resolve problems (both academic and industrial) currently beyond the scope of either individual field. Formally, it is also a European Horizon 2020 Coordination and Support Action, running from July 2016 to September 2018: see http://www.sc-square.org.…”

Section: Introductionmentioning

confidence: 99%

SC-square: when Satisfiability Checking and Symbolic Computation join forces

Ábrahám

Abbott

Becker

et al.

EPiC Series in Computing

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Symbolic Computation and Satisfiability Checking are two research areas, both having their individual scientific focus but with common interests, e.g., in the development, implementation and application of decision procedures for arithmetic theories. Despite their commonalities, the two communities are rather weakly connected. The aim of the SC 2 initiative is to strengthen the connection between these communities by creating common platforms, initiating interaction and exchange, identifying common challenges, and developing a common roadmap from theory along the way to tools and (industrial) applications.

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Comparing Machine Learning Models to Choose the Variable Ordering for Cylindrical Algebraic Decomposition

England

Florescu

2019

Lecture Notes in Computer Science

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There has been recent interest in the use of machine learning (ML) approaches within mathematical software to make choices that impact on the computing performance without affecting the mathematical correctness of the result. We address the problem of selecting the variable ordering for cylindrical algebraic decomposition (CAD), an important algorithm in Symbolic Computation. Prior work to apply ML on this problem implemented a Support Vector Machine (SVM) to select between three existing human-made heuristics, which did better than anyone heuristic alone. Here we extend this result by training ML models to select the variable ordering directly, and by trying out a wider variety of ML techniques. We experimented with the NLSAT dataset and the Regular Chains Library CAD function for Maple 2018. For each problem, the variable ordering leading to the shortest computing time was selected as the target class for ML. Features were generated from the polynomial input and used to train the following ML models: k-nearest neighbours (KNN) classifier, multi-layer perceptron (MLP), decision tree (DT) and SVM, as implemented in the Python scikit-learn package. We also compared these with the two leading human-made heuristics for the problem: the Brown heuristic and sotd. On this dataset all of the ML approaches outperformed the human-made heuristics, some by a large margin.

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$$\mathsf {SC}^\mathsf{2} $$ : Satisfiability Checking Meets Symbolic Computation

Cited by 24 publications

References 49 publications

What Does “Without Loss of Generality” Mean, and How Do We Detect It

What Does “Without Loss of Generality” Mean, and How Do We Detect It

SC-square: when Satisfiability Checking and Symbolic Computation join forces

Comparing Machine Learning Models to Choose the Variable Ordering for Cylindrical Algebraic Decomposition

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