# Jasper Nalbach: Algebraic Methods in SMT-Solving

Algorithms and tools for checking the satisfiability of quantifier-free first-order logic formulas over different theories have many applications in e.g. verification, planning and numerous other fields and enjoy increasing interest. The theory of non-linear real arithmetic (also called real algebra), whose formulas are Boolean combinations of (in)equalities between polynomial expressions evaluated over the real numbers, admits a high expressive power at the cost of high computational costs for satisfiability checking. A subset of this theory, linear arithmetic, where the polynomial expressions are all linear, can be solved more efficiently. In particular, these theories are expressive enough for encoding complex properties about uncertainties. These could be safety properties of systems with linear and non-linear behaviour such as neural networks, and more generally non-linear probability distributions This project is about the general problem of solving (non-)linear arithmetic rather than specific applications. For this, several algorithms are developed and extended, which are implemented and evaluated in our SMT solver SMT-RAT [1,2] which builds on top of our computer algebra library CArL.
Non-linear arithmetic. Although Tarski [3] proved in 1948 that non-linear arithmetic is decidable, the cylindrical algebraic decomposition (CAD) method published in 1975 by Collins [4] was the first complete decision procedure for its solution. Recently, several novel approaches have been developed; namely the model-constructing satisfiability calculus (MCSAT) [5], the one-cell construction method [6] and the cylindrical algebraic coverings method (CAlC) [7]. MCSAT and the one-cell construction can be used in a symbiotic way to solve existential real-arithmetic problems. This new approach is still based on the CAD idea, but instead of a full decomposition it uses the CAD idea to generalize a non-satisfying sample point to a non-satisfying region. The cylindrical algebraic covering methods generates a covering of unsatisfying regions using similar ideas. We developed and implemented a more flexible variant of the original one-cell construction algorithm. This work allows future improvements of both theoretical as well as heuristic nature.
Currently, a publication with a formal proof of the one-cell algorithm algorithm and its experimental evaluation is in progress. In the future, we will develop further improvements of this method and will re-implement the cylindrical algebraic coverings to benefit from these ideas as well.

Linear arithmetic. Linear arithmetic is of interest as it is not only a subset of non-linear arithmetic but also (incomplete) reductions from non-linear arithmetic to linear arithmetic exist. Thus, improving our linear arithmetic solver also benefits the non-linear solver.
The general Simplex algorithm [8] is the most common method for solving linear arithmetic in SMT solving. Despite its exponential running time in worst case, it is efficient in practical instances, heavily depending on chosen heuristics. We are working on improving our Simplex implementation using state-of-the-art heuristics.
Furthermore, we are developing a novel approach that could be promising in the SMT solving context based on the Fourier-Motzkin variable elimination [9] procedure. Extensions of this novel method for learning combinatorial properties of the problem as well as deeper interleaving with the Boolean structure of formulas are conceivable.
While working on these problems, we proved the extension of the Simplex method and others for strict inequalities, which is currently under review. Although a proof already exists, we think that our publication provides more insights into the nature of the problem.

SMT-RAT and CArL. For several reasons, we maintain our own library for arithmetic operations. We are currently evaluating our library against other libraries with regards to efficiency and examine possible extensions or integrations of our library.

1. Kremer, Gereon, and Erika Abraham. Modular strategic SMT solving with SMT-RAT. Acta Universitatis ´ Sapientiae, Informatica 10.1: 5-25, 2018.
2. Kremer, Gereon. Cylindrical Algebraic Decomposition for Nonlinear Arithmetic Problems. Dissertation RWTH Aachen, 2020.
3. Tarski, Alfred. A decision method for elementary algebra and geometry. Quantifier elimination and cylindrical algebraic decomposition. Springer, pages 24–84, 1998.
4. Collins, George E. Quantifier elimination for real closed fields by cylindrical algebraic decompostion. Automata Theory and Formal Languages, pages 134–183, Springer, 1975.
5. Jovanovic, Dejan, and Leonardo De Moura. Solving non-linear arithmetic. International Joint Conference ´ on Automated Reasoning. Springer, 2012.
6. Brown, Christopher W., and Marek Kosta. Constructing a single cell in cylindrical algebraic decomposition Journal of Symbolic Computation 70: 14–48, 2015.
7. Abraham, Erika, et al. Deciding the consistency of non-linear real arithmetic constraints with a conflict ´ driven search using cylindrical algebraic coverings. Journal of Logical and Algebraic Methods in Programming 119: 100633, 2021.
8. Dutertre, Bruno, and Leonardo De Moura. A fast linear-arithmetic solver for DPLL (T). Proc. of CAV. Springer, 2006.
9. Fourier, Jean Baptiste Joseph. Solution d’une question particuliere du calcul des inegalit ´ es. Nouveau ´ Bulletin des Sciences par la Societ ´ e Philomatique de Paris 99: 100, 1826.

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