Solving Satisfiability of Polynomial Formulas By Sample-Cell Projection

💡 Research Summary

The paper introduces a novel algorithm for deciding the satisfiability of polynomial formulas over the real numbers, targeting the nonlinear arithmetic (NRA) theory within SMT solving. Traditional approaches rely on Cylindrical Algebraic Decomposition (CAD), which guarantees sign‑invariance of each polynomial on each cell but suffers from doubly‑exponential time complexity because the projection phase generates a massive number of intermediate polynomials. Recent work by Jovanović and de Moura integrated CAD into a CDCL‑style search, using CAD only for conflict explanation, yet still required a full CAD projection on the conflict core, leading to prohibitive costs.

To overcome these limitations, the authors propose a custom CAD‑like projection operator called sample‑cell projection. The key idea is to compute only the cell that contains a given sample (i.e., a partial variable assignment produced during CDCL search) rather than the entire decomposition of the space. For a sample a and a set of polynomials F, the operator extracts:

1. The non‑zero coefficients of each polynomial evaluated at a (s‑coeff);
2. The discriminants and resultants that become relevant for the sample;
3. The “sample polynomials” (s‑poly) that determine the interval in which the sample’s last variable lies.

These components form Proj_sc(F, x_n, a), a compact projection set that is provably order‑invariant on the submanifold defined by the already assigned variables. The authors prove (Theorem 2) that every polynomial in F is either identically zero on that submanifold or analytically delineable, and that each section of a polynomial either coincides with or is disjoint from the sample interval. Consequently, the cell containing the sample can be described by a conjunction of interval constraints derived from the s‑poly set.

Complexity analysis shows that constructing the sample cell requires O((r² + rⁿ)(2 + n)ⁿ⁻¹) operations, where r is the number of input polynomials and n the number of variables. This is singly exponential, a dramatic improvement over the doubly exponential bound of full CAD. Moreover, the operator can be applied recursively: after fixing the first n‑2 variables, the sample‑cell projection is used to compute the interval for the next variable, and so on, yielding a full sample cell without ever enumerating all cells of a full CAD.

The sample‑cell projection is then embedded into a CDCL‑style search framework similar to that of Jovanović and de Moura. During search, each decision assigns a value to a variable, producing a new sample. The projection operator is invoked to compute the local cell, which is used to check consistency of the remaining literals (often only one unassigned variable remains, allowing real‑root isolation in polynomial time). When a conflict is detected, the conflict core is projected using the same operator, producing a minimal cell that explains the conflict; a learned clause is then added, guiding the search away from the conflicting region.

The authors implemented a prototype solver named LiMbS on top of Mathematica 12. Experimental evaluation on a benchmark suite of well‑known nonlinear arithmetic problems demonstrates that LiMbS outperforms existing SMT‑NL solvers, especially on instances with 5–7 variables where traditional CAD‑based conflict explanation becomes a bottleneck. The results show reduced solving time and memory consumption, confirming that focusing on the sample cell rather than the whole decomposition yields practical benefits.

In summary, the paper makes three major contributions:

1. Definition of the sample‑cell projection operator, a CAD‑like projection that isolates only the cell containing a given sample.
2. Theoretical proof that the operator preserves order‑invariance and analytic delineability, leading to a singly exponential overall complexity.
3. Integration of the operator into a CDCL‑style SMT search, with a working prototype and empirical evidence of its effectiveness.

The work opens a promising direction for scalable nonlinear real arithmetic solving within SMT, suggesting future extensions to higher‑dimensional problems, richer theories (e.g., transcendental functions), and tighter integration with mainstream SMT frameworks.