GiNaCRA: A C++ Library for Real Algebraic Computations

Journal of Logical and Algebraic Methods in Programming, 2021

We present a new algorithm for determining the satisfiability of conjunctions of non-linear polynomial constraints over the reals, which can be used as a theory solver for satisfiability modulo theory (SMT) solving for non-linear real arithmetic. The algorithm is a variant of Cylindrical Algebraic Decomposition (CAD) adapted for satisfiability, where solution candidates (sample points) are constructed incrementally, either until a satisfying sample is found or sufficient samples have been sampled to conclude unsatisfiability. The choice of samples is guided by the input constraints and previous conflicts. The key idea behind our new approach is to start with a partial sample; demonstrate that it cannot be extended to a full sample; and from the reasons for that rule out a larger space around the partial sample, which build up incrementally into a cylindrical algebraic covering of the space. There are similarities with the incremental variant of CAD, the NLSAT method of Jovanović and de Moura, and the NuCAD algorithm of Brown; but we present worked examples and experimental results on a preliminary implementation to demonstrate the differences to these, and the benefits of the new approach.

Journal of Automated Reasoning, 2012

Polynomial constraint solving plays a prominent role in several areas of hardware and software analysis and verification, e.g., termination proving, program invariant generation and hybrid system verification, to name a few. In this paper we propose a new method for solving non-linear constraints based on encoding the problem into an SMT problem considering only linear arithmetic. Unlike other existing methods, our method focuses on proving satisfiability of the constraints rather than on proving unsatisfiability, which is more relevant in several applications as we illustrate with several examples. Nevertheless, we also present new techniques based on the analysis of unsatisfiable cores that allow one to efficiently prove unsatisfiability too for a broad class of problems. The power of our approach is demonstrated by means of extensive experiments comparing our prototype with state-of-the-art tools on benchmarks taken both from the academic and the industrial world.

Lecture Notes in Computer Science, 2012

We present SMT-RAT, a C++ toolbox offering theory solver modules for the development of SMT solvers for nonlinear real arithmetic (NRA). NRA is an important but hard-to-solve theory and only fragments of it can be handled by some of the currently available SMT solvers. Our toolbox contains modules implementing the virtual substitution method, the cylindrical algebraic decomposition method, a Gröbner bases simplifier and a general simplifier. These modules can be combined according to a user-defined strategy in order to exploit their advantages.

Frontiers of Computer Science in China, 2008

We discuss the verification of mathematical software solving polynomial systems symbolically by way of triangular decomposition. Standard verification techniques are highly resource consuming and apply only to polynomial systems which are easy to solve. We exhibit a new approach which manipulates constructible sets represented by regular systems. We provide comparative benchmarks of different verification procedures applied to four solvers on a large set of well-known polynomial systems. Our experimental results illustrate the high efficiency of our new approach. In particular, we are able to verify triangular decompositions of polynomial systems which are not easy to solve.

In this paper we present a method for developing correct programs for exact real numbers based on proof theoretic methods. The basic idea in general is to extract from a formal constructive proof of a specification a realisation of it, in the technical sense of a realisability interpretation. The reason why we believe that this method is worth pursuing is the fact that it offers a path to provably correct programs requiring–contrary to common belief–less effort than traditional methods based on verification.

Lecture Notes in Computer Science, 2008

Many highly sophisticated tools exist for solving linear arithmetic optimization and feasibility problems. Here we analyze why it is difficult to use these tools inside systems for SAT Modulo Theories (SMT) for linear arithmetic: one needs support for disequalities, strict inequalities and, more importantly, for dealing with incorrect results due to the internal use of imprecise floating-point arithmetic. We explain how these problems can be overcome by means of result checking and error recovery policies. Second, by means of carefully designed experiments with, among other tools, the newest version of ILOG CPLEX and our own new Barcelogic T -solver for arithmetic, we show that, interestingly, the cost of result checking is only a small fraction of the total T -solver time. Third, we report on extensive experiments running exactly the same SMT search using CPLEX and Barcelogic as T -solvers, where CPLEX tends to be slower than Barcelogic. We analyze these at first sight surprising results, explaining why tools such as CPLEX are not very adequate (nor designed) for this kind of relatively small incremental problems. Finally, we show how our result checking techniques can still be very useful in combination with inexact floating-point-based T -solvers designed for incremental SMT problems.

CEUR Workshop Proceedings, 2019

Satisfiability-modulo-theories (SMT ) solving is a technique for checking the satisfiability of logical formulas. In this context, recently a framework called model-constructing satisfiability calculus (MCSAT ) has been introduced which relaxes some design restrictions of the classical SMT setting and allows more freedom to construct an efficient interplay between the search for models in the Boolean and the theory domains. In this paper we discuss issues of dynamic variable orderings to drive the MCSAT model search for the theory of non-linear real arithmetic.