124 research outputs found
Cylindrical algebraic decomposition with equational constraints
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.Comment: Accepted into the Journal of Symbolic Computation. arXiv admin note:
text overlap with arXiv:1501.0446
PolyARBerNN: A Neural Network Guided Solver and Optimizer for Bounded Polynomial Inequalities
Constraints solvers play a significant role in the analysis, synthesis, and
formal verification of complex embedded and cyber-physical systems. In this
paper, we study the problem of designing a scalable constraints solver for an
important class of constraints named polynomial constraint inequalities (also
known as non-linear real arithmetic theory). In this paper, we introduce a
solver named PolyARBerNN that uses convex polynomials as abstractions for
highly nonlinear polynomials. Such abstractions were previously shown to be
powerful to prune the search space and restrict the usage of sound and complete
solvers to small search spaces. Compared with the previous efforts on using
convex abstractions, PolyARBerNN provides three main contributions namely (i) a
neural network guided abstraction refinement procedure that helps selecting the
right abstraction out of a set of pre-defined abstractions, (ii) a Bernstein
polynomial-based search space pruning mechanism that can be used to compute
tight estimates of the polynomial maximum and minimum values which can be used
as an additional abstraction of the polynomials, and (iii) an optimizer that
transforms polynomial objective functions into polynomial constraints (on the
gradient of the objective function) whose solutions are guaranteed to be close
to the global optima. These enhancements together allowed the PolyARBerNN
solver to solve complex instances and scales more favorably compared to the
state-of-art non-linear real arithmetic solvers while maintaining the soundness
and completeness of the resulting solver. In particular, our test benches show
that PolyARBerNN achieved 100X speedup compared with Z3 8.9, Yices 2.6, and
NASALib (a solver that uses Bernstein expansion to solve multivariate
polynomial constraints) on a variety of standard test benches
Combined decision procedures for nonlinear arithmetics, real and complex
We describe contributions to algorithmic proof techniques for deciding the satisfiability
of boolean combinations of many-variable nonlinear polynomial equations and
inequalities over the real and complex numbers.
In the first half, we present an abstract theory of Grobner basis construction algorithms
for algebraically closed fields of characteristic zero and use it to introduce
and prove the correctness of Grobner basis methods tailored to the needs of modern
satisfiability modulo theories (SMT) solvers. In the process, we use the technique of
proof orders to derive a generalisation of S-polynomial superfluousness in terms of
transfinite induction along an ordinal parameterised by a monomial order. We use this
generalisation to prove the abstract (âstrategy-independentâ) admissibility of a number
of superfluous S-polynomial criteria important for efficient basis construction. Finally,
we consider local notions of proof minimality for weak Nullstellensatz proofs and give
ideal-theoretic methods for computing complex âunsatisfiable coresâ which contribute
to efficient SMT solving in the context of nonlinear complex arithmetic.
In the second half, we consider the problem of effectively combining a heterogeneous
collection of decision techniques for fragments of the existential theory of real
closed fields. We propose and investigate a number of novel combined decision methods
and implement them in our proof tool RAHD (Real Algebra in High Dimensions).
We build a hierarchy of increasingly powerful combined decision methods, culminating
in a generalisation of partial cylindrical algebraic decomposition (CAD) which we
call Abstract Partial CAD. This generalisation incorporates the use of arbitrary sound
but possibly incomplete proof procedures for the existential theory of real closed fields
as first-class functional parameters for âshort-circuitingâ expensive computations during
the lifting phase of CAD. Identifying these proof procedure parameters formally
with RAHD proof strategies, we implement the method in RAHD for the case of
full-dimensional cell decompositions and investigate its efficacy with respect to the
Brown-McCallum projection operator.
We end with some wishes for the future
Using Machine Learning to Improve Cylindrical Algebraic Decomposition
Cylindrical Algebraic Decomposition (CAD) is a key tool in computational
algebraic geometry, best known as a procedure to enable Quantifier Elimination over real-closed fields. However, it has a worst case complexity doubly exponential in the size of the input, which is often encountered in practice. It has been observed that for many problems a change in algorithm settings or problem formulation can cause huge differences in runtime costs, changing problem instances from intractable to easy. A number of heuristics have been developed to help with such choices, but the complicated nature of the geometric relationships involved means these are imperfect and can sometimes make poor choices. We investigate the use of machine learning (specifically
support vector machines) to make such choices instead. Machine learning is the process of fitting a computer model to a complex
function based on properties learned from measured data. In this paper we apply it in two case studies: the first to select between heuristics for choosing a CAD variable ordering; the second to identify when a CAD problem instance would benefit from Groebner Basis preconditioning. These appear to be the first such applications of machine learning to Symbolic Computation. We demonstrate in both cases that the machine learned choice outperforms human developed heuristics.This work was supported by EPSRC grant EP/J003247/1; the European Unionâs Horizon 2020 research and innovation programme under grant agreement No 712689 (SC2); and the China Scholarship
Council (CSC)
Q(sqrt(-3))-Integral Points on a Mordell Curve
We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M Ìuller,combined with a sieving technique, to determine the integral points overQ(ââ3) on the Mordell curve y2 = x3 â 4
When Less Is More: Consequence-Finding in a Weak Theory of Arithmetic
This paper presents a theory of non-linear integer/real arithmetic and
algorithms for reasoning about this theory. The theory can be conceived as an
extension of linear integer/real arithmetic with a weakly-axiomatized
multiplication symbol, which retains many of the desirable algorithmic
properties of linear arithmetic. In particular, we show that the conjunctive
fragment of the theory can be effectively manipulated (analogously to the usual
operations on convex polyhedra, the conjunctive fragment of linear arithmetic).
As a result, we can solve the following consequence-finding problem: given a
ground formula F, find the strongest conjunctive formula that is entailed by F.
As an application of consequence-finding, we give a loop invariant generation
algorithm that is monotone with respect to the theory and (in a sense)
complete. Experiments show that the invariants generated from the consequences
are effective for proving safety properties of programs that require non-linear
reasoning
Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System
Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics
Automated Reasoning
This volume, LNAI 13385, constitutes the refereed proceedings of the 11th International Joint Conference on Automated Reasoning, IJCAR 2022, held in Haifa, Israel, in August 2022. The 32 full research papers and 9 short papers presented together with two invited talks were carefully reviewed and selected from 85 submissions. The papers focus on the following topics: Satisfiability, SMT Solving,Arithmetic; Calculi and Orderings; Knowledge Representation and Jutsification; Choices, Invariance, Substitutions and Formalization; Modal Logics; Proofs System and Proofs Search; Evolution, Termination and Decision Prolems. This is an open access book
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