Gr\"obner Bases over Algebraic Number Fields

Although Buchberger's algorithm, in theory, allows us to compute Gr\"obner bases over any field, in practice, however, the computational efficiency depends on the arithmetic of the ground field. Consider a field $K = \mathbb{Q}(\alpha)$, a simple extension of $\mathbb{Q}$, where $\alpha$ is an algebraic number, and let $f \in \mathbb{Q}[t]$ be the minimal polynomial of $\alpha$. In this paper we present a new efficient method to compute Gr\"obner bases in polynomial rings over the algebraic number field $K$. Starting from the ideas of Noro [Noro, 2006], we proceed by joining $f$ to the ideal to be considered, adding $t$ as an extra variable. But instead of avoiding superfluous S-pair reductions by inverting algebraic numbers, we achieve the same goal by applying modular methods as in [Arnold, 2003; B\"ohm et al., 2015; Idrees et al., 2011], that is, by inferring information in characteristic zero from information in characteristic $p > 0$. For suitable primes $p$, the minimal polynomial $f$ is reducible over $\mathbb{F}_p$. This allows us to apply modular methods once again, on a second level, with respect to the factors of $f$. The algorithm thus resembles a divide and conquer strategy and is in particular easily parallelizable. At current state, the algorithm is probabilistic in the sense that, as for other modular Gr\"obner basis computations, an effective final verification test is only known for homogeneous ideals or for local monomial orderings. The presented timings show that for most examples, our algorithm, which has been implemented in SINGULAR, outperforms other known methods by far.Comment: 16 pages, 1 figure, 1 tabl

Involutive Bases Algorithm Incorporating F5 Criterion

Faugere's F5 algorithm is the fastest known algorithm to compute Groebner bases. It has a signature-based and an incremental structure that allow to apply the F5 criterion for deletion of unnecessary reductions. In this paper, we present an involutive completion algorithm which outputs a minimal involutive basis. Our completion algorithm has a nonincremental structure and in addition to the involutive form of Buchberger's criteria it applies the F5 criterion whenever this criterion is applicable in the course of completion to involution. In doing so, we use the G2V form of the F5 criterion developed by Gao, Guan and Volny IV. To compare the proposed algorithm, via a set of benchmarks, with the Gerdt-Blinkov involutive algorithm (which does not apply the F5 criterion) we use implementations of both algorithms done on the same platform in Maple.Comment: 24 pages, 2 figure

Function Verification of Combinational Arithmetic Circuits

Hardware design verification is the most challenging part in overall hardware design process. It is because design size and complexity are growing very fast while the requirement for performance is ever higher. Conventional simulation-based verification method cannot keep up with the rapid increase in the design size, since it is impossible to exhaustively test all input vectors of a complex design. An important part of hardware verification is combinational arithmetic circuit verification. It draws a lot of attention because flattening the design into bit-level, known as the bit-blasting problem, hinders the efficiency of many current formal techniques. The goal of this thesis is to introduce a robust and efficient formal verification method for combinational integer arithmetic circuit based on an in-depth analysis of recent advances in computer algebra. The method proposed here solves the verification problem at bit level, while avoiding bit-blasting problem. It also avoids the expensive Groebner basis computation, typically employed by symbolic computer algebra methods. The proposed method verifies the gate-level implementation of the design by representing the design components (logic gates and arithmetic modules) by polynomials in Z2n . It then transforms the polynomial representing the output bits (called “output signature”) into a unique polynomial in input signals (called “input signature”) using gate-level information of the design. The computed input signature is then compared with the reference input signature (golden model) to determine whether the circuit behaves as anticipated. If the reference input signature is not given, our method can be used to compute (or extract) the arithmetic function of the design by computing its input signature. Additional tools, based on canonical word-level design representations (such as TED or BMD) can be used to determine the function of the computed input signature represents. We demonstrate the applicability of the proposed method to arithmetic circuit verification on a large number of designs

A New Algorithm For The Generation Of Unitarity-Compatible Integration By Parts Relations

Many multi-loop calculations make use of integration by parts relations to reduce the large number of complicated Feynman integrals that arise in such calculations to a simpler basis of master integrals. Recently, Gluza, Kajda, and Kosower argued that the reduction to master integrals is complicated by the presence of integrals with doubled propagator denominators in the integration by parts relations and they introduced a novel reduction procedure which eliminates all such integrals from the start. Their approach has the advantage that it automatically produces integral bases which mesh well with generalized unitarity. The heart of their procedure is an algorithm which utilizes the weighty machinery of computational commutative algebra to produce complete sets of unitarity-compatible integration by parts relations. In this paper, we propose a conceptually simpler algorithm for the generation of complete sets of unitarity-compatible integration by parts relations based on recent results in the mathematical literature. A striking feature of our algorithm is that it can be described entirely in terms of straightforward linear algebra.Comment: 20 pages; My apologies to Krzysztof Kajda for misspelling his name in v1; in v3: the labeling of the variables in (4.5) and eqs. (4.20) and (4.21) was adjusted to match the notation used in the rest of Section 4. I thank York Schroeder for pointing out the notational inconsistenc

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 (“strategy-independent”) 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 “unsatisfiable cores” 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 “short-circuiting” 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 Union’s Horizon 2020 research and innovation programme under grant agreement No 712689 (SC2); and the China Scholarship Council (CSC)