Quantifier Elimination Method Research Articles

Interval Quadratic Equations: A Review

In this study, we tackle the subject of interval quadratic equations and we aim to accurately determine the root enclosures of quadratic equations, whose coefficients constitute interval variables. This study focuses on interval quadratic equations that contain only one coefficient considered as an interval variable. The four methods reviewed here in order to solve this problem are: (i) the method of classic interval analysis used by Elishakoff and Daphnis, (ii) the direct method based on minimizations and maximizations also used by the same authors, (iii) the method of quantifier elimination used by Ioakimidis, and (iv) the interval parametrization method suggested by Elishakoff and Miglis and again based on minimizations and maximizations. We will also compare the results yielded by all these methods by using the computer algebra system Mathematica for computer evaluations (including quantifier eliminations) in order to conclude which method would be the most efficient way to solve problems relevant to interval quadratic equations.

Reasoning Method between Polynomial Error Assertions

Error coefficients are ubiquitous in systems. In particular, errors in reasoning verification must be considered regarding safety-critical systems. We present a reasoning method that can be applied to systems described by the polynomial error assertion (PEA). The implication relationship between PEAs can be converted to an inclusion relationship between zero sets of PEAs; the PEAs are then transformed into first-order polynomial logic. Combined with the quantifier elimination method, based on cylindrical algebraic decomposition, the judgment of the inclusion relationship between zero sets of PEAs is transformed into judgment error parameters and specific error coefficient constraints, which can be obtained by the quantifier elimination method. The proposed reasoning method is validated by proving the related theorems. An example of intercepting target objects is provided, and the correctness of our method is tested through large-scale random cases. Compared with reasoning methods without error semantics, our reasoning method has the advantage of being able to deal with error parameters.

English

Computing powers of interval matrices is a computationally hard problem. Indeed, it is NP-hard even when the exponent is 3 and the matrices only have interval components in one row and one column. Motivated by this result, we consider special types of interval matrices where the interval components occupy specific positions. We show that computing the third power of matrices with only one column occupied by interval components can be solved in cubic time; so the asymptotic time complexity is the same as for the real case (considering the textbook matrix product method). We further show that for a fixed exponent k and for each interval matrix (of an arbitrary size) whose kth power has components that can be expressed as polynomials in a fixed number of interval variables, the computation of the kth power is polynomial up to a given accuracy. Polynomiality is shown by using the Tarski method of quantifier elimination. This result is used to show the polynomiality of computing the cube of interval band matrices, among others. Additionally, we study parametric matrices and prove NP-hardness already for their squares. We also describe one specific class of interval parametric matrices that can be squared by a polynomial algorithm.

A Logic Based Approach to Finding Real Singularities of Implicit Ordinary Differential Equations

We discuss the effective computation of geometric singularities of implicit ordinary differential equations over the real numbers using methods from logic. Via the Vessiot theory of differential equations, geometric singularities can be characterised as points where the behaviour of a certain linear system of equations changes. These points can be discovered using a specifically adapted parametric generalisation of Gaussian elimination combined with heuristic simplification techniques and real quantifier elimination methods. We demonstrate the relevance and applicability of our approach with computational experiments using a prototypical implementation in Reduce.

Automatic generation of bounds for polynomial systems with application to the Lorenz system

This study covers an analytical approach to calculate positive invariant sets of dynamical systems. Using Lyapunov techniques and quantifier elimination methods, an automatic procedure for determining bounds in the state space as an enclosure of attractors is proposed. The available software tools permit an algorithmizable process, which normally requires a good insight into the systems dynamics and experience. As a result we get an estimation of the attractor, whose conservatism only results from the initial choice of the Lyapunov candidate function. The proposed approach is illustrated on the well-known Lorenz system.

JavaScript Algorithm for Quantifier Elimination in Epidemiology.

Introduction:Quantifier Elimination gives us perfect insight into the most basic world of the computer, its origin, its primer functions and it basic operations. Carefully designed and programmed Algorithm for Quantifier Elimination makes the quantifier elimination from the quantified formulas much easier and much more comprehensiveAim:This paper explains how Quantifier Elimination algorithm can be used in the field of Biology, or to be more specific, in the field of Epidemiology.Material and methods:Exemplary formulas needed for the algorithm are all the formulas from the Mathematical Logic field. JavaScript programming language was used in order to program fast and effective algorithm for Quantifier Elimination.Results:Solving the certain problems from the field of Epidemiology using the Quantifier Elimination method, proved to be very successful in the past, because it made possible for the results to be extracted very fast. Doing the exact thing using the newer generation algorithm might be even more effective.Conclusion:The most basic concepts of Mathematical Logic can be implemented in order to solve the one of the most important questions in Epidemiology.

Bounding quantification in parametric expansions of Presburger arithmetic

Generalizing Cooper’s method of quantifier elimination for Presburger arithmetic, we give a new proof that all parametric Presburger families $$\{S_t : t \in \mathbb {N}\}$$ [as defined by Woods (Electron J Comb 21:P21, 2014)] are definable by formulas with polynomially bounded quantifiers in an expanded language with predicates for divisibility by f(t) for every polynomial $$f \in \mathbb {Z}[t]$$ . In fact, this quantifier bounding method works more generally in expansions of Presburger arithmetic by multiplication by scalars $$\{\alpha (t): \alpha \in R, t \in X\}$$ where R is any ring of functions from X into $$\mathbb {Z}$$ .

The energy method in problems of buckling of bars with quantifier elimination

The classical energy method for the approximate determination of critical buckling loads of bars is revisited. This method is based on the stability condition of the bar and on the appropriate selection of an approximation to the deflection of the bar. Moreover, it is frequently related to the Rayleigh quotient or to the Timoshenko quotient for the determination of the critical buckling load. Here we will use again the energy method for the determination of critical buckling loads of bars but now on the basis of a new computational approach. This new approach consists of using the modern computational method of quantifier elimination efficiently implemented in the computer algebra system Mathematica instead of partial differentiations when we use the stability condition of the bar or essentially equivalently when we minimize the Rayleigh quotient or the Timoshenko quotient. This approach, which avoids partial differentiations, is also more rigorous than the classical approach based on partial derivatives because it does not require the use of the conditions for a minimum based on second partial derivatives, which are generally ignored in practice. Moreover, it is very simple to use inside the powerful computational environment offered by Mathematica. The present approach is illustrated in several buckling problems of bars including parametric buckling problems. Buckling problems of bars with two internal unilateral constraints, where the classical energy method is difficult to apply, are also studied. Even in this rather difficult application the critical buckling load is directly determined with a sufficient accuracy.

Application of quantifier elimination to inverse buckling problems

The inverse buckling problem for a column is the problem where both the loading and the buckling mode are defined in advance (the latter generally in a polynomial form), and the flexural rigidity of the column is sought in a similar form with the help of the related ordinary differential equation. This problem was proposed and studied in many buckling problems by Elishakoff and his collaborators. A serious difficulty in its solution is that the resulting flexural rigidity should be positive along the column. Here in order to check this positivity, the modern computational method of quantifier elimination is proposed and used inside the computational environment offered by the computer algebra system Mathematica and mainly based on the Collins cylindrical algebraic decomposition algorithm. At first, the simple inverse buckling problem of an inhomogeneous column under a concentrated load is studied with respect to the aforementioned positivity requirement. Next, the much more difficult problem concerning a variable distributed loading is also studied both in the case of one parameter and in the case of two parameters in this loading. Parametric rational and trigonometric forms of the flexural rigidity are also studied. Naturally, the resulting conditions for the positivity of the flexural rigidity are rather simple for one loading parameter, but they may become sufficiently complicated for two loading parameters. The present computational approach constitutes a simple, efficient and mathematically rigorous way for the derivation of positivity conditions for the flexural rigidity of a column in a variety of inverse buckling problems.

Caustics, pseudocaustics and the related illuminated and dark regions with the computational method of quantifier elimination

The method of caustics is a powerful experimental method in elasticity and particularly in fracture mechanics for crack problems. The related method of pseudocaustics is also of interest. Here we apply the computational method of quantifier elimination implemented in the computer algebra system Mathematica in order to determine (i) the non-parametric equation and two properties of the caustic at a crack tip and especially (ii) the illuminated and the dark regions related to caustics and pseudocaustics in plane elasticity and plate problems. The present computations concern: (i) The derivation of the non-parametric equation of the classical caustic about a crack tip through the elimination of the parameter involved (here the polar angle) as well as two geometrical properties of this caustic. (ii) The derivation of the inequalities defining the illuminated region on the screen in the problem of an elastic half-plane loaded normally by a concentrated load with the boundary of this illuminated region related to some extent to the caustic formed. (iii) Similarly for the problem of a clamped circular plate under a uniform loading with respect to the caustic and the pseudocaustic formed. (iv) Analogously for the problem of an equilateral triangular plate loaded by uniformly distributed moments along its whole boundary, which defines the related pseudocaustic. (v) The determination of quantities of interest in mechanics from the obtained caustics or pseudocaustics. The kind of computations in the applications (ii) to (iv), i.e. the derivation of inequalities defining the illuminated region on the screen, seems to be completely new independently of the use here of the method of quantifier elimination. Additional applications are also possible, but some of them require the expansion of the present somewhat limited power of the quantifier elimination algorithms in Mathematica. This is expected to take place in the future.

Proving Theorems by Program Transformation

In this paper we present an overview of the unfold/fold proof method, a method for proving theorems about programs, based on program transformation. As a metalanguage for specifying programs and program properties we adopt constraint logic programming CLP, and we present a set of transformation rules including the familiar unfolding and folding rules which preserve the semantics of CLP programs. Then, we show how program transformation strategies can be used, similarly to theorem proving tactics, for guiding the application of the transformation rules and inferring the properties to be proved. We work out three examples: i the proof of predicate equivalences, applied to the verification of equality between CCS processes, ii the proof of first order formulas via an extension of the quantifier elimination method, and iii the proof of temporal properties of infinite state concurrent systems, by using a transformation strategy that performs program specialization.

Parametric equation solving and quantifier elimination in finite fields with the characteristic set method

For a parametric algebraic system in finite fields, this paper presents a method for computing the cover and the refined cover based on the characteristic set method. From the cover, the author knows for what parametric values the system has solutions and at the same time presents the solutions in the form of proper chains. By the refined cover, the author gives a complete classification of the number of solutions for this system, that is, the author divides the parameter space into several disjoint components, and on every component the system has a fix number of solutions. Moreover, the author develops a method of quantifier elimination for first order formulas in finite fields.

The Ackermann approach for modal logic, correspondence theory and second-order reduction

The problem of eliminating second-order quantification over predicate symbols is in general undecidable. Since an application of second-order quantifier elimination is correspondence theory in modal logic, understanding when second-order quantifier elimination methods succeed is an important problem that sheds light on the kinds of axioms that are equivalent to first-order correspondence properties and can be used to obtain complete axiomatizations for modal logics. This paper introduces a substitution-rewrite approach based on Ackermannʼs Lemma to second-order quantifier elimination in modal logic. Compared to related approaches, the approach includes a number of enhancements: The quantified symbols that need to be eliminated can be flexibly specified. The inference rules are restricted by orderings compatible with the elimination order, which provides more control and reduces non-determinism in derivations thereby increasing the efficiency and success rate. The approach is equipped with a powerful notion of redundancy, allowing for the flexible definition of practical simplification and optimization techniques. We present correctness, termination and canonicity results, and consider two applications: (i) computing first-order frame correspondence properties for modal axioms and rules, and (ii) rewriting second-order modal problems to equivalent simpler forms. The approach allows us to define and characterize two new classes of formulae, which are elementary and canonical, and subsume the class of Sahlqvist formulae and the class of monadic inductive formulae.

An analytic solution to the alibi query in the space–time prisms model for moving object data

Moving objects produce trajectories, which are stored in databases by means of finite samples of time-stamped locations. When speed limitations in these sample points are also known, space–time prisms (also called beads) (Pfoser and Jensen 1999, Egenhofer 2003, Miller 2005) can be used to model the uncertainty about an object's location in between sample points. In this setting, a query of particular interest that has been studied in the literature of geographic information systems (GIS) is the alibi query. This boolean query asks whether two moving objects could have physically met. This adds up to deciding whether the chains of space–time prisms (also called necklaces of beads) of these objects intersect. This problem can be reduced to deciding whether two space–time prisms intersect. The alibi query can be seen as a constraint database query. In the constraint database model, spatial and spatiotemporal data are stored by boolean combinations of polynomial equalities and inequalities over the real numbers. The relational calculus augmented with polynomial constraints is the standard first-order query language for constraint databases and the alibi query can be expressed in it. The evaluation of the alibi query in the constraint database model relies on the elimination of a block of three exªistential quantifiers. Implementations of general purpose elimination algorithms, such as those provided by QEPCAD, Redlog, and Mathematica, are, for practical purposes, too slow in answering the alibi query for two specific space–time prisms. These software packages completely fail to answer the alibi query in the parametric case (i.e., when it is formulated in terms of parameters representing the sample points and speed constraints). The main contribution of this article is an analytical solution to the parametric alibi query, which can be used to answer the alibi query on two specific space–time prisms in constant time (a matter of milliseconds in our implementation). It solves the alibi query for chains of space–time prisms in time proportional to the sum of the lengths of the chains. To back this claim up, we implemented our method in Mathematica alongside the traditional quantifier elimination method. The solutions we propose are based on the geometric argumentation and they illustrate the fact that some practical problems require creative solutions, where at least in theory, existing systems could provide a solution.

Annotation Theories over Finite Graphs

In the current paper we consider theories with vocabulary containing a number of binary and unary relation symbols. Binary relation symbols represent labeled edges of a graph and unary relations represent unique annotations of the graph’s nodes. Such theories, which we call annotation theories, can be used in many applications, including the formalization of argumentation, approximate reasoning, semantics of logic programs, graph coloring, etc. We address a number of problems related to annotation theories over finite models, including satisfiability, querying problem, specification of preferred models and model checking problem.We show that most of considered problems are NPTime- or co-NPTime-complete. In order to reduce the complexity for particular theories, we use second-order quantifier elimination. To our best knowledge none of existing methods works in the case of annotation theories. We then provide a new second-order quantifier elimination method for stratified theories, which is successful in the considered cases. The new result subsumes many other results, including those of [2, 28, 21].

Combining logical and algebraic techniques for natural style proving in elementary analysis

PCS (Proving-Computing-Solving) introduced by B. Buchberger in 2001 and S-Decomposition, introduced by T. Jebelean in 2001, are strategies for handling proof problems by combining logic inference steps (e.g., modus ponens, Skolemization, instantiation) with rewriting steps (application of definitions) and solving procedures based on algebraic techniques (e.g., Groebner Bases, Cylindrical Algebraic Decomposition). If one formalizes the main notions of elementary analysis like continuity, convergence, etc., usually a sequence of alternating quantifier blocks appears in the quantifier prefix of the corresponding formula. This makes the proof problems involving these notions difficult. The S-Decomposition strategy is especially suitable for property-preserving problems like continuity of sum, because it is designed for handling problems where the goal and the main assumptions have a similar structure. During proof deduction, existentially quantified goals and universal assumptions are handled by introducing metavariables, if no suitable ground instance is known in advance. For finalizing proof attempts, the metavariables must be instantiated in such a way that they satisfy the cumulated algebraic constraints collected during the proof attempt. The instantiation problem is considered to be difficult in a purely logical calculus. Appropriate instances can be often found using quantifier elimination (QE) over real closed fields. In order to obtain witness terms we utilize the QE method based on cylindrical algebraic decomposition (CAD) invented by G. Collins in 1973. However, the QE method alone is not sufficient. One needs to pre-process the (closed, quantified) conjectured formula and post-process the resulting CAD-structure after the call of the QE algorithm.

Bounds on the automata size for Presburger arithmetic

Automata provide a decision procedure for Presburger arithmetic. However, until now only crude lower and upper bounds were known on the sizes of the automata produced by the automata-based approach for Presburger arithmetic. In this article, we give an upper bound on the number of states of the minimal deterministic automaton for a Presburger arithmetic formula. This bound depends on the length of the formula and the quantifiers occurring in it. We establish the upper bound by comparing the automata for Presburger arithmetic formulas with the formulas produced by a quantifier-elimination method. We show that our bound is tight, also for nondeterministic automata. Moreover, we provide automata constructions for atomic formulas and establish lower bounds for the automata for linear equations and inequations.

Towards incorporating background theories into quantifier elimination

In the paper we present a technique for eliminating quantifiers of arbitrary order, in particular of first-order. Such a uniform treatment of the elimination problem has been problematic up to now, since techniques for eliminating first-order quantifiers do not scale up to higher-order contexts and those for eliminating higher-order quantifiers are usually based on a form of monotonicity w.r.t implication (set inclusion) and are not applicable to the first-order case. We make a shift to arbitrary relations “ordering” the underlying universe. This allows us to incorporate background theories into higher-order quantifier elimination methods which, up to now, has not been achieved. The technique we propose subsumes many other results, including the Ackermann's lemma and various forms of fixpoint approaches when the “ordering” relations are interpreted as implication and reveals the common principle behind these approaches.

First-order queries on structures of bounded degree are computable with constant delay

A relational structure is d -degree-bounded, for some integer d , if each element of the domain belongs to at most d tuples. In this paper, we revisit the complexity of the evaluation problem of not necessarily Boolean first-order ( FO ) queries over d -degree-bounded structures. Query evaluation is considered here as a dynamical process. We prove that any FO query on d -degree-bounded structures belongs to the complexity class constant-Delay lin , that is, can be computed by an algorithm that has two separate parts: it has a precomputation step of time linear in the size of the structure and then, it outputs all solutions (i.e., tuples that satisfy the formula) one by one with a constant delay (i.e., depending on the size of the formula only) between each. Seen as a global process, this implies that queries on d -degree-bounded structures can be evaluated in total time f (|φ|).(| S | + |φ( S )|) and space g (|φ|).| S | where S is the structure, φ is the formula, φ( S ) is the result of the query and f , g are some fixed functions. Among other things, our results generalize a result of Seese on the data complexity of the model-checking problem for d -degree-bounded structures. Besides, the originality of our approach compared to related results is that it does not rely on the Hanf's model-theoretic technique and is simple and informative since it essentially rests on a quantifier elimination method.

Quantifier Elimination over Algebraically Closed Fields in a Proof Assistant using a Computer Algebra System

We propose a decision procedure for algebraically closed fields based on a quantifier elimination method. The procedure is intended to build proofs for systems of polynomial equations and inequations. We describe how this procedure can be carried out in a proof assistant using a Computer Algebra system in a purely skeptical way. We present an implementation in the particular framework of Coq and Maple giving some details regarding the interface between the two tools. This allows us to show that a Computer Algebra system can be used not only to bring additional computational power to a proof assistant but also to enhance the automation of such tools.