Quantifier Elimination Research Articles

Generating invariants for non-linear loops by linear algebraic methods

Abstract We present new computational methods that can automate the discovery and the strengthening of non-linear interrelationships among the variables of programs containing non-linear loops, that is, that give rise to multivariate polynomial and fractional relationships. Our methods have complexities lower than the mathematical foundations of the previous approaches, which used Gröbner basis computations, quantifier eliminations or cylindrical algebraic decompositions. We show that the preconditions for discrete transitions can be viewed as morphisms over a vector space of degree bounded by polynomials. These morphisms can, thus, be suitably represented by matrices. We also introduce fractional and polynomial consecution, as more general forms for approximating consecution. The new relaxed consecution conditions are also encoded as morphisms represented by matrices. By so doing, we can reduce the non-linear loop invariant generation problem to the computation of eigenspaces of specific morphisms. Moreover, as one of the main results, we provide very general sufficient conditions allowing for the existence and computation of whole loop invariant ideals. As far as it is our knowledge, it is the first invariant generation methods that can handle multivariate fractional loops.

MODEL COMPLETENESS OF O-MINIMAL FIELDS WITH CONVEX VALUATIONS

AbstractWe let R be an o-minimal expansion of a field, V a convex subring, and (R0,V0) an elementary substructure of (R,V). Our main result is that (R,V) considered as a structure in a language containing constants for all elements of R0 is model complete relative to quantifier elimination in R, provided that kR (the residue field with structure induced from R) is o-minimal. Along the way we show that o-minimality of kR implies that the sets definable in kR are the same as the sets definable in k with structure induced from (R,V). We also give a criterion for a superstructure of (R,V) being an elementary extension of (R,V).

Worst-case results for positive semidefinite rank

We present various worst-case results on the positive semidefinite (psd) rank of a nonnegative matrix, primarily in the context of polytopes. We prove that the psd rank of a generic $$n$$n-dimensional polytope with $$v$$v vertices is at least $$(nv)^{\frac{1}{4}}$$(nv)14 improving on previous lower bounds. For polygons with $$v$$v vertices, we show that psd rank cannot exceed $$4 \left\lceil v/6 \right\rceil $$4v/6 which in turn shows that the psd rank of a $$p \times q$$p×q matrix of rank three is at most $$4\left\lceil \min \{p,q\}/6 \right\rceil $$4min{p,q}/6. In general, a nonnegative matrix of rank $${k+1 \atopwithdelims ()2}$$k+12 has psd rank at least $$k$$k and we pose the problem of deciding whether the psd rank is exactly $$k$$k. Using geometry and bounds on quantifier elimination, we show that this decision can be made in polynomial time when $$k$$k is fixed.

On Optimal Quadratic Lagrange Interpolation: Extremal Node Systems with Minimal Lebesgue Constant via Symbolic Computation

We consider optimal Lagrange interpolation with polynomials of degree at most two on the unit interval [−1, 1]. In a largely unknown paper, Schurer (1974, Stud. Sci. Math. Hung. 9, 77-79) has analytically described the infinitely many zero-symmetric and zero-asymmetric extremal node systems −1 ≤ x1 < x2 < x3 ≤ 1 which all lead to the minimal Lebesgue constant 1.25 that had already been determined by Bernstein (1931, Izv. Akad. Nauk SSSR 7, 1025-1050). As Schurer’s proof is not given in full detail, we formally verify it by providing two new and sound proofs of his theorem with the aid of symbolic computation using quantifier elimination. Additionally, we provide an alternative, but equivalent, parameterized description of the extremal node systems for quadratic Lagrange interpolation which seems to be novel. It is our purpose to bring the computer-assisted solution of the first nontrivial case of optimal Lagrange interpolation to wider attention and to stimulate research of the higher-degree cases. This is why our style of writing is expository.

Galois stratification and ACFA

We prove a direct image theorem stating that the direct image of a Galois formula by a morphism of difference schemes is equivalent to a Galois formula modulo the theory ACFA of existentially closed difference fields. As a consequence, we obtain an effective quantifier elimination procedure and a precise algebraic-geometric description of definable sets over existentially closed difference fields in terms of twisted Galois formulae associated with finite Galois difference ring/scheme covers.

Quantifier elimination in quasianalytic structures via non-standard analysis

The paper is a continuation of our earlier article where we developed a theory of active and non-active infinitesimals and intended to establish quantifier elimination in quasianalytic structures. That article, however, did not attain full generality, which refers to one of its results, namely the theorem on an active infinitesimal, playing an essential role in our non-standard analysis. The general case was covered in our subsequent preprint, which constitutes a basis for the approach presented here. We also provide a quasianalytic exposition of the results concerning rectilinearization of terms and of definable functions from our earlier research. It will be used to demonstrate a quasianalytic structure corresponding to a Denjoy-Carleman class which, unlike the classical analytic structure, does not admit quantifier elimination in the language of restricted quasianalytic functions augmented merely by the reciprocal function. More precisely, we construct a plane definable curve, which indicates both that the classical theorem by J. Denef and L. van den Dries as well as \L{}ojasiewicz's theorem that every subanalytic curve is semianalytic are no longer true for quasianalytic structures. Besides rectilinearization of terms, our construction makes use of some theorems on power substitution for Denjoy-Carleman classes and on non-extendability of quasianalytic function germs. The last result relies on Grothendieck's factorization and open mapping theorems for (LF)-spaces. Note finally that this paper comprises our earlier preprints on the subject from May 2012.

Generalized Three-body Problem and the Instability of the Core-halo Objects in Binary Systems

Goal of the presented research is to construct simplified model of the core-halo structures in binary systems. Examples are provided by Thorne-Zytkov objects, hot Jupiters, protoplanets with large moons, red supergiants in binaries and globular clusters with central black hole. Instability criteria due to resonance between internal and orbital frequencies in such a systems has been derived. To achieve assumed goals, generalized planar circular restricted three body problem is investigated with one of the point masses, $M$, replaced with spherical body of finite size. Mechanical system under consideration includes two large masses $m$ and $M$ and the test body with small mass $\mu$. Only gravitational interactions are considered. Equations of motion are presented, and linear instability criteria are derived using quantifier elimination. Motion of the test mass $\mu$ is shown to be unstable due to resonance between orbital and internal frequencies if $\frac{M}{d^3} < \frac{4}{3} \pi \rho < \frac{ M + 3 m \left( 1+\mu/M \right)^{-1}}{d^3}$, where $\rho$ is the central density of mass $M$, and $d$ distance between masses $m$ and $M$ (circular orbit diameter). The above result is important for core-collapse supernova theory, with mass $\mu$ identified with helium core of the exploding massive star. The instability cause off-center supernova "ignition" relative to the center-of-mass of the hydrogen envelope. The instability is also inevitable during protoplanet growth, with hypothetical ejection of the rocky core from gas giants and formation of the "puffy planets" due to resonance with orbital frequency. Hypothetical central intermediate black holes of the globular clusters are also in unstable position with respect to perturbations caused by the Galaxy.

Tensor products of valued fields

We give a short argument why the tensor product semi-norm on K ⊗ k L is multiplicative when k is an algebraically closed valued field and K and L are valued extensions (valued in R). When the valuation on k is non-trivial we use the fact that ACVF, the theory of algebraically closed (non-trivially) valued fields, has quantifier elimination.

On polymorphism-homogeneous relational structures and their clones

A relational structure is homomorphism-homogeneous if every homomorphism between finite substructures extends to an endomorphism of the structure. This notion was introduced recently by Cameron and Ne\v{s}et\v{r}il. In this paper we consider a strengthening of homomorphism-homogeneity --- we call a relational structure polymorphism-homogeneous if every partial polymorphism with a finite domain extends to a global polymorphism of the structure. It turns out that this notion (under various names and in completely different contexts) has been existing in algebraic literature for at least 30 years. Motivated by this observation, we dedicate this paper to the topic of polymorphism-homogeneous structures. We study polymorphism-homogeneity from a model-theoretic, an algebraic, and a combinatorial point of view. E.g., we study structures that have quantifier elimination for positive primitive formulae, and show that this notion is equivalent to polymorphism-homogeneity for weakly oligomorphic structures. We demonstrate how the Baker-Pixley theorem can be used to show that polymorphism-homogeneity is a decidable property for finite relational structures. Eventually, we completely characterize the countable polymorphism-homogeneous graphs, the polymorphism-homogeneous posets of arbitrary size, and the countable polymorphism-homogeneous strict posets.

A Complexity Theory of Constructible Functions and Sheaves

In this paper we introduce constructible analogs of the discrete complexity classes $\mathbf{VP}$ and $\mathbf{VNP}$ of sequences of functions. The functions in the new definitions are constructible functions on $\mathbb{R}^n$ or $\mathbb{C}^n$. We define a class of sequences of constructible functions that play a role analogous to that of $\mathbf{VP}$ in the more classical theory. The class analogous to $\mathbf{VNP}$ is defined using Euler integration. We discuss several examples, develop a theory of completeness, and pose a conjecture analogous to the $\mathbf{VP}$ vs. $\mathbf{VNP}$ conjecture in the classical case. In the second part of the paper we extend the notions of complexity classes to sequences of constructible sheaves over $\mathbb{R}^n$ (or its one point compactification). We introduce a class of sequences of simple constructible sheaves, that could be seen as the sheaf-theoretic analog of the Blum-Shub-Smale class $\mathbf{P}_{\mathbb{R}}$. We also define a hierarchy of complexity classes of sheaves mirroring the polynomial hierarchy, $\mathbf{PH}_{\mathbb{R}}$, in the B-S-S theory. We prove a singly exponential upper bound on the topological complexity of the sheaves in this hierarchy mirroring a similar result in the B-S-S setting. We obtain as a result an algorithm with singly exponential complexity for a sheaf-theoretic variant of the real quantifier elimination problem. We pose the natural sheaf-theoretic analogs of the classical $\mathbf{P}$ vs. $\mathbf{NP}$ question, and also discuss a connection with Toda's theorem from discrete complexity theory in the context of constructible sheaves. We also discuss possible generalizations of the questions in complexity theory related to separation of complexity classes to more general categories via sequences of adjoint pairs of functors.

Generating property-directed potential invariants by quantifier elimination in a k-induction-based framework

This paper addresses the issue of potential invariant generation in the formal analysis of transition systems with k-induction, in the linear real/integer arithmetic fragment. First, quantifier elimination is used to find parameters for generic templates such that the said templates become inductive. Second, a backward analysis, also using quantifier elimination, outputs preimages of the negation of the proof objective, viewed as unauthorized states, or gray states. Two heuristics are proposed to take advantage of this source of information and generate potential invariants: a thorough exploration of the possible partitionings of the gray state space, and an inexact exploration regrouping and over-approximating disjoint areas of the gray state space. Both aim at discovering hidden relations between state variables. K-induction is used to isolate actual invariants and to check if they make the proof objective inductive. These heuristics can be used on the first preimage of the backward exploration, and each time a new one is output, refining the information on the gray states. We show, on examples of interest in the application field of critical embedded systems, that our approach is able to prove properties for which other academic or commercial tools fail. The different methods are introduced as components of a collaborative formal verification framework based on k-induction and are motivated through two examples, one of which was provided by Rockwell Collins.

Special algorithm for stability analysis of multistable biological regulatory systems

We consider the problem of counting (stable) equilibriums of an important family of algebraic differential equations modeling multistable biological regulatory systems. The problem can be solved, in principle, using real quantifier elimination algorithms, in particular real root classification algorithms. However, it is well known that they can handle only very small cases due to the enormous computing time requirements. In this paper, we present a special algorithm which is much more efficient than the general methods. Its efficiency comes from the exploitation of certain interesting structures of the family of differential equations.

Quantifier elimination for a class of exponential polynomial formulas

Quantifier elimination is a foundational issue in the field of algebraic and logic computation. In first-order logic, every formula is well composed of atomic formulas by negation, conjunction, disjunction, and introducing quantifiers. It is often made quite complicated by the occurrences of quantifiers and nonlinear functions in atomic formulas. In this paper, we study a class of quantified exponential polynomial formulas extending polynomial ones, which allows the exponential to appear in the first variable. We then design a quantifier elimination procedure for these formulas. It adopts the scheme of cylindrical decomposition that consists of four phases—projection, isolation, lifting, and solution formula construction. For the non-algebraic representation, the triangular systems are introduced to define transcendental coordinates of sample points. Based on that, our cylindrical decomposition produces projections for input variables only. Hence the procedure is direct and effective.

Quantifier elimination by dependency sequents

We consider the problem of existential quantifier elimination for Boolean CNF formulas. We present a new method for solving this problem called Derivation of Dependency-Sequents (DDS). A Dependency-sequent (D-sequent) is used to record that a set of quantified variables is redundant under a partial assignment. We introduce the join operation that produces new D-sequents from existing ones. We show that DDS is compositional, i.e., if our input formula is a conjunction of independent formulas, DDS automatically recognizes and exploits this information. We introduce an algorithm based on DDS and present experimental results demonstrating its potential.

Solving strong controllability of temporal problems with uncertainty using SMT

Temporal Problems (TPs) represent constraints over the timing of activities, as arising in many applications such as scheduling and temporal planning. A TP with uncertainty (TPU) is characterized by activities with uncontrollable duration. Different classes of TPU are possible, depending on the Boolean structure of the constraints: we have simple (STPU), constraint satisfaction (TCSPU), and disjunctive (DTPU) temporal problems with uncertainty. In this paper we tackle the problem of strong controllability, i.e. finding an assignment to all the controllable time points, such that the constraints are fulfilled under any possible assignment of uncontrollable time points. Our approach casts the problem in the framework of Satisfiability Modulo Theory (SMT), where the uncertainty of durations can be modeled by means of universal quantifiers. The use of quantifier elimination techniques leads to quantifier-free encodings, which are in turn solved with efficient SMT solvers. We obtain the first practical and comprehensive solution for strong controllability. We provide a family of efficient encodings, that are able to exploit the specific structure of the problem. The approach has been implemented, and experimentally evaluated over a large set of benchmarks. The results clearly demonstrate that the proposed approach is feasible, and outperforms the best state-of-the-art competitors, when available.

Existential type systems between Church and Curry style (type-free style)

We study type checking, typability, and type inference problems for second-order existential systems in type-free style. The type-free style is an intermediate representation of terms between the Curry style and the Church style. Terms in the type-free style contain information on where the existential quantifier elimination and introduction take place, but omit the information on which types are involved. We show that all the problems are undecidable employing reduction of second-order unification for considerably restricted special instances in flat form. These results hold even in a fragment of the existential system where →,∃ are only used and in case of predicative fragments of the existential systems.

Safety verification of semi-algebraic dynamical systems via inductive invariant

To verify the safety of nonlinear dynamical systems based on inductive invariants, key issues include defining the most complete inductive condition and discovering an inductive invariant that satisfies the specified inductive condition. In this paper, to lay a solid foundation for future research into the safety verification of semi-algebraic dynamical systems, we first establish a formal framework for evaluating the quality of continuous inductive conditions. In addition, we propose a new complete and computable inductive condition for verifying the safety of semi-algebraic dynamical systems. Compared with the existing complete and computable inductive condition, this new inductive condition can be easily adapted to achieve a set of sufficient inductive conditions with different level of conservativeness and computational complexity, which provides us with a means to trade off between the verification power and complexity. These inductive conditions can be solved by quantifier elimination and SMT solvers.

Model Theories of Set Theories and Type Theory

This paper is divided into three parts. In the first part, we review the historical background of a system of logic devised by Henry S. Leonard to allow for reasoning using existence as a predicate. In the second part, we consider various directions in which his logic could be further developed, syntactically, semantically, and as an adjunct to quantifier elimination and set theory. In the third and final part, we develop proofs of some underlying results of his logic, using modern notation but retaining his axioms and rules of inference.

Quantifier-free encoding of invariants for hybrid systems

Hybrid systems are a clean modeling framework for embedded systems, which feature integrated discrete and continuous dynamics. A well-known source of complexity comes from the time invariants, which represent an implicit quantification of a constraint over all time points of a continuous transition. Emerging techniques based on Satisfiability Modulo Theory (SMT) have been found promising for the verification and validation of hybrid systems because they combine discrete reasoning with solvers for first-order theories. However, these techniques are efficient for quantifier-free theories and the current approaches have so far either ignored time invariants or have been limited to hybrid systems with linear constraints. In this paper, we propose a new method that encodes a class of hybrid systems into transition systems with quantifier-free formulas. The method does not rely on expensive quantifier elimination procedures. Rather, it exploits the sequential nature of the transition system to split the continuous evolution enforcing the invariants on the discrete time points. This way, we can encode all hybrid systems whose invariants can be expressed in terms of polynomial constraints. This pushes the application of SMT-based techniques beyond the standard linear case.

Inductive invariant generation via abductive inference

This paper presents a new method for generating inductive loop invariants that are expressible as boolean combinations of linear integer constraints. The key idea underlying our technique is to perform a backtracking search that combines Hoare-style verification condition generation with a logical abduction procedure based on quantifier elimination to speculate candidate invariants. Starting with true, our method iteratively strengthens loop invariants until they are inductive and strong enough to verify the program. A key feature of our technique is that it is lazy: It only infers those invariants that are necessary for verifying program correctness. Furthermore, our technique can infer arbitrary boolean combinations (including disjunctions) of linear invariants. We have implemented the proposed approach in a tool called HOLA. Our experiments demonstrate that HOLA can infer interesting invariants that are beyond the reach of existing state-of-the-art invariant generation tools.