Nicolai Vorobjov

Complexity of null- and Positivstellensatz proofs

Abstract We introduce two versions of proof systems dealing with systems of inequalities: Positivstellensatz refutations and Positivstellensatz calculus. For both systems we prove the lower bounds on degrees and lengths of derivations for the example due to Lazard, Mora and Philippon. These bounds are sharp, as well as they are for the Nullstellensatz refutations and for the polynomial calculus. The bounds demonstrate a gap between the Null- and Positivstellensatz refutations on one hand, and the polynomial calculus and Positivstellensatz calculus on the other.

Betti Numbers of Semialgebraic and Sub-Pfaffian Sets

Let X be a subset in [-1, 1](n0) subset of R-n0 defined by the formula X = {x(0) \ Q(1)x(1) Q(2)x(2) ... Q(v)x(v) ((x(0), x(1), ...,x(v)) is an element of X-v)}, where Q(i) is an element of {There Exists, For All}, Q(i) not equal Q(i+1), x(i) is an element of [-1, 1](ni), and X-v may be either an open or a closed set in being the difference between a finite CW-complex and its subcomplex. An upper bound on each Betti number of X is expressed via a sum of Betti numbers of some sets defined by quantifier-free formulae involving X-v. In important particular cases of semialgebraic and semi-Pfaffian sets defined by quantifier-free formulae with polynomials and Pfaffian functions respectively, upper bounds on Betti numbers of X-v are well known. The results allow to extend the bounds to sets defined with quantifiers, in particular to sub-Pfaffian sets.

Approximation of definable sets by compact families, and upper bounds on homotopy and homology

We prove new upper bounds on homotopy and homology groups of o-minimal sets in terms of their approximations by compact o-minimal sets. In particular, we improve the known upper bounds on Betti numbers of semialgebraic sets defined by quantifier-free formulae and obtain, for the first time, a singly exponential bound on Betti numbers of sub-Pfaffian sets.

Betti Numbers of Semialgebraic Sets Defined by Quantifier-Free Formulae

Abstract Let X be a semialgebraic set in Rn defined by a Boolean combination of atomic formulae of the kind h * 0 where * \in { >, \ge, = }, deg(h) < d, and the number of distinct polynomials h is k. We prove that the sum of Betti numbers of X is less than O(k2d)n.

Pfaffian Hybrid Systems

It is well known that in an o-minimal hybrid system the continuous and discrete components can be separated, and therefore the problem of finite bisimulation reduces to the same problem for a transition system associated with a continuous dynamical system. It was recently proved by several authors that under certain natural assumptions such finite bisimulation exists. In the paper we consider o-minimal systems defined by Pfaffian functions, either implicitly (via triangular systems of ordinary differential equations) or explicitly (by means of semi-Pfaffian maps). We give explicit upper bounds on the sizes of bisimulations as functions of formats of initial dynamical systems. We also suggest an algorithm with an elementary (doubly-exponential) upper complexity bound for computing finite bisimulations of these systems.

Bounds on numers of vectors of multiplicities for polynomials which are easy to compute

Let F be an algebraically closed field of zero characteristic, a polynomial @@@@ ∈ F[<italic>X</italic><subscrpt>1</subscrpt>, … , <italic>X<subscrpt>n</subscrpt></italic> have a multiplicative complexity <italic>r</italic> and ƒ<subscrpt>1</subscrpt>, … ƒ<subscrpt><italic>k</italic></subscrpt> ∈ F[<italic>X</italic><subscrpt>1</subscrpt>, … , <italic>X<subscrpt>n</subscrpt></italic>] be some polynomials of degrees not exceeding <italic>d</italic>, such that @@@@ = ƒ<subscrpt>1</subscrpt> = ··· = ƒ<subscrpt><italic>k</italic></subscrpt> = 0 has a finite number of roots. We show that the number of possible distinct vectors of multiplicities of these roots is small when <italic>r, d</italic> and <italic>k</italic> are small. As technical tools we design algorithms which produce Gröbner bases and vectors of multiplicities of the roots for a parametric zero-dimensional system. The complexities of these algorithms are singly exponential. We also describe an algorithm for parametric absolute factorization of multivariate polynomials. This algorithm has subexponential complexity in the case of a small (relative to the number of variables) degree of the polynomials.

Semi-monotone sets

A coordinate cone in R^n is an intersection of some coordinate hyperplanes and open coordinate half-spaces. A semi-monotone set is a defnable in an o-minimal structure over the reals, open bounded subset of R^n such that its intersection with any translation of any coordinate cone is connected. This can be viewed as a generalization of the convexity property. Semi-monotone sets have a number of interesting geometric and combinatorial properties. The main result of the paper is that every semi-monotone set is a topological regular cell.

New complexity bounds for cylindrical decompositions of sub-pfaffian sets

Tarski-Seidenberg principle plays a key role in many applications and algorithm of computer algebra. Moreover it is constructive, and some very efficient quantifier elimination algorithms appeared recently. However, Tarski-Seidenberg principle is wrong for first-order theories involving some real analytic functions (e.g. an exponential function). In this case a weaker statement is sometimes true, a possibility to eliminate one sort of quantifiers (either ∀ or ∃). We construct a new algorithm for a cylindrical cell decomposition of a closed cube In ⊄ Rn compatible with a semianalytic subset S ⊄ In, defined by Pfaffian functions. In particular the algorithm is able to eliminate one sort of quantifiers from a first-order formula. The complexity bound of the algorithm is doubly exponential in n2.

Upper and lower bounds on sizes of finite bisimulations of pfaffian hybrid systems

In this paper we study a class of hybrid systems defined by Pfaffian maps. It is a sub-class of o-minimal hybrid systems which capture rich continuous dynamics and yet can be studied using finite bisimulations. The existence of finite bisimulations for o-minimal dynamical and hybrid systems has been shown by several authors (see e.g. [3,4,13]). The next natural question to investigate is how the sizes of such bisimulations can be bounded. The first step in this direction was done in [10] where a double exponential upper bound was shown for Pfaffian dynamical and hybrid systems. In the present paper we improve this bound to a single exponential upper bound. Moreover we show that this bound is tight in general, by exhibiting a parameterized class of systems on which the exponential bound is attained. The bounds provide a basis for designing efficient algorithms for computing bisimulations, solving reachability and motion planning problems.

Computing the complexification of a semi-algebraic set

We describe an algorithm for producing the smallest complex algebralc variety containing a given semi-algebraic set S, and all the irreducible components of S. Let S be defined by s polynomials of degrees less than d with integer coefficients of bit lengths less than A4. Then the complexity of the algorithm is bounded from above by a polynomial in M, Sn, dn’. The degree of the complexification is less than sndo ‘n), while the degrees of polynomials defining the complexification and irreducible components are less than do(n)