Volker Weispfenning

Quantifier Elimination for Real Algebra — the Quadratic Case and Beyond

Abstract. We present a new, “elementary” quantifier elimination method for various special cases of the general quantifier elimination problem for the first-order theory of real numbers. These include the elimination of one existential quantifier ∃x in front of quantifier-free formulas restricted by a non-trivial quadratic equation in x (the case considered also in [7]), and more generally in front of arbitrary quantifier-free formulas involving only polynomials that are quadratic in x. The method generalizes the linear quantifier elimination method by virtual substitution of test terms in [9]. It yields a quantifier elimination method for an arbitrary number of quantifiers in certain formulas involving only linear and quadratic occurrences of the quantified variables. Moreover, for existential formulas ϕ of this kind it yields sample answers to the query represented by ϕ. The method is implemented in REDUCE as part of the REDLOG package (see [4, 5]). Experiments show that the method is applicable to a range of benchmark examples, where it runs in most cases significantly faster than the QEPCAD package of Collins and Hong. An extension of the method to higher degree polynomials using Thom’s lemma is sketched.

Constructing Universal Groebner Bases

A universal Grobner basis is a finite basis for a polynomial ideal that has the Grobner property with respect to all admissible term-orders. Let R be a commutative polynomial ring over a field K, or more generally a non-commutative polynomial ring of solvable type over K (see [KRW]). We show, how to construct and characterize left, right, two-sided, and reduced universal Grobner bases in R. Moreover, we extend the upper complexity bounds in [We4] to the construction of universal Grobner bases. Finally, we prove the stability of universal Grobner bases under specialization of coefficients. All these results have counterparts for polynomial rings over commutative regular rings (comp. [We3]).

A note on ℵ0-categorical model-companions

Let T be a countable first-order theory. This note is concerned with necessary and sufficient conditions on the Lindenbaum algebras B,(T) associated with T, in order that T has an No-categorical model-companion. The problem is motivated by a theorem of Saracino [9] saying that any N0-categorical theory which has only infinite models has an N0-categorical model-companion. Unfortunately, this remarkable result seems to be of little practical value for finding No-categorical model-companions. Since the proof of the theorem is based on the classical characterization of No-categorical theories in terms of the finiteness of the Lindenbaum algebras B,(T), it is natural to look for weaker conditions on B,(T) that are necessary and sufficient for the existence of an No-categorical modelcompanion. Such conditions have been obtained by Pouzet [7] and Simmons [11] (compare also Hirschfetd-Wheeler [6], Theorem 6.4). We give various other equivalent conditions and new proof for the equivalence which is simpler than the above, and gives a more explicit description of axioms for the model-companion. As a corollary we obtain a very handy condition for the case of a universal theory T in a finite language such that Mod(T) has the amalgamation property. This condition provides an easy proof for the existence of an N0-categorical modelcompletion e.g. for the following theories: graphs, ordered sets, partially ordered sets, distributive lattices, Boolean algebras, Stone algebras, pseudocomplemented distributive lattices satisfying the Lee-identity L z (see Gr~itzer, Lattice theory, [5]). Let L be a countable first-order language and let Tbe a consistent theory in L. We denote the set ofalI formulas (all 3k-formulas, all ¥k-formulas in L with free variables among x I . . . . . x., n~0 , by Qn(Ek,..4k. ). For ~ps Q., let (pT denote the equivalence class of q~ in Q. with respect to the equivalence relation (p~ tpc~Tl---q).~--W. We denote the set of all q)r with q)eQ.(cpe Ek., ~o~ Ak.) by B.(T)(Ek.(T), Ak.(T))and we put Qk.(T)= Ek.(T)c~ Ak.(T). Each of these sets forms a distributive lattice with 0 and 1 under the operations c~, w induced by A, V ; the sets B. (T)and Qk.(T) even form Boolean algebras with complementation ~ induced by 7 . [Whenever we consider Qo0(T), we require that L contains at least one individual constant so that Qoo is not empty,] The algebras B.(T) are known as the Lindenbaum algebras associated with T. For every keN, Ekn(Y), Akn(T), Qkn(T) form 0 1-sublattices of

Semilinear Motion Planning in REDLOG

Abstract. We study a new type of motion planning problem in dimension 2 and 3 via linear and quadratic quantifier elimination. The object to be moved and the free space are both semilinear sets with no convexity assumptions. The admissible motions are finite continuous sequences of translations along prescribed directions. When the number of translations is bounded in advance, then the corresponding path finding problem can be modelled and solved as a linear quantifier elimination problem. Moreover the problem to find a shortest or almost shortest admissible path can be modelled as a special quadratic quantifier elimination problem. We give upper complexity bounds on these problems, report experimental results using the elimination facilities of the REDLOG package of REDUCE, and indicate a possible application.

Model Theory of Abelian l-Groups

All groups in this chapter will be commutative, written additively. So an l-group is an Abelian lattice-ordered group and an o-group is an Abelian linearly ordered group.

Existential equivalence of ordered abelian groups with parameters

SummaryIn [GK], Gurevich and Kokorin proved that any two non-trivial ordered abelian groups (o-groups, for short) satisfy the same existential sentences. Let nowG, H be non-trivialo-groups with a commono-subgroupG0. We determine whetherG andH are existentially equivalent overG0. As a corollary, we obtain algebraic criteria for deciding, whether ano-subgroupG is existentially closed in ano-groupH. Corresponding results are proved foro-groups in which congruences are regarded as atomic relations.

The complexity of the word problem for abelian l -groups

Abstract The word problem for abelian lattice-ordered groups and the universal theory of these groups is co-NP-complete; the existential theory of nontrivial abelian l -groups is NP-complete. Corresponding results hold for abelian l -groups of specified dimension.

On the Number of Term Orders

AbstractBy an admissible order on a finite subsetQ of ℚn we mean the restriction toQ of a linear order on ℚn compatible with the group structure of ℚn and such that ℕn is contained in the positive cone of the order. We first derive upper and lower bounds on the number of admissible orders on a given setQ in terms of the dimensionn and the cardinality ofQ. Better estimates are possible if the setQ enjoys symmetry properties and in the case whereQ is a discrete hyperbox of the form

Quantifier elimination for modules.

\mathop \Pi \limits_{k = 1}^n [1,d_k ].