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The starthng success of the Rabm-Strassen-Solovay pnmahty algorithm, together with the intriguing foundattonal posstbthty that axtoms of randomness may constttute a useful fundamental source of mathemaucal truth independent of the standard axmmaUc structure of mathemaUcs, suggests a wgorous search for probabdisuc algonthms In dlustratmn of this observaUon, vanous fast probabdlsttc algonthms, with probability of correctness guaranteed a prion, are presented for testing polynomial ldentmes and propemes of systems of polynomials. Ancdlary fast algorithms for calculating resultants and Sturm sequences are given. Probabilistlc calculatton in real anthmetlc, prewously considered by Davis, is justified ngorously, but only in a special case. Theorems of elementary geometry can be proved much more efficiently by the techmques presented than by any known arttficml-mtelhgence approach KEY WORDS AND PHRASES: polynommls, polynomml algonthms, probabdtsuc algorithms CR CATEGORIES. 5.21, 5.7 Integer Probabilistic Calculations for Multivariate PolynomialsThe startling success of the Rabin-Strassen-Solovay algorithm (see Rabin [17]), together with the intriguing foundational possibility that axioms of randomness may constitute a useful fundamental source of mathematical truth independent of, but supplementary to, the standard axiomatic structure of mathematics (see Chaitin and Schwartz [3l), suggests that probabilistic algorithms ought to be sought vigorously. As an illustration of what may be possible, this paper presents probabilistic algorithms for testing asserted multivariable polynomial identities Q = R, as well as other asserted or conjectured relationships between sets of polynomials, e.g., the assertion that one polynomial Q belongs to the ideal generated by finitely many others.The technique that we use is essentially elementary. Given a purported polynomial identity, we can always write it as Q ffi 0. We do not suppose that the Q presented to us for testing is given in standard simplified polynomial form. For example, if we did not immediately recognize its truth, we might wish to test the identity (x + y)(x -y) -x 2 + y2 = 0. Indeed, if we write Q for the standard simplified form of Q, what we want is precisely a test to determine whether all the coefficients of Q are zero.We allow our polynomials to have coefficients in any field or integral domain F. At some points in our argument the condition that F should be infinite will play an essential role. We write deg(Q) for the degree of Q and [ S[ for the cardinality of a set S.Note that it will generally be trivial to develop upper bounds for deg(Q) directly from the expression structure of Q.

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We study the criteria under which an object can be gripped by a multifingered dexterous hand, assuming no static friction between the object and the fingers; such grips are called positive grips. We study three cases in detail: (i) the body is at equilibrium, (ii) the body is under some constant external force/torque, and (iii) the body is under a varying external force/torque. In each case we obtain tight bounds on the number of fingers needed to obtain grip.We also present efficient algorithms to synthesize such positive grips for bounded polyhedral/polygonal objects; the number of fingers employed in the grips synthesized by our algorithms match the above bounds. The algorithms run in time linear in the number of faces/sides. The paper may be of independent interest for its presentation of algorithms arising in the study of positive linear spaces.

Coordinated motion planning for a large number af three-di mensional objects in the presence of obstacles is a computa tional problem whose complexity is important to calibrate. In this paper we show that even the restricted two-dimensional problem for arbitrarily many rectangles in a rectangular region is PSPACE-hard. This result should be viewed as a guide to the difficulty, of the general problem and should lead researchers to consider more tractable restricted classes of motion problems of practical interest.

We present an algorithm that solves a two-dimensional case of the following problem which arises in robotics: Given a body B, and a region bounded by a collection of "walls", either find a continuous motion connecting two given positions and orientations of B during which B avoids collision with the walls, or else establish that no such motion exists. The algorithm is polynomial in the number of walls ( O h S ) if n is the number of walls), but for typical wall configurations can run more efficiently. It is somewhat related to a technique outlined by Reif.

Abstruct-We describe new techniques for model-based recognition of flat objects in 3-D space. The recognition is performed from single gray scale images taken from unknown viewpoints. The objects in the scene may be overlapping and partially occluded. An eficient matching algorithm, which assumes affine approximation to the perspective viewing transformation, is proposed. The algorithm has an off-line model preprocessing (shape representation) phase, which is independent of the scene information, and a recognition phase, based on efficient indexing. It has a Straightforward parallel implementation. The algorithm was successfully tested in recognition of industrial objects appearing in composite occluded scenes.