Results 1  10
of
21
InductiveDataType Systems
, 2002
"... In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schem ..."
Abstract

Cited by 821 (23 self)
 Add to MetaCart
In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schema", whichgeneral39I theusual recursor definitions fornatural numbers and simil9 "basic inductive types". This combined lmbined was shown to bestrongl normalIk39f The purpose of this paper is toreformul33 and extend theGeneral Schema in order to make it easil extensibl3 to capture a more general cler of inductive types, cals, "strictly positive", and to ease the strong normalgAg9Ik proof of theresulGGg system. Thisresul provides a computation model for the combination of anal"DAfGI specification language based on abstract data types and of astrongl typed functional language with strictly positive inductive types.
An efficient surface intersection algorithm based on the lower dimensional formulation
 ACM TRANSACTIONS ON GRAPHICS
, 1997
"... We present an efficient algorithm to compute the intersection of algebraic and NURBS surfaces. Our approach is based on combining the marching methods with the algebraic formulation. In particular, we propose a matrix representation for the intersection curve and compute it accurately using matrix c ..."
Abstract

Cited by 72 (18 self)
 Add to MetaCart
We present an efficient algorithm to compute the intersection of algebraic and NURBS surfaces. Our approach is based on combining the marching methods with the algebraic formulation. In particular, we propose a matrix representation for the intersection curve and compute it accurately using matrix computations. We present algorithms to compute a start point oneach component of the intersection curve (both open and closed components), detect the presence of singularities, and find all the curve branches near the singularity. We also suggest methods to compute the step size during tracing to prevent component jumping. The algorithm runs an order of magnitude faster than previously published robust algorithms. The complexity of the algorithm is output sensitive.
Improved Algorithms for Sign Determination and Existential Quantifier Elimination
 THE COMPUTER JOURNAL
, 1993
"... Recently there has been a lot of activity in ... In this paper we describe a new sign determination method based on the earlier algorithm, but with two advantages: (i) It is faster in the univariate case, and (ii) In the general case, it allows purely symbolic quantifier elimination in pseudopolyno ..."
Abstract

Cited by 37 (1 self)
 Add to MetaCart
(Show Context)
Recently there has been a lot of activity in ... In this paper we describe a new sign determination method based on the earlier algorithm, but with two advantages: (i) It is faster in the univariate case, and (ii) In the general case, it allows purely symbolic quantifier elimination in pseudopolynomial time. By purely symbolic, we mean that it is possible to eliminate a quantified variable from a system of polynomials no matter what the coefficient values are. The previous methods required the coefficients to be themselves polynomials in other variables. Our new
Firstorder queries on finite structures over the reals
"... We investigate properties of finite relational structures over the reals expressed by firstorder sentences whose predicates are the relations of the structure plus arbitrary polynomial inequalities, and whose quantifiers can range over the whole set of reals. In constraint programming terminology, ..."
Abstract

Cited by 32 (2 self)
 Add to MetaCart
We investigate properties of finite relational structures over the reals expressed by firstorder sentences whose predicates are the relations of the structure plus arbitrary polynomial inequalities, and whose quantifiers can range over the whole set of reals. In constraint programming terminology, this corresponds to Boolean real polynomial constraint queries on finite structures. The fact that quantifiers range over all reals seems crucial � however, we observe that each sentence in the firstorder theory of the reals can be evaluated by letting each quantifier range over only a finite set of real numbers without changing its truth value. Inspired by this observation, we then show that when all polynomials used are linear, each query can be expressed uniformly on all finite structures by a sentence of which the quantifiers range only over the finite domain of the structure. In other words, linear constraint programming on finite structures can be reduced to ordinary query evaluation as usual in finite model theory and databases. Moreover, if only "generic" queries are taken into consideration, we show that this can be reduced even further by proving that such
Linear constraint query languages: Expressive power and complexity
 Logic and Computational Complexity
, 1994
"... Abstract. We giveanAC 0 upper bound on the complexity of rstoder queries over (in nite) databases de ned by restricted linear constraints. This result enables us to deduce the nonexpressibility ofvarious usual queries, such as the parity of the cardinality of a set or the connectivity of a graph i ..."
Abstract

Cited by 26 (12 self)
 Add to MetaCart
(Show Context)
Abstract. We giveanAC 0 upper bound on the complexity of rstoder queries over (in nite) databases de ned by restricted linear constraints. This result enables us to deduce the nonexpressibility ofvarious usual queries, such as the parity of the cardinality of a set or the connectivity of a graph in rstorder logic with linear constraints. 1
Quantifier Elimination by Lazy Model Enumeration
 of Lecture Notes in Computer Science
, 2010
"... Abstract We propose a quantifier elimination scheme based on nested lazy model enumeration through SMTsolving, and projections. This scheme may be applied to any logic that fulfills certain conditions; we illustrate it for linear real arithmetic. The quantifier elimination problem for linear real ..."
Abstract

Cited by 22 (3 self)
 Add to MetaCart
Abstract We propose a quantifier elimination scheme based on nested lazy model enumeration through SMTsolving, and projections. This scheme may be applied to any logic that fulfills certain conditions; we illustrate it for linear real arithmetic. The quantifier elimination problem for linear real arithmetic is doubly exponential in the worst case, and so is our method. We have implemented it and benchmarked it against other methods from the literature.
Motion Planning for Multiple Robots
, 1998
"... We study the motionplanning problem for pairs and triples of robots operating in a shared workspace containing n obstacles. A standard way to solve such problems is to view the collection of robots as one composite robot, whose number of degrees of freedom is d, the sum of the numbers of degrees o ..."
Abstract

Cited by 19 (1 self)
 Add to MetaCart
We study the motionplanning problem for pairs and triples of robots operating in a shared workspace containing n obstacles. A standard way to solve such problems is to view the collection of robots as one composite robot, whose number of degrees of freedom is d, the sum of the numbers of degrees of freedom of the individual robots. We show that it is sufficient to consider a constant number of robot systems whose number of degrees of freedom is at most d \Gamma 1 for pairs of robots, and d \Gamma 2 for triples. (The result for a pair assumes that the sum of the number of degrees of freedom of the robots constituting the pair reduces by at least one if the robots are required to stay in contact; for triples a similar assumption is made. Moreover, for triples we need to assume that a solution with positive clearance exists.) We use this to obtain an O(n d ) time algorithm to solve the motionplanning problem for a pair of robots; this is one order of magnitude faster than what the st...
Complexity of cylindrical decompositions of subPfaffian sets
 J. Pure Appl. Algebra
, 2001
"... Abstract. We construct an algorithm for a cylindrical cell decomposition of aclosedcubeI n ⊂ R n compatible with a “restricted ” subPfaffian subset Y ⊂ I n, provided an oracle deciding consistency of a system of Pfaffian equations and inequalities is given. In particular, the algorithm produces the ..."
Abstract

Cited by 18 (8 self)
 Add to MetaCart
(Show Context)
Abstract. We construct an algorithm for a cylindrical cell decomposition of aclosedcubeI n ⊂ R n compatible with a “restricted ” subPfaffian subset Y ⊂ I n, provided an oracle deciding consistency of a system of Pfaffian equations and inequalities is given. In particular, the algorithm produces the complement ˜Y = I n \ Y. The complexity bound of the algorithm, the number and formats of cells are doubly exponential in n 3.
NumericSymbolic Algorithms for Evaluating OneDimensional Algebraic Sets
 APPEARED IN PROCEEDINGS OF ISSAC'95
, 1995
"... We present efficient algorithms based on a combination of numeric and symbolic techniques for evaluating onedimensional algebraic sets in a subset of the real domain. Given a description of a onedimensional algebraic set, we compute its projection using resultants. We represent the resulting plane ..."
Abstract

Cited by 17 (5 self)
 Add to MetaCart
We present efficient algorithms based on a combination of numeric and symbolic techniques for evaluating onedimensional algebraic sets in a subset of the real domain. Given a description of a onedimensional algebraic set, we compute its projection using resultants. We represent the resulting plane curve as a singular set of a matrix polynomial as opposed to roots of a bivariate polynomial. Given the matrix formulation, we make use of algorithms from numerical linear algebra to compute start points on all the components, partition the domain such that each resulting region contains only one component and evaluate it accurately using marching methods. We also present techniques to handle singularities for wellconditioned inputs. The resulting algorithm is iterative and its complexity is output sensitive. It has been implemented in oatingpoint arithmetic and we highlight its performance in the context of computing intersection of highdegree algebraic surfaces.
Reachability and Connectivity Queries in Constraint Databases
"... It is known that standard query languages for constraint databases lack the power to express connectivity properties. Such properties are important in the context of geographical databases, where one naturally wishes to ask queries about connectivity (what are the connected components of a given set ..."
Abstract

Cited by 17 (2 self)
 Add to MetaCart
It is known that standard query languages for constraint databases lack the power to express connectivity properties. Such properties are important in the context of geographical databases, where one naturally wishes to ask queries about connectivity (what are the connected components of a given set?) or reachability (is there a path from A to B that lies entirely in a given region?). No existing constraint query languages that allow closed form evaluation can express these properties. In the rst part of the paper, we show that in principle there is no obstacle to getting closed languages that can express connectivity and reachability queries. In fact, we show that adding any topological property to standard languages like FO+Lin and FO+Poly results in a closed language. In the second part of the paper, we look for tractable closed languages for expressing reachability and connectivity queries. We introduce path logic, which allows one to state properties of paths with respect to given regions. We show that it is closed, has polynomial time data complexity for linear and polynomial constraints, and can express a large number of reachability properties beyond simple connectivity. Query evaluation in the logic involves obtaining a discrete abstraction of a continuous path,