Efficient Solving of Quantified Inequality Constraints over the Real Numbers

Let a quantified inequality constraint over the reals be a formula in the first-order predicate language over the structure of the real numbers, where the allowed predicate symbols are $\leq$ and $<$. Solving such constraints is an undecidable problem when allowing function symbols such $\sin$ or $\cos$. In the paper we give an algorithm that terminates with a solution for all, except for very special, pathological inputs. We ensure the practical efficiency of this algorithm by employing constraint programming techniques

Automated sequence and motion planning for robotic spatial extrusion of 3D trusses

While robotic spatial extrusion has demonstrated a new and efficient means to fabricate 3D truss structures in architectural scale, a major challenge remains in automatically planning extrusion sequence and robotic motion for trusses with unconstrained topologies. This paper presents the first attempt in the field to rigorously formulate the extrusion sequence and motion planning (SAMP) problem, using a CSP encoding. Furthermore, this research proposes a new hierarchical planning framework to solve the extrusion SAMP problems that usually have a long planning horizon and 3D configuration complexity. By decoupling sequence and motion planning, the planning framework is able to efficiently solve the extrusion sequence, end-effector poses, joint configurations, and transition trajectories for spatial trusses with nonstandard topologies. This paper also presents the first detailed computation data to reveal the runtime bottleneck on solving SAMP problems, which provides insight and comparing baseline for future algorithmic development. Together with the algorithmic results, this paper also presents an open-source and modularized software implementation called Choreo that is machine-agnostic. To demonstrate the power of this algorithmic framework, three case studies, including real fabrication and simulation results, are presented.Comment: 24 pages, 16 figure

Innovative systems for the transportation disadvantaged: towards more efficient and operationally usable planning tools

When considering innovative forms of public transport for specific groups, such as demand responsive services, the challenge is to find a good balance between operational efficiency and 'user friendliness' of the scheduling algorithm even when specialized skills are not available. Regret insertion-based processes have shown their effectiveness in addressing this specific concern. We introduce a new class of hybrid regret measures to understand better why the behaviour of this kind of heuristic is superior to that of other insertion rules. Our analyses show the importance of keeping a good balance between short- and long-term strategies during the solution process. We also use this methodology to investigate the relationship between the number of vehicles needed and total distance covered - the key point of any cost analysis striving for greater efficiency. Against expectations, in most cases decreasing fleet size leads to savings in vehicle mileage, since the heuristic solution is still far from optimality

Using Restarts in Constraint Programming over Finite Domains - An Experimental Evaluation

The use of restart techniques in complete Satisfiability (SAT) algorithms has made solving hard real world instances possible. Without restarts such algorithms could not solve those instances, in practice. State of the art algorithms for SAT use restart techniques, conflict clause recording (nogoods), heuristics based on activity variable in conflict clauses, among others. Algorithms for SAT and Constraint problems share many techniques; however, the use of restart techniques in constraint programming with finite domains (CP(FD)) is not widely used as it is in SAT. We believe that the use of restarts in CP(FD) algorithms could also be the key to efficiently solve hard combinatorial problems. In this PhD thesis we study restarts and associated techniques in CP(FD) solvers. In particular, we propose to including in a CP(FD) solver restarts, nogoods and heuristics based in nogoods as this should improve search algorithms, and, consequently, efficiently solve hard combinatorial problems. We thus intend to: a) implement restart techniques (successfully used in SAT) to solve constraint problems with finite domains; b) implement nogoods (learning) and heuristics based on nogoods, already in use in SAT and associated with restarts; and c) evaluate the use of restarts and the interplay with the other implemented techniques. We have conducted the study in the context of domain splitting backtrack search algorithms with restarts. We have defined domain splitting nogoods that are extracted from the last branch of the search algorithm before the restart. And, inspired by SAT solvers, we were able to use information within those nogoods to successfully help the variable selection heuristics. A frequent restart strategy is also necessary, since our approach learns from restarts

Abstract Logical Model Checking of Infinite-State Systems Using Narrowing

A concurrent system can be naturally specified as a rewrite theory R = (Sigma, E, R) where states are elements of the initial algebra of terms modulo E and concurrent transitions are axiomatized by the rewrite rules R. Under simple conditions, narrowing with rules R modulo equations E can be used to symbolically represent the system\u27s state space by means of terms with logical variables. We call this symbolic representation a "logical state space" and it can also be used for model checking verification of LTL properties. Since in general such a logical state space can be infinite, we propose several abstraction techniques for obtaining either an over-approximation or an under-approximation of the logical state space: (i) a folding abstraction that collapses patterns into more general ones, (ii) an easy-to-check method to define (bisimilar) equational abstractions, and (iii) an iterated bounded model checking method that can detect if a logical state space within a given bound is complete. We also show that folding abstractions can be faithful for safety LTL properties, so that they do not generate any spurious counterexamples. These abstraction methods can be used in combination and, as we illustrate with examples, can be effective in making the logical state space finite. We have implemented these techniques in the Maude system, providing the first narrowing-based LTL model checker we are aware of

Probabilistic constraint reasoning with Monte Carlo integration to robot localization

This work studies the combination of safe and probabilistic reasoning through the hybridization of Monte Carlo integration techniques with continuous constraint programming. In continuous constraint programming there are variables ranging over continuous domains (represented as intervals) together with constraints over them (relations between variables) and the goal is to find values for those variables that satisfy all the constraints (consistent scenarios). Constraint programming “branch-and-prune” algorithms produce safe enclosures of all consistent scenarios. Special proposed algorithms for probabilistic constraint reasoning compute the probability of sets of consistent scenarios which imply the calculation of an integral over these sets (quadrature). In this work we propose to extend the “branch-and-prune” algorithms with Monte Carlo integration techniques to compute such probabilities. This approach can be useful in robotics for localization problems. Traditional approaches are based on probabilistic techniques that search the most likely scenario, which may not satisfy the model constraints. We show how to apply our approach in order to cope with this problem and provide functionality in real time

Interval linear constraint solving in constraint logic programming.

by Chong-kan Chiu.Thesis (M.Phil.)--Chinese University of Hong Kong, 1994.Includes bibliographical references (leaves 97-103).Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Related Work --- p.2Chapter 1.2 --- Organizations of the Dissertation --- p.4Chapter 1.3 --- Notations --- p.4Chapter 2 --- Overview of ICLP(R) --- p.6Chapter 2.1 --- Basics of Interval Arithmetic --- p.6Chapter 2.2 --- Relational Interval Arithmetic --- p.8Chapter 2.2.1 --- Interval Reduction --- p.8Chapter 2.2.2 --- Arithmetic Primitives --- p.10Chapter 2.2.3 --- Interval Narrowing and Interval Splitting --- p.13Chapter 2.3 --- Syntax and Semantics --- p.16Chapter 3 --- Limitations of Interval Narrowing --- p.18Chapter 3.1 --- Computation Inefficiency --- p.18Chapter 3.2 --- Inability to Detect Inconsistency --- p.23Chapter 3.3 --- The Newton Language --- p.27Chapter 4 --- Design of CIAL --- p.30Chapter 4.1 --- The CIAL Architecture --- p.30Chapter 4.2 --- The Inference Engine --- p.31Chapter 4.2.1 --- Interval Variables --- p.31Chapter 4.2.2 --- Extended Unification Algorithm --- p.33Chapter 4.3 --- The Solver Interface and Constraint Decomposition --- p.34Chapter 4.4 --- The Linear and the Non-linear Solvers --- p.37Chapter 5 --- The Linear Solver --- p.40Chapter 5.1 --- An Interval Gaussian Elimination Solver --- p.41Chapter 5.1.1 --- Naive Interval Gaussian Elimination --- p.41Chapter 5.1.2 --- Generalized Interval Gaussian Elimination --- p.43Chapter 5.1.3 --- Incrementality of Generalized Gaussian Elimination --- p.47Chapter 5.1.4 --- Solvers Interaction --- p.50Chapter 5.2 --- An Interval Gauss-Seidel Solver --- p.52Chapter 5.2.1 --- Interval Gauss-Seidel Method --- p.52Chapter 5.2.2 --- Preconditioning --- p.55Chapter 5.2.3 --- Increment ality of Preconditioned Gauss-Seidel Method --- p.58Chapter 5.2.4 --- Solver Interaction --- p.71Chapter 5.3 --- Comparisons --- p.72Chapter 5.3.1 --- Time Complexity --- p.72Chapter 5.3.2 --- Storage Complexity --- p.73Chapter 5.3.3 --- Others --- p.74Chapter 6 --- Benchmarkings --- p.76Chapter 6.1 --- Mortgage --- p.78Chapter 6.2 --- Simple Linear Simultaneous Equations --- p.79Chapter 6.3 --- Analysis of DC Circuit --- p.80Chapter 6.4 --- Inconsistent Simultaneous Equations --- p.82Chapter 6.5 --- Collision Problem --- p.82Chapter 6.6 --- Wilkinson Polynomial --- p.85Chapter 6.7 --- Summary and Discussion --- p.86Chapter 6.8 --- Large System of Simultaneous Equations --- p.87Chapter 6.9 --- Comparisons Between the Incremental and the Non-Incremental Preconditioning --- p.89Chapter 7 --- Concluding Remarks --- p.93Chapter 7.1 --- Summary and Contributions --- p.93Chapter 7.2 --- Future Work --- p.95Bibliography --- p.9