Searching Feasible Design Space by Solving Quantified Constraint Satisfaction Problems

2013 ◽  
Vol 136 (3) ◽  
Author(s):  
Jie Hu ◽  
Masoumeh Aminzadeh ◽  
Yan Wang
Keyword(s):  
Monte Carlo ◽  
Constraint Satisfaction ◽  
Design Space ◽  
Constraint Satisfaction Problem ◽  
Constraint Satisfaction Problems ◽  
Design Solution ◽  
New Approach ◽  
Solution Sets ◽  
Branch And Prune Algorithm ◽  
Branch And Prune

In complex systems design, multidisciplinary constraints are imposed by stakeholders. Engineers need to search feasible design space for a given problem before searching for the optimum design solution. Searching feasible design space can be modeled as a constraint satisfaction problem (CSP). By introducing logical quantifiers, CSP is extended to quantified constraint satisfaction problem (QCSP) so that more semantics and design intent can be captured. This paper presents a new approach to formulate searching design problems as QCSPs in a continuous design space based on generalized interval, and to numerically solve them for feasible solution sets, where the lower and upper bounds of design variables are specified. The approach includes two major components. One is a semantic analysis which evaluates the logic relationship of variables in generalized interval constraints based on Kaucher arithmetic, and the other is a branch-and-prune algorithm that takes advantage of the logic interpretation. The new approach is generic and can be applied to the case when variables occur multiple times, which is not available in other QCSP solving methods. A hybrid stratified Monte Carlo method that combines interval arithmetic with Monte Carlo sampling is also developed to verify the correctness of the QCSP solution sets obtained by the branch-and-prune algorithm.

Statistical Tolerance Analysis Based on Constraint Satisfaction Problems and Monte Carlo Simulation

2012 ◽  
Vol 433-440 ◽  
pp. 6616-6621
Author(s):  
Yong Jun Jiang
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Mathematical Formulation ◽  
Tolerance Analysis ◽  
Constraint Satisfaction Problems ◽  
Statistical Tolerance ◽  
Problem Solvers ◽  
Statistical Tolerance Analysis

This paper deals with the mathematical formulation of tolerance analysis. The mathematical formulation presented simulates the influences of geometrical deviations on the geometrical behavior of the mechanism, and integrates the quantifier notion. We propose a mathematical formulation of tolerance analysis which simulates the influences of geometrical deviations on the geometrical behavior of the mechanism, and integrates the quantifier notion. To compute this mathematical formulation, two approaches based on Quantified Constraint Satisfaction Problem solvers and Monte Carlo simulation are proposed and tested.

scholarly journals Constraint Satisfaction Problems and Their Reduction to Directed Graphs

2021 ◽  
Author(s):  
Muhanda Stella Mbaka Muzalal
Keyword(s):  
Constraint Satisfaction ◽  
Cell Phone ◽  
Constraint Satisfaction Problem ◽  
Directed Graphs ◽  
Oriented Graph ◽  
Constraint Satisfaction Problems ◽  
Systems Of Equations ◽  
Relational Structures ◽  
Algorithmic Problems ◽  
Boolean Formulas

Constraint satisfaction problems present a general framework for studying a large class of algorithmic problems such as satisfaction of Boolean formulas, solving systems of equations over finite fields, graph colourings, as well as various applied problems in artificial intelligence (scheduling, allocation of cell phone frequencies, among others.) CSP (Constraint Satisfaction Problems) bring together graph theory, complexity theory and universal algebra. It is a well known result, due to Feder and Vardi, that any constraint satisfaction problem over a finite relational structure can be reduced to the homomorphism problem for a finite oriented graph. Until recently, it was not known whether this reduction preserves the type of the algorithm which solves the original constraint satisfaction problem, so that the same algorithm solves the corresponding digraph homomorphism problem. We look at how a recent construction due to Bulin, Deli´c, Jackson, and Niven can be used to show that the polynomial solvability of a constraint satisfaction problem using Datalog, a programming language which is a weaker version of Prolog, translates from arbitrary relational structures to digraphs.

Equations in oligomorphic clones and the constraint satisfaction problem for ω-categorical structures

2019 ◽  
Vol 19 (02) ◽  
pp. 1950010 ◽  
Author(s):  
Libor Barto ◽  
Michael Kompatscher ◽  
Miroslav Olšák ◽  
Van Pham Trung ◽  
Michael Pinsker
Keyword(s):  
Constraint Satisfaction ◽  
Short Proof ◽  
Constraint Satisfaction Problem ◽  
Constraint Satisfaction Problems ◽  
Ramsey Property ◽  
Categorical Structure ◽  
Homogeneous Structures ◽  
Identity Modulo ◽  
Rational Order ◽  
Its Polymorphism

There exist two conjectures for constraint satisfaction problems (CSPs) of reducts of finitely bounded homogeneous structures: the first one states that tractability of the CSP of such a structure is, when the structure is a model-complete core, equivalent to its polymorphism clone satisfying a certain nontrivial linear identity modulo outer embeddings. The second conjecture, challenging the approach via model-complete cores by reflections, states that tractability is equivalent to the linear identities (without outer embeddings) satisfied by its polymorphisms clone, together with the natural uniformity on it, being nontrivial. We prove that the identities satisfied in the polymorphism clone of a structure allow for conclusions about the orbit growth of its automorphism group, and apply this to show that the two conjectures are equivalent. We contrast this with a counterexample showing that [Formula: see text]-categoricity alone is insufficient to imply the equivalence of the two conditions above in a model-complete core. Taking another approach, we then show how the Ramsey property of a homogeneous structure can be utilized for obtaining a similar equivalence under different conditions. We then prove that any polymorphism of sufficiently large arity which is totally symmetric modulo outer embeddings of a finitely bounded structure can be turned into a nontrivial system of linear identities, and obtain nontrivial linear identities for all tractable cases of reducts of the rational order, the random graph, and the random poset. Finally, we provide a new and short proof, in the language of monoids, of the theorem stating that every [Formula: see text]-categorical structure is homomorphically equivalent to a model-complete core.

Symbolic Solutions for Symbolic Constraint Satisfaction Problems

2020 ◽  
Author(s):  
Alexsander Andrade de Melo ◽  
Mateus De Oliveira Oliveira
Keyword(s):  
Constraint Satisfaction ◽  
Existence Of Solutions ◽  
Constraint Satisfaction Problem ◽  
Constant Width ◽  
Computable Function ◽  
Constraint Satisfaction Problems ◽  
Decision Diagrams ◽  
Turing Machines ◽  
Branching Programs ◽  
Hard Problems

A fundamental drawback that arises when one is faced with the task of deterministically certifying solutions to computational problems in PSPACE is the fact that witnesses may have superpolynomial size, assuming that NP is not equal to PSPACE. Therefore, the complexity of such a deterministic verifier may already be super-polynomially lower-bounded by the size of a witness. In this work, we introduce a new symbolic framework to address this drawback. More precisely, we introduce a PSPACE-hard notion of symbolic constraint satisfaction problem where both instances and solutions for these instances are implicitly represented by ordered decision diagrams (i.e. read-once, oblivious, branching programs). Our main result states that given an ordered decision diagram D of length k and width w specifying a CSP instance, one can determine in time f(w,w')*k whether there is an ODD of width at most w' encoding a solution for this instance. Intuitively, while the parameter w quantifies the complexity of the instance, the parameter w' quantifies the complexity of a prospective solution. We show that CSPs of constant width can be used to formalize natural PSPACE hard problems, such as reachability of configurations for Turing machines working in nondeterministic linear space. For such problems, our main result immediately yields an algorithm that determines the existence of solutions of width w in time g(w)*n, where g:N->N is a suitable computable function, and n is the size of the input.

scholarly journals Nature-Inspired Techniques For Dynamic Constraint Satisfaction Problems

2021 ◽  
Author(s):  
Mehdi Bidar ◽  
Malek Mouhoub
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Dynamic Environment ◽  
Iterative Algorithms ◽  
Constraint Satisfaction Problems ◽  
Environmental Stability ◽  
Final State ◽  
Dynamic Constraint ◽  
Consistent Solution ◽  
Constraint Problem

Abstract Combinatorial applications such as configuration, transportation and resource allocation, often operate under highly dynamic and unpredictable environments. In this regard, one of the main challenges is to maintain a consistent solution anytime constraints are (dynamically) added. While many solvers have been developed to tackle these applications, they often work under idealized assumptions of environmental stability. In order to address limitation, we propose a methodology, relying on nature-inspired techniques, for solving constraint problems when constraints are added dynamically. The choice for nature-inspired techniques is motivated by the fact that these are iterative algorithms, capable of maintaining a set of promising solutions, at each iteration. Our methodology takes advantage of these two properties, as follows. We first solve the initial constraint problem and save the final state (and the related population) after obtaining a consistent solution. This saved context will then be used as a resume point for finding, in an incremental manner, new solutions to subsequent variants of the problem, anytime new constraints are added. More precisely, once a solution is found, we resume from the current state to search for a new one (if the old solution is no longer feasible), when new constraints are added. This can be seen as an optimization problem where we look for a new feasible solution satisfying old and new constraints, while minimizing the differences with the solution of the previous problem, in sequence. This latter objective ensures to find the least disruptive solution, as this is very important in many applications including scheduling, planning and timetabling. Following on our proposed methodology, we have developed the dynamic variant of several nature-inspired techniques to tackle dynamic constraint problems. Constraint problems are represented using the well-known Constraint Satisfaction Problem (CSP) paradigm. Dealing with constraint additions in a dynamic environment can then be expressed as a series of static CSPs, each resulting from a change in the previous one by adding new constraints. This sequence of CSPs is called the Dynamic CSP (DCSP). To assess the performance of our proposed methodology, we conducted several experiments on randomly generated DCSP instances, following the RB model. The results of the experiments are reported and discussed.

Constraint Reasoning in Concurrent Design

1990 ◽  
Author(s):  
James R. Rinderle ◽  
V. Krishnan
Keyword(s):  
Life Cycle ◽  
Constraint Satisfaction ◽  
Design Space ◽  
Design Solution ◽  
Concurrent Design ◽  
Design Consideration ◽  
Simultaneous Treatment ◽  
Critical Design ◽  
Constraint Reasoning ◽  
Fan Blade

Abstract One paradigm of concurrent design is based on the simultaneous consideration of a broad range of life-cycle constraints including those arising from function, manufacturing and maintenance. This simultaneous treatment of life-cycle issues results in a multitude of constraints, which not only increase the complexity of finding a design solution, but also make it difficult to understand the trends and interactions underlying the design. It is our goal to enhance the designer’s ability to identify and discriminate those constraints that critically impact the design from those that are irrelevant. We propose an interval analysis based approach, which is augmented with monotonicity and dominance principles. The approach helps in identifying regions of the design space where constraints possess certain desirable properties. It also enables reasoning with constraints in these regions. The regional inferences can then be reassembled to obtain global results. These ideas have been applied in the concurrent design of a fan blade, to identify the dominant, active and redundant constraints, enabling the designer to more clearly perceive and base his decisions on the critical design consideration. Furthermore, the identification of dominant constraints permits the easy evaluation of the significance of newly asserted constraints and frequently facilitates the automatic formulation of noniterative constraint satisfaction methods which guarantee a globally optimal design.

PREPROCESSING QUANTIFIED CONSTRAINT SATISFACTION PROBLEMS WITH VALUE REORDERING AND DIRECTIONAL ARC AND PATH CONSISTENCY

2008 ◽  
Vol 17 (02) ◽  
pp. 321-337 ◽  
Author(s):  
KOSTAS STERGIOU
Keyword(s):  
Constraint Satisfaction ◽  
Early Stage ◽  
Constraint Satisfaction Problem ◽  
Constraint Satisfaction Problems ◽  
Combinatorial Problems ◽  
Arc Consistency ◽  
Solution Methods ◽  
Speed Up ◽  
Phase Transition Region ◽  
The Impact

The Quantified Constraint Satisfaction Problem (QCSP) is an extension of the CSP that can be used to model combinatorial problems containing contingency or uncertainty. It allows for universally quantified variables that can model uncertain actions and events, such as the unknown weather for a future party, or an opponent's next move in a game. Although interest in QCSPs is increasing in recent years, the development of techniques for handling QCSPs is still at an early stage. For example, although it is well known that local consistencies are of primary importance in CSPs, only arc consistency has been extended to quantified problems. In this paper we contribute towards the development of solution methods for QCSPs in two ways. First, by extending directional arc and path consistency, two popular local consistencies in constraint satisfaction, to the quantified case and proposing an algorithm that achieves these consistencies. Second, by showing how value ordering heuristics can be utilized to speed up computation in QCSPs. We study the impact of preprocessing QCSPs with value reordering and directional quantified arc and path consistency by running experiments on randomly generated problems. Results show that our preprocessing methods can significantly speed up the QCSP solving process, especially on hard instances from the phase transition region.

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