Worst-case and statistical tolerance analysis based on quantified constraint satisfaction problems and Monte Carlo simulation

The constraint satisfaction probem (CSP) is a well-acknowledged framework in which many combinatorial search problems can be naturally formulated. The CSP may be viewed as the problem of deciding the truth of a logical sentence consisting of a conjunction of constraints, in front of which all variables are existentially quantified. The quantified constraint satisfaction problem (QCSP) is the generalization of the CSP where universal quantification is permitted in addition to existential quantification. The general intractability of these problems has motivated research studying the complexity of these problems under a restricted constraint language, which is a set of relations that can be used to express constraints. This paper introduces collapsibility, a technique for deriving positive complexity results on the QCSP. In particular, this technique allows one to show that, for a particular constraint language, the QCSP reduces to the CSP. We show that collapsibility applies to three known tractable cases of the QCSP that were originally studied using disparate proof techniques in different decades: Quantified 2-SAT (Aspvall, Plass, and Tarjan in 1979), Quantified Horn-SAT (Karpinski, Kleine Büning, and Schmitt in 1987), and Quantified Affine-SAT (Creignou, Khanna, and Sudan in 2001). This reconciles and reveals common structure among these cases, which are describable by constraint languages over a two-element domain. In addition to unifying these known tractable cases, we study constraint languages over domains of larger size.

This study focuses on the robust design optimization of permanent magnet linear synchronous motors (PMLSM) considering the effect of uncertainties (noise factors) such as manufacturing errors, installation errors and working wears. A robust design optimization approach based on quantified constraint satisfaction problem (QCSP) is proposed. It establishes a robust PMLSM thrust ripple model considering noise factors by QCSP, and obtains the optimal design parameters that have lower thrust ripple and smaller thrust ripple fluctuation within their error range by branch-and-prune algorithm. First, an analytical model of thrust ripple for PMLSM is established. Second, the robust QCSP model of thrust ripple is established by quantifier concept based on analytical model. It defines the design parameters and noise variables as existential and universal quantifiers, respectively, defines the variables of QCSP model as quantifier parameters that are composed of existential and universal quantifiers, then QCSP model are established by quantifier parameters. Third, the algorithm of branch-and-prune based on interval arithmetic is proposed to optimize the QCSP model. This approach is proven to be robust and superior compared with traditional deterministic approach and another robust approach. Finally, Finite element analysis and motor prototype experiments confirm the feasibility and validity of the proposed method.

This paper reviews four major methods for tolerance analysis and compares them. The methods discussed are: (1) one-dimensional tolerance charts; (2) parametric tolerance analysis, especially parametric analysis based on the Monte Carlo simulation; (3) vector loop (or kinematic) based tolerance analysis; and (4) ASU Tolerance-Map® (T-Map®) (Patent pending; nonprovisional patent application number: 09/507, 542 (2002)) based tolerance analysis. Tolerance charts deal with worst-case tolerance analysis in one direction at a time and ignore possible contributions from the other directions. Manual charting is tedious and error prone, hence, attempts have been made for automation. The parametric approach to tolerance analysis is based on parametric constraint solving; its inherent drawback is that the accuracy of the simulation results are dependent on the user-defined modeling scheme, and its inability to incorporate all Y14.5 rules. The vector loop method uses kinematic joints to model assembly constraints. It is also not fully consistent with Y14.5 standard. The ASU T-Map® based tolerance analysis method can model geometric tolerances and their interaction in truly three-dimensional context. It is completely consistent with Y14.5 standard but its use by designers may be quite challenging. The T-Map® based tolerance analysis method is still under development. Despite the shortcomings of each of these tolerance analysis methods, each may be used to provide reasonable results under certain circumstances. Through a comprehensive comparison of these methods, this paper will offer some recommendations for selecting the best method to use for a given tolerance accumulation problem.

The quantified constraint satisfaction problem (QCSP) is a natural and useful generalization of the constraint satisfaction problem (CSP) in which both universal and existential quantification of variables is permitted. Because the CSP and QCSP are in general intractable, much effort has been directed towards identifying restricted cases of these problems that are tractable in polynomial time. In this paper, we investigate restricted cases of the QCSP having 2-semilattice polymorphisms. We prove a complete classification of 2-semilattice polymorphisms, demonstrating that each gives rise to a case of the QCSP that is either tractable in polynomial time, or coNP-hard.

Successful products are those presenting the highest quality at a fair cost. Although different approaches can be used to define the concept of quality, functional reliability is always a major requirement, due to implications such as safety and user losses regarding maintenance expenses, and product availability. Intelligent designs are robust and result in a fair cost. Robust designs are those insensitive to sources of variation occurring during the product life, keeping their performance under variable use conditions, like thermal and stress effects. The robustness approach is a function of two main design criteria: low complexity and tolerance design. Design for manufacture and assembly is closely related to decreasing complexity. Tolerance design is a tool in which the unavoidable manufacturing variations are considered during product development. This work presents a proposal for an intelligent design in an actual application by considering design simplification through the reduction of parts for an automotive water pump. The tolerance analysis is performed by means of a powerful statistical approach—a Monte Carlo simulation—in which process behavior is randomly simulated representing a high production volume. Additionally, service thermal effects are also contemplated, and assembly tests are proposed for automatic rejection of non-conforming parts, assuring high reliability and full compliance with functional requirements. This is an example of integrated design–manufacturing work aiming at both cost saving and improved reliability.

Tolerance analysis methods, which are important to achieve a balance between manufacturing costs and functional requirements, have been studied and improved over the last 30 years. This paper proposes a new method to improve the precision of assembly simulation for statistical assembly simulation; by investigating the effect of the geometric deviations and mechanical behaviors of assembly contact on the stack-up assembly deviation by combining the recent theories on the tolerance analysis of rigid bodies and local contact deformation. To improve the precision of the tolerance analysis, real parts with form defects were simulated with the help of non-nominal skin model shapes. The SMSs were first considered to be rigid, and a quadratic optimization-based method for the rigid body displacement was used to calculate the initial contact points of all the SMSs. Next, contact deformation, following Hertzian contact theory, occurred at the initial contact points, which changed the relative positions of the SMSs in the subsequent interactions. The iteration process was convergent, and the final simulation for tolerance analysis could be generated after several iterations. Case studies demonstrated the practicality of the proposed method. Their actual assembly deviations were simulated by combining the effects of geometric deviations and local surface deformations. Since the proposed tolerance analysis framework based on statistical assembly simulation, it is expected to control the functional requirements of mechanical assemblies with much rough tolerance design and help optimize the tolerance in the future.

A paper by Dag Westerståhl and myself twenty years ago introduced operators that are both connectives and quantifiers. We introduced two binary operators that are classically interdefinable: one that fuses conjunction and existential quantification and one that fuses implication and universal quantification. We called the system PFO. A complete Gentzen-Prawitz style Natural Deduction axiomatization of classical PL was provided. For intuitionistic PL, however, it seemed that existential quantification should be fused with disjunction rather than with conjunction. Whether this was true, and if so why, were questions not answered at the time. Also, it seemed that there is no uniform definition of such a disjunctive-existential operator in classical PFO. This, too, remained a conjecture. In this paper, I return to these previously unresolved questions, and resolve them.

This research aims to create an efficient way to guide designers in selecting tolerances more cost-effectively through the Jacobian-Torsor unified model by performing an iterative statistical and deterministic analysis of geometric tolerances. This model is created based on the assembly's Functional Requirements (FR) and the starting tolerance values of the Functional Elements (FE) that comprise the tolerance chain. The model initially produces deterministic results. However, the Monte Carlo simulation allows converting it into a statistical model. The model is assessed iteratively, with each simulation iteration employing random values of the torsor components that correspond to the functional elements as inputs. The simulation ends when the number of iterations approaches the maximum value set at the star, and a collection of values representing the components of the Functional Requirement (FR) torsor is created based on repeated multiplications of the model. Finally, the percentage calculation defines the contribution of each Functional Element (FE) to the total Functional Requirement (FR). A numerical example is provided to demonstrate the use of the suggested method. A comparison of the two approaches, deterministic and statistical, reveals that the latter has a more significant economic impact.

Quality and cost are among the most important concerns for a product. While tolerance is one of the typical metrics for quality, it is a trade-off between tolerances and costs in product development though the two factors often conflict with each other. This paper describes a systematic approach to compute linear tolerance accumulation for cylindrical parts by machining operations based on the tolerance chart. A computer-aided tolerance chart system is developed to assist the construction of the corresponding tolerance chart and the computation of accumulated linear tolerances for a given process plan. Tolerance distribution to each machining operation by optimization techniques is also investigated. The goal is to minimize machining cost subject to constraints on tolerance accumulation and process capability. It shows that the machining cost of a sample part with the worst-case tolerance analysis can be reduced by 39%, compared to that by experience, and can be further reduced if statistical tolerance analysis applies.

This paper presents a practical method for redesigning statistical tolerances by combining the unified Jacobian-Torsor model and Monte Carlo simulation. After having identified the assembly functional requirement (FR) and determined the functional elements (FEs) encountered in tolerance chain, the specific expression of Jacobian-Torsor model can be obtained. Monte Carlo simulation is employed to repeatedly evaluate the model using sets of random values as inputs to make this deterministic model become a statistical model. Once the real multiplication operations have been performed on this model, we can get large amount of simulated data, from which we can obtain the statistical limits of the assembly FR in the desired direction. The percentage contribution of each FE to the total FR can help the designer redesign the tolerances of the critical FEs to respect the specified FR. This redesign process may need to be repeated several times until the calculated FR is consistent with the specified FR, so the tolerance values that need to be determined are obtained, and the redesign of statistical tolerances is achieved. The effectiveness of the proposed method is illustrated by taking the crank-connecting rod mechanism of piston aeroengine assembly as an example.

\nThe unified Jacobian-Torsor model has been developed for deterministic (worst case) tolerance analysis. This paper presents a comprehensive model for performing statistical tolerance analysis by integrating the unified Jacobian-Torsor model and Monte Carlo simulation. In this model, an assembly is sub-divided into surfaces, the Small Displacements Torsor (SDT) parameters are used to express the relative position between any two surfaces of the assembly. Then, 3D dimension-chain can be created by using a surface graph of the assembly and the unified Jacobian-Torsor model is developed based on the effect of each functional element on the whole functional requirements of products. Finally, Monte Carlo simulation is implemented for the statistical tolerance analysis. A numerical example is given to demonstrate the capability of the proposed method in handling three-dimensional assembly tolerance analysis.\n

BOOK REVIEWS Prior, A.N., Formal Logic. Oxford: The Clarendon Press, 1955. Pp. ix—329. $5.60. In the preface, the author indicates that this book is designed primarily as a textbook, though he hopes that it will prove to be of interest to others who are not Logic students. The content, which includes; Part I — The Classical Prepositional and Functional Calculi; Part II — The Traditional Logic of Terms; and Part HI — Modal, Three-valued and Extensional Systems ; is of such a nature that it should attract a variety of critico-analytical, perceptive readers other than those solely interested in Logic. Part I of the triparted arrangement commences with 'Truth-Functions and Tautologies' and is a systematic study of tautologous implications derived from fixed rules of transformation or inference and their relation to systems of tautologous alterations and conjunctions. In the axiomatizations of the Propositional Calculus to obtain a mechanical accuracy, a rigorously fixed symbolism is employed to express the laws laid down and established, as well as to represent the processes of proof. Using axioms and rules the author shows that these axioms are 'complete' and sufficient, not only to prove all tautologous formulae in the propositional calculus, but also to disprove all non-tautologous ones. Besides the fundamental axioms found in the Principia Mathematica of Whitehead and Russell, the newer and more economical ones derived by Lukasiewicz, and Hubert and Ackerman are employed. In answer to the question 'Why should the logician, whose primary interest is in inference, study other tautologous formulae beside those which are implications ?', the chapter entitled 'System with Propositional Constants and Functional Variables', is an attempt to show that all other tautologous formulae are either implications or definitional abbreviations for them. In defense of this thesis there are sections about The Implicational Calculus and Wajsberg's System in C and 0; Quine's Completeness Proof for the Full Calculus; Wajsberg's Completeness Proof for the Restricted Calculus; and Logic, Metalogic, and the Functorial Variables. Especially well expressed and explained in connection with the Theory of Quantification is the portion of the text on propositional operators, which form propositions, not out of other propositions, but out of names. The author illucidates the manner in which these new types of propositional formulae are introduced, and substituted for propositional variables in theses of the propositional calculus. With keen and astute logical acumen, he draws attention to the fact that there is a close resemblance (1) between the laws of equipollence and the principles now known as de Morgan Laws ; and (2) between universal quantification and conjunction, and existential quantification and alteration. 407 4o8Book-Reviews A. N. Prior's critical ability is especially pronounced in connection with 'Quantification and Equivalential Calculi: Definition Re-examined.' The work on extension, by the use of quantifiers and functional variables, of versions of the propositional calculus in which the only undefined truth-factor is E, 'If and only if,' includes the interesting note that Leániewski's special interest in a protothenic with an 'equivalential' basis arose from the personal view that he held about the nature of definition in formal systems. The author's account of definitions as assertions of equivalence include the contributions of Leániewski, Tarski and Sobocinski, who espoused the theory in its entirety, and of Lukasiewicz and Meredith, who partially adopted it. Despite the opinions of this distinguished company, the author clearly states that the conclusion of the account is wrong-headed and he shows that it makes it difficult to distinguish the definitions of a system from additional axioms. Indeed worthy of critical study, is Part II which is a truly remarkable synthesis of the Traditional Logic of Terms, that commences with the Aristotelian Syllogistic; is followed by a comprehensive treatment of Categorical Forms with Negative, Complex, and Quantified Terms; and concludes with Singular and Existential Propositions. When Professor Prior discusses Singulars and Generals in Traditional and Modern Logic, he explains with keen perception that there is a large element of anachronism in attributing to Aristotle any satisfactory way of interpreting his forms in the modern functional calculus and at the same time 'saving' all or most of his laws. The elaborate body of thought of...

The system A oo In this chapter we construct the formal system A oo . It is a very simple arithmetic with familiar fundamental concepts. These are: the natural number zero , the successor function , the operation of repeatedly applying a function, the operation of forming functions by abstraction, equality and inequality between numerical expressions, and the logical connectives, conjunction and disjunction. The atomic statements are equations and inequations between numerical terms, compound statements are built up from atomic statements by conjunction and disjunction. Negation, material implication and material equivalence are definable, but existential quantification and universal quantification are unrepresentable. We give definitions of A oo - truth and of A oo - falsity for closed A oo -statements, and show that they are exclusive properties. We also show that a closed A oo -statement is A oo -true if and only if it is an A oo -theorem. Thus the system A oo is consistent in the sense that A oo -theorems are A oo -true; and is complete in the sense that A oo -true A oo -statements are A oo -theorems. We give a procedure which applied to a closed A oo -statement will terminate and tell us whether it is A oo -true or is A oo -false. Thus the system A oo is decidable. The A oo -rules of formation To construct the system A oo we first list the A oo -signs and attach a type to each proper A oo -symbol and give each A oo -symbol a name which will assist the reader in understanding how the system was first conceived. Parentheses round type symbols are usually omitted by association to the left as explained in Ch. 1. (See table overleaf.)

We consider the extension of Boolean circuits to quantified Boolean circuits by adding universal and existential quantifier nodes with semantics adopted from quantified Boolean formulas (\(\mbox{\rm QBF}\)). The concept allows not only prenex representations of the form \(\forall x_{1}\exists y_{1}...\forall x_{n}\exists y_{n}\ c\) where c is an ordinary Boolean circuit with inputs x 1,...,x n ,y 1,...,y n . We also consider more general non-prenex normal forms with quantifiers inside the circuit as in non-prenex \(\mbox{\rm QBF}\), including circuits in which an input variable may occur both free and bound. We discuss the expressive power of these classes of circuits and establish polynomial-time equivalence-preserving transformations between many of them. Additional polynomial-time transformations show that various classes of quantified circuits have the same expressive power as quantified Boolean formulas and Boolean functions represented as finite sequences of nested definitions (\(\mbox{\rm NBF}\)). In particular, universal quantification can be simulated efficiently by circuits containing only existential quantifiers if overlapping scopes of variables are allowed.KeywordsNormal FormBoolean FunctionFree VariableExpressive PowerOutgoing EdgeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Fuzzy relational compositions have been extensively studied by many authors.Especially, we would like to express studies of the fuzzy relational compositions motivated by their applications to medical diagnosis by Willis Bandler and Ladislav Kohout.These types of compositions use only two quantifiers: universal quantifier and existential quantifier.There exists a huge gap between these quantifiers.This is a natural motivation for studying fuzzy relational compositions defined based on generalized quantifiers.In this paper, we revisit and provide definitions of these compositions as well as those of derived images and preimages.Furthermore, we investigate their mathematical properties.