Beginningl     In this paper, we focus on the need for l     This paper proceeds as      The structure of the paper is as      In this paper, we shall first briefly introduce fuzzy sets and related conceptsl     To begin with we will provide a brief background on the Introductionl     This will be followed by a description of the fuzzy nature of the problem and a detailed presentation of how the required membership functions are      Details on xx and xx are discussed in later      In the next section, after a statement of the basic problem, various situations involving possibility knowledge are investigated: first, an entirely possibility model is proposed; then the cases of a fuzzy service time with stochastic arrivals and non‑fuzzy service rule is studied; lastly, fuzzy service rule are      This review is followed by an introduction.  l     A brief summary of some of the relevant concepts in xxx and xxx is presented in Section      In the next section a brief view of the .... is given.  l     In the next section, a short review of ... is given with special regard to      Section2reviews relevant research related to      Section briefly surveys the motivation for a methodology of action, while looks at the difficulties posed by the complexity of systems and outlines the need for development of possibility      Section1defines the notion of robustness, and argues for its      Section1devoted to the basic aspects of the FLC decision‑making      Section2gives the background of the problem which includes xxxl     Section2discusses some problems with and approaches to, natural language      Section2explains how flexibility which often ... can be expressed in terms of fuzzy time‑windowl     Section3discusses the aspects of fuzzy set theory that are used in the      Section3describes the system itself in a general way, including the also discusses how to evaluate system      Section3describes a new measure of      Section3demonstrates the use of fuzzy possibility theory in the analysis of      Section3is a fine description of fuzzy formulation of human      Section3, is developed to the modeling and processing of fuzzy decision rulesl     The main idea of the FLC is described in Section3while Section4describes the xx      Section3and4show experimental studies for verifying the proposed      Section4discusses a previous fuzzy set‑based approach to cost variance      Section4gives a specific example of      Section4is the experimental study to make a fuzzy model of memory      Section4contains a discussion of the implication of the results of Section2and      Section4applies this fuzzy measure to the analysis of xx and illustrate its use on experimental      Section5presents the primary results of the paper: a fuzzy set model      Section5contains some conclusions plus some ideas for further      Section6illustrate the model with an      Various ways of fuzzification and the reasons for their choice are discussed very briefly in Section      In Section2are presented the block diagram expression of a whole model of human DM systeml     In Section2we shall list a collection of basic assumptions which a ... scheme must      In Section2of this paper, we present representation and uniqueness theorems for the fundamental measurement of fuzziness when the domain of discourse is order‑     In Section3, we describe the preliminary results of an empirical study currently in progress to verify the measurement model and to construct membership      In Section5is analyzed the inference process through the two kinds of inference Sectionl     In this section, the characteristics and environment under which MRP is designed are      We will provide in this section basic terminologies and notations which are necessary for the understanding of subsequent Sectionl     The next section describes the mathematics that goes into the computer implementation of such fuzzy logic      However, it is cumbersome for this purpose and in practical applications the formulae were rearranged and simplified as discussed in the next      The three components will be described in the next two section, and an example of xx analysis of a computer information system will then illustrate their      We can interpret the results of Experiments I and II as in the following      The next section summarizes the method in a from that is useful for arguments based on xxSummaryl     This paper concludes with a discussion of future research consideration in section      Section5summarizes the results of this      Section5gives the conclusions and future directions of      Section7provides a summary and a discussion of some extensions of the      Finally, conclusions and future work are summarizedl     The basic questions posed above are then discussed and conclusions are      Section7is the conclusion of the 0. Abstractl     A basic problem in the design of xx is presented by the choice of a xx rate for the measurement of experimental      This paper examines a new measure of xx in xx based on fuzzy mathematics which overcomes the difficulties found in other xx      This paper describes a system for the analysis of the      The method involves the construction of xx from fuzzy      The procedure is useful in analyzing how groups reach a      The technique used is to employ a newly developed and versatile xx      The usefulness of xx is also      A brief methodology used in xx is      The analysis is useful in xx and xx      A model is developed for a xx analysis using fuzzy      Algorithms to combine these estimates and produce a xx are presented and      The use of the method is discussed and an example is      Results of an experimental applications of this xx analysis procedure are given to illustrate the proposed      This paper analyses problems in l     This paper outlines the functions carried out by      This paper includes an illustration of the      This paper provides an overview and information useful for approachingl     Emphasis is placed on the construction of a criterion function by which the xx in achieving a hierarchical system of objectives are      The main emphasis is placed on the problem of xxl     Our proposed model is verified through experimental      The experimental results reveal interesting examples of fuzzy phases of : xx,xxl     The compatibility of a project in terms of cost, and xx are likewise represented by linguistic      A didactic example is included to illustrate the computational procedureChapter1. IntroductionTimel     Over the course of the past30 years, .. has emerged form intuitive l     Technological revolutions have recently hit the industrial worldl     The advent of ... systems for  has had a significant impact on the l     The development of ... is exploredl     During the past decade, the theory of fuzzy sets has developed in a variety of directions,l     The concept of xx was investigated quite intensively in recent yearsl     There has been a turning point in ... methodology in accordance with the advent of      A major concern in ... today is to continue to      A xx is a latecomer in the part representation      At the time of this writing, there is still no standard way of xxl     Although a lot of effort is being spent on improving these weaknesses, the efficient and effective method has yet to be      The pioneer work can be traced to xx [1965].l     To date,  none of the methods developed is perfect and all are far from ready to be used in commercial / Goal / Purposel     The purpose of the inference engine can be outlined as follows:l     The ultimate goal of the xx system is to allow the non‑experts to utilize the existing knowledge in the area of manual handling of loads, and to provide intelligent, computer‑aided instruction for      The paper concerns the development of a xxl     The scope of this research lies inl     The main theme of the paper is the application of rule‑based decision      These objectives are to be met with such thoroughness and confidence as to permit      The objectives of the  ... operations study are as follows:l     The primary purpose/consideration/objective ofl     The ultimate goal of this concept is to providel     The main objective of such a ... system is tol     The aim of this paper is to provide methods to construct such probability     In order to achieve these objectives, an xx must meet the following requirements:l     In order to take advantage of their similarityl     more research is still required before final goal of ... can be completedl     In this trial, the objective is to      for the sake of concentrating on ... research issuesl     A major goal of this report is to extend the utilization of a recently developed procedure for the      For an illustrative purpose, four well‑known OR problems are studied in presence of fuzzy data:      A major thrust of the paper is to discuss approaches and strategies for structuring      This illustration points out the need to specifyl     The ultimate goal is both descriptive and      Chapter2. Literature Reviewl     A wealth of information is to be found in the statistics literature, for example, regarding xxl     A considerable amount of research has been done .. during the last decadel     A great number of studies report on the treatment of uncertainties associated with      There is considerable amount of literature on planningl     However, these studies do not provide much attention to undertainty in      Since then, the subject has been extensively explored and it is still under investigation as well in methodological aspects as in concrete      Many research studies have been carried out on this      Problem of xx draw recently more and more attention of system      Attempts to resolve this dilemma have resulted in the development of l     Many complex processes unfortunately, do not yield to this design procedure and have, therefore, not yet been      Most of the methods developed so far are deterministic and /or probabilistic in      The central issue in all these studies is to l     The problem of xx has been studied by other investigators, however, these studies have been based upon classical statistical      Applied ... techniques tol     Characterized the ... system asl     Developed an algorithm tol     Developed a system called ... whichl     Uses an iterative algorithm to deducel     Emphasized the need tol     Identifies six key issues surrounding high technologyl     A comprehensive study of the .. has been undertaken l     Much work has been reported recently in these filedl     Proposedl     Presentedl     State thatl     Point out that the problem of l     Describedl     Illustratedl     Indicatedl     Has shown / showedl     Addressl     Highlightsl     A study on done / developed by []l     Previous work, such as [] and [], deal only withl     The approach taken by [] isl     The system developed by [] consistsl     A paper relevant to this research was published by []l     []"s model requires consideration of      []" model draws attention to evolution in human developmentl     []"s model focuses      Little research has been conducted in applying ... tol     The published information that is relevant to this      This study further shows thatl     Their work is based on the principle of l     More history of ... can be found in xx et al. [1979].l     Studies have been completed to establishedl     The   indicated that l     Though application of xx in the filed of xx has proliferated in recent years, effort in analyzing xx, especially xx, is / Issue / Questionl     Unfortunately, real-world engineering problems such as manufacturing planning do not fit well with this narrowly defined model.  They tend to span broad activities and require consideration of multiple      Remedy / solve / alleviate these problemsl     ... is a difficult problem, yet to be adequately resolvedl     Two major problems have yet to be addressedl     An unanswered questionl     This problem in essence involves using x to obtain a      An additional research issue to be tackled is      Some important issues in developing a ... system are discussedl     The three prime issues can be summarized:l     The situation leads to the problem of how to determine the      There have been many attempts to l     It is expected to be serious barrier to l     It offers a simple solution in a limited domain for a complex      There are several ways to get around this      As difficult as it seems to be, xx is by no means      The problem is to recognize xx from a design      A xx problem can trace its roots to      xx [1987] used a heuristic approach to simplify the complexity of the      Several problems are associated with      Although some progress has been made in this area, at least two major obstacles must be overcome before a fully automated system can be      Most problems in practice are complicatedl     More problem surface      Hamper effort toward a xx systeml     In order to overcome the limitations due to incomplete and imprecise xx knowledge, a xx program has been developed, which bases its knowledge upon the statistical analysis of a sample population of xxl     The above difficulties are real challenges faced by researchers attempting to developl     This type of mapping raises no controversy to the issue of membership function      However, attempts to quantify the xx have met both theoretical and empirical      It has become apparent that in order to apply this new methodological framework to real‑world problems and data, we have to pay attention to the problems of xx and 3. Proposed methodologyAssumptionl     In the case when the assumption of a xx seems to be too restrictive or inadequate, the formulation with Fuzzy termination time, . given by a fuzzy set in the space of control stages, may be      We assume here the fuzzy constraints to be state‑dependent, and the fuzzy goal to be the same for all the control states, xx, which stems from the problem"s      An approach to the solution of this problem is presented under the assumption that the sampling rate Decision can be made prior to the execution of the experiment, as opposed to being made while the experiment is in      Another assumption made above is that there are precise odds at which the expert is      Main simplifying assumptions are:l     This, in our view, is a questionable / Structure / Modulel     An outline of the researchl     Information is incorporated within the schemel     Is built into ... structurel     A nice modular      The principles of ... are applied as modularized criteriaClassificationl     A xx system comprises three main components:l     Must decompose the original .. into a set of      Consists of the following steps:l     This is summarized in the following steps:l     Can be broadly classified into the following areas:l     Can be characterized by its function of effectively processing thel     Can allow further breadth of application of morel     The following steps should be followedl     xx can be classified by a different      Based on the xx, one may classify xx into the following:l     This catalog may change due to wear, breakage, and      Unlike many conventional program, expert systems do not usually deal with problem for which there is clearly a right or wrong      The system consists of both ... and      The system has a hierarchical modular architecture organized on three      expert system domains are area of expertisel     To develop a xx system for xx, the following factors must be considered:l     The system has been developed / designed to determinel     The system has proven to be able to l     The domain in which an expert system operates is a particular domainl     The system comprises a ... withl     The system is [feature-oriented ] / based on the ... techniquel     The system environment must be relatively stablel     The system is utilized to generate, load, store, update and retrieve      The development of a xx system has two stages:  xx stage and xx      The most essential part of .. system is the      The successful developments in ESs have made them an important tool in the development of l     An automated system was developed forl     In this case, the system can be considered to be      An interactive automatic ... systeml     A .. is commonly thought of as a truly integrated .. systeml     Should be capable of being generated from a ... systeml     xx is an important part of the integrated      The model consists of four rule bases, each of which addresses a separate problem in the hierarchy of scheduling      The rule bases are linked to each other in a chin‑like manner in the sense that the consequent of one rule base constitutes a part of the antecedent of the next rule      The rule base consists of all possible combinations of the linguistic terms associated with the linguistic variable of the antecedent of a Systeml     The system has been implemented using Prolog language in an MS‑DOS environment.  Prolog was chosen because it offers a well known and flexible environment in which fuzzy reasoning may be easily      The current version of the xx program when compiled with WATFOR77results in an executable code of about270K bytes.  Typical run time, when run on a XX computer (an IBM compatible machine) operating at Mhz with640K RAM, ranges from10 min to2h, depending on the size (or complexity) of the      Time consuming procedures have been implemented in C‑language and directly linked to the Prolog      The xx process, once the xx"s data has been entered, requires approximately180      It should be noted that the computation was done with a20 Hhz,80386‑based microcomputer equipped with a80387math co‑     The computer programs used for the analyses, one based on the xx method and the other based on the new method, were written in FORTRAN with a compiler that supports the math co‑     Lisp, Prolog give maximum flexibility but also maximizes development      Internal representation is the way a model is represented in the      An interactive menu-driven procedure is used in this studyl     Shell can be develop very fast at the cost of time fairly severe      While there is no measurable saving of time for the case involving five criteria, the saving is dramatic for the case involving10 criteria -- the computation time reduces from10 hr40 min to about1     This combination is being implemented in an objected‑oriented programming environment (Smalltalk‑80 system) to solve problems encountered in construction / Approach / Study / Process Model / Equation /Algorithm / Rule / Formula / Techniquel     A discussion is presented of a problem-solving systeml     To improve the efficiency of the method, the following approach may be      In order to an investigation was made to find the causes of thel     Although large collections of rules and equations have been complied, none are generally accepted l     This approach will be explained and discussed thoroughly in the body of the      This can be accomplished byl     This algorithm to compute the total cost can be described step by step as follows:l     The above preliminary analysis has provided important informationl     Various methods have been proposed for selecting an      These concepts have been applied to l     On the basis of the concept mentioned above,l     This can be achieved byl     This fact suggests that a new conceptl     This was accomplished by taking      The preparatory stage is very time consuming      Test are performed for validity, completeness, and compatibilityl     There is little hope of achieving successful      There has been an increasing awareness of the potential of using most far made have not taken this approach, with the exception ofl     Only a few studies can be      It is a very tedious process to go throughl     It is only when .. has been completed that .. may be effectedl     The entire interpretation process is conducted in one"s      These approaches are sometimes very      Several techniques can be usedl     A polynomial parametric model can be written as [the following]/[follows]:l     A xx model is constructed/formulated using      A xx model represents an xx by its      A process decision model captures the logic essential tol     From the equation above, xx is equal to the summation of xx times the      The validity of a xx model can be checked using Euler"s      Given a model, one can mathematically determine whether ... or      Equations for xx need to be derived and implemented in the      A number of heuristic rules have been developed forl     Optimum .. techniques can be made more reliable by ... so thatl     An algorithm based on the characteristic ... is used to determinel     Euler"s formula states the following:l     The completed model should agree with the      For manufacturing purposes, a detailed and precise model of the object is necessaryl     Engineering design models are very well defined; therefore, l     To keep the domain narrow enough to be implementable, yet wide enough to be of Viewl     from an implementation standpoint,l     From the point of view of this application,l     From this point of view, Zadeh suggested an inference rule named xxx (CRI for short).l    Information is the meaningful interpretation and correlation of some aggregation of data in order to allow one to make      From a practical point of view, the computational aspects of an FLC require a simplification of the fuzzy control      The use of a hammer to insert screws, although partly effective, tends to distort, destroy, and generally defeat the purpose of using a screw [Kusiak AI Implications for CIM ]Justificationl     We choose the so called xx in our experiment because it has received wide acceptance and canl     Prolog was chosen because it offers a well known and flexible environment in which fuzzy reasoning may be easily      The rationale behind this is that it can be much easier for an estimator to rate a cost as high than to attempt to place a dollar value on the      This strategy has been widely used in fuzzy control applications since it is natural and easy to      A function definition expresses the membership function of a fuzzy set in a functional form, typically a bell‑shaped function, etc.  Such functions are used in FLC because they lead themselves to manipulation through the use of fuzzy      It should be noted that in our daily life most of the information on which our decisions are based is linguistic rather than numerical in nature.  Seen in this perspective, fuzzy control rules provide a natural framework for the characterization of human behavior and decisions analysis.  l     Many experts have found that fuzzy control rules provide a convenient way to express their domain knowledge.  This explains why most FLCs are based on the knowledge and experience which are expressed in the language of fuzzy "if‑the" 4. ExamplesExample/ Datal     The data used in the following example was taken from an experiment in which xx was measured between x and x using a xx      The data consists of over xx      An example of xx is discussed and the control rules of xx are compared with a xxl     Examples of complex processes to which this technique may be applied are xx, xx,      The following example is constructed only for the purpose of illustrating the computational procedure      This example clearly demonstrates that the profile of an individual xx, or a very small group of xx, with no enough data to be studied statistically, can be meaningfully analyzed by fuzzy possibilistic      There is no space here to go into detail on all these methods, but deserve a mention and the bibliography will point to detailed references for those wishing this level of      Note that the golf ball spotting example is used throughout the      As well, the pros and cons of these representations from a process planning point of view will be      The method of using xx to implement xx described by Zadeh (1973) appeared more suitablel     As discussed [in the previous section]/[preciously],Relationl     We can not invert F" directly because it defines a many-to-one      The relationships appear very complicatel     Lifting tasks involve complex and imprecise relationship between the task variables and the human operator"s      These methods are based on the relationship between ... and      The fundamental concept of a fuzzy rating language is that we can establish a relationship among terms such as high, medium, and low, and then modify these      This article will thus mention the latter as well as the      The former two bear a close relation to a fuzzy Cartesian      The emphasis is on an implementation of a general approach to rule based decision / Attentionl     Careful evaluation is necessary to ensure l     Such a formulation does not change further      Considerable attention has been paid tol     Attention should be paid to an important finding of this      Caution should be exercised in this process to avoid     Primary consideration is given to ... components, though others can be accommodatedl     After ... has been defined by ..., a carefully analysis is carried out/performed to determinel     A number of factors such as to be taken into consideration before making the appropriate      It should be noted thatl     It is important to point out that      These considerations have heightened interest in the possibility of providing      We should stress the fundamental importance of the xxChapter5. / Disadvantagel     One of the major advantages of this new measure of xx is that it can be applied to the experimental study ofl     One advantage of using a .. is the ease of preparing      The xx system is versatilel     It has a very fast decision making processl     All the algorithms involve mostly logical   It can be easily and without additional cost implemented in a microprocessor‑based      It can reduce the waste of designing from      The advantages of using a xx to represent xx are the following:l     However, xx is not without its      In most cases, the xxx shows an improvement over the existing      Compared to the existing xx, the impacts of the xx are generally reduced by5% to9%.l     The "best case" results shows a savings of6% to9%.l     Most of the existing works based on xx approach can only recognize a xx .l     Most of the above methods are computational expansive and limited to      Some other advantages of xx are the following:l     The problem is the limitation of this method to a limited domain of      It proved limited in application because it demanded precision in system modeling that was impossible in      There are advantages to be gained in the structuring of costs and benefits, the use of xx, l     The disadvantages of this method are also disadvantages of conventional xx      This combines the best features of both techniquesl     Hopefully, this tool can be as the reference framework of for developing a xx platform, and helping the administration, marketing, and knowledge management activities in virtual      An improvement on the result shown above can be made by based on the data providedl     Discussion of these theories is beyond the scope of this reviewl     Based on the information contained in thisl     The result can be categorized into nine classesl     The results are illustrated by an examplel     The experimental results for each xx time are reported in Table      From the results obtained so far, it seem thatl     Because of the inaccuracy of the ..., a conclusion cannot be drawn as l     Although much effort has been made to., this reality is far from      The results indicate that the total benefits are higher than the total      Their results may then serve as guidelines for lower level models, less fuzzy and more 6. Conclusionl     From the discussion, one may conclude that      Form the above discussion, the conclusion can be reached that l     The conclusions drawn are also validl     In conclusion to this, it becomes obvious that the problem of xx lies not only      We have attempted to introduce some concepts associated with a theory of xx based on fuzzy      Considerable more work, hopefully, will be done in this areal     A fuzzy set procedure is proposed to solve xx selection problems interwoven with imprecise datal     Employing the compositional rule of inference, the assessment of the xx compatibility in achieving prescribed xx projectiles in any level of the hierarchy is made     This paper has presented a theoretical and experimental study of the xx process and xx     The experimental research results will hopefully serve as useful feedback information for improvements for xx     The scope of this contribution was to introduce a xx     In general, fuzzy sets theory provides an alternative foundation for xx analysis in a fuzzy Researchl     Thus, first extension of the approach could be, l     Present some cues for a further approach from Fuzzy Sets Theory application to l     Some improvements to the scheduling aspect of the model may be brought through additional levels in the hierarchy for more detailed representation of the scheduling and Figuresl     Figure7-1sketches these      The graphical representation of these functions is shown in Figure      The xx may be depicted as in Figure      Figure x shows the schematic diagram of  the l     Figure1though2provide a ... thatl     the architecture of this expert system for .... is illustrated in Figure      Figure2gives the outline of an ... systeml     Table shows thel     as shown in Table1and2l     This concept is illustrated in Figure2l     At the top of Table xx are shown two blocks of data. l     Each table or matrix has constructs xx through xx as row‑headings, xx through xx as column‑     A table of .. is developed and significant recommendations are Indicate Additionl     additionally, again, also, and then, as can be easily understood, besides, equally important, especially, finally, for the same reason, first, further, furthermore, in addition, last, likewise, moreover, next, second, third, too, evidently, obviously, roughly speaking, broadly speakingTo Indicate Cause and Effectl     accordingly, as a result, consequently, for this reason, hence, in short, otherwise, then, therefore, thus, trulyTo indicate Comparisonl     in a like manner, likewise, similarly , alternativelyTo Indicate Concessionl     after all, although this may be true, at the same time, even though, even so , I admit, naturally, of courseTo Indicate Contrastl     and yet, at the same time, but, for all that, however, yet, in fact, in contrast, in the real life, in spite of, nevertheless, notwithstanding, normally, on the contrary, on the other hand, still, traditionally, rather, unfortunately,To Indicate Time Relationshipsl     after a short time, afterwards, as indicated earlier, as long as, as soon as, at last, at length, at the moment, at that time, at the same time, before, earlier, currently, immediately, in the meantime, in recent years, lately, later, meanwhile, often, of late, presently, recently, soon, shortly, since, thereupon, temporarily, therefore, until, when, whileTo Indicate Special Features or Examplesl     for example, for instance, incidentally, indeed, in fact, in other words, in particular, in practice, specifically, that is, to illustrate, in this respect, theoretically, as mentioned before / aboveTo Indicate Summaryl     in brief, in conclusion, in short, in summary, on the whole, to conclude, in general, to summarize, to sum up, as a result, ultimately,VERB PHASEl     build a .. modell     build up the key linkl     began a new era in      can be     regarded as / achieved / used to/for / found / obtained throughl     can result inl     carries out  ... tasks l     production information in order to simultaneouslyl     contains all information necessary to describel     do not make use of production informationl     deals withl     end with failurel     fetch the information from the model directlyl     has great potential / yet to be resolved / spurred the development of  / been recognized as l     BE   aimed at / built up / carried out by / essentially concerned with / considered to be the key technology / associated with each feature of a composite component / cable of / currently implemented for / demonstrated by an example / finally reached / made equal to / equivalent to  / more suitable to / oriented to / interpreted as / pointed out / potentially of great benefit in the complex task of  / shown in  / used to effectively guide the searchl     makes use of l     make upl     meets the needs of real life production,/ the current demands placed upon itl     must be justifiedl     point outl     play an important rolel     relates tol     rely on l     satisfy the needsl     determine the total requirements for the      uses ... as a key to search      without relying onl     will be available/ performed/ overlookedNOUN PHASEl     a basic technical function of l     a critical needl     a key / principle feature ofl     a substantial impact onl     an intensive review was conductedl     an increasing need for expanding the application of      an important component / function / aspect / issuel     each rule is numbered in sequencel     each of these involvesl     for this calculation, it is necessary to definel     in the physical environment / integration ofl     in the reality of situations where …l     many aspects of l     most past efforts have been spent on      common sense to a well studied and documented technical      sources of additional information on ... are listedl     systematic and rationally structured formatl     the basis on which a  range of can be established is shownl     THE   basic philosophy / principles of / key element / general hypothesis / candidate list of / concept of  ... has attracted wide interest /  function is concerned with / heart / impact / nature / role / task of / kernel functions l     the number of parts needed to l     the above statement means thatl     the output data is passed to l     the proposed method /  underlying principlel     the recommendations made in this report, if implemented, shouldl     this information resides inl     this process is composed of ... different ... operationl     along with the use of l     concerning general aspects ofl     due tol     for later use in generating...,l     in turn,l     IT IS     believed / noted / assumed / suggested / shown / quite evident / appears / implies that / intended for / of no important tol     it can be claimed/concluded thatl     it demonstrates the decisions required of l     it also provides information to      it becomes essential to l     let ... be the probability thatl     once... is written, it is compiled      suppose it is observe