Sufficient theory

Sufficiency. Given the example, the theory we employ to deduce the conditions must guarantee that the conditions on equivalence or dominance will be satisfied, since the example itself is simply an instance of the particular problem structure, and may not capture all the possible variations. The sufficient theory will thus end up being specialized by the example, but since we have asserted the need for validity of this process, it will still be guaranteed to satisfy the abstract necessary and sufficient conditions on dominance and equivalence. 

Before showing how the logical analysis will be carried out, it is useful to describe the sufficient theory we will be using for the specific flowshop example. This theory is not restricted to flowshop scheduling, but applies to many state-space problems. 

In these cases there is no well defined notion of a looser constraint, the choice is then either to force those variables to be equal in x and y, or to find some path from their value to a constraint on another inter- or intrasituational variable and thus be able to show that their values in jc, y should obey some ordering based on these other constraints. This topic is the subject of current research, but is not limiting in the flowshop example, since no such constraints exist. Lastly, it is not enough to assert conditions on the state variables in x and y, since we have made no reference to the discrete space of alternatives that the two solutions admit. Our definition of equivalence and dominance constrains us to have the same set of possible completions. For equivalence relationships the previous statement requires that the partial solutions, x and y, contain the same set of alphabet symbols, and for dominance relations the symbols of JC have to be equal to, or a subset of those of y. Thus our sufficient theory can be informally stated as follows ... 

In general, we would like our sufficient theory to have the following qualities ... 

Efficacy. We must balance making the sufficient theories simple versus making them effective. If, for example, the relative ordering of the intrasituational variables or some function of the state variables, such as the difference between the end-times of successive units, is the driving... 

Modularity. Since we would like to use the sufficient theory in a variety of contexts and problems, we need a theory that was easy to extend and modify depending on the context. In our state-space formulation the sufficient theory is couched in terms of constraints on variables. This theory gives us the opportunity to modularize its representation, partitioning the information necessary to prove the looseness of one type of constraint from that required to prove the looseness of a different constraint type. The ability to achieve modularity is a function not only of the theory but also of the representation, which should have sufficient granularity to support the natural partitioning of the components of the theory. 

The next section will focus on the representation necessary to express this sufficient theory to the computer, so that it can automatically carry out the reasoning associated with analyzing the examples selected by the syntactic criteria presented in this section. Section V will describe the learning methodology, which, using the representation of Section IV, will generate the new dominance and equivalence conditions. 

We need to introduce two more types of predicates to support the sufficient theory used in the analysis. As we have stated in Section III, D, the sufficient theory rests on being able to prove that the constraints on one state are looser than those on another. The predicate that is used to express this is looser - constraint — on — variable , which takes the form ... 

With the basic predicate types in place we can now define the various implications that will allow us to express the sufficient theory. There are two key steps that have to be made. The first is to take an intersituational variable and figure out what the constraint on the variable is, which of the... 

We can now use the preceding implications (rules) to help build the general sufficient theory. As we stated in Section III, D, we have to ensure that all the intersituational variables of the next state are more loosely constrained in x than in y. Thus, our top-level implication (rule) is simply... 

This completes the representation of the sufficient theory required for the flowshop example. It consists of about 10 different predicates listed in Table II and configured in four different implications (rules). These predicates have an intuitive appeal, and are not complex to evaluate, thus the sufficient theory could be thought of as being simple. The theory is capable of deriving the equivalence-dominance condition in flowshop problem. It is, however, expressed in terms that could be applied to any problem with that type of constraint. Thus it has generality, and since we can add new implications to deal with new constraint types, it has modularity. 

The general implications of the sufficient theory (i.e., the state-space sufficient theory). 

Since we are not solely dependent on IR spectra for identification, a detailed analysis of the spectrum will not be required. Following our general plan, we shall present only sufficient theory to accomplish our purpose utilization of IR spectra in conjunction with other spectral data in order to determine molecular structure. 

Each monograph places emphasis on the practical application of the technique it covers, yet also includes sufficient theory and detail on instrumentation to enable new graduates and non-specialists alike to grasp fully the fundamentals concerned. 

The idea that the behavior and properties of a molecule are concealed in the fundamental structural formulae has been aroimd for a long time. Modern physical chemistry has become increasingly orientated toward understanding how these properties can be decoded from the structure. Ideally, all properties of a chemical compound would be calculated from first principles. This is, however, unlikely in the foreseeable future because of a number of reasons, including the lack of sufficient theory and limits of available computational power. An alternative approach to finding qualitative mathematical relationships between the intrinsic molecular structure and observable properties of a chemical compound will be extremely valuable to both industrial and academic chemists. 

The aim of this chapter is to equip the pharmacometrician with sufficient theory and application to confidently approach the PK/PD-based analysis of count data and thus derive the maximum return on investment from clinical study data. Section 27.2 provides a motivating example and Section 27.3 presents relevant definitions and theory. Section 27.4 applies the theory to the example and introduces diagnostics methods. Throughout the chapter, the focus is on population approaches using nonlinear mixed effects models. Code segments of NONMEM control files are presented in the appendix. Mixed effects analysis methodology is described in detail in Chapter 4 of this text. 

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