AXIOMATIC APPROACH

In this chapter we discuss techniques for program verification and their mathematical justification. The basic idea behind these methods was originally presented by Floyd mathematical formulations and logical justifications were developed by Cooper and Manna, and others, and continued in King s Ph.D. thesis in which he presented the development of a partial implementation for these techniques. A sanewhat different axiomatic approach has been pursued by Hoare et al. The reader who has never made acquaintance with the formalism of the first order predicate calculus should at this point turn to Appendix A for a brief and unrigorous exposition of the material relevant to this chapter. 

Hbare, C. A. R., "Procedures and Parameters An Axiomatic Approach," in Symposium on Semantics of Algorithmic Languages, E. Engeler (editor),... 

Probability can be defined as a limiting case of a frequency ratio, and from this view the various rules of probability can be derived. An alternative approach is an axiomatic one that states that there is a quantity called probability associated with events and that it possesses assigned properties. The former is largely the frequentist point of view, the axiomatic approach is shared by Bayesians and non-Bayesians alike. 

Probability values lie continuously in the range 0 to 1 inclusive, where the endpoints zero and unity are identified with impossibility and certainty, respectively. This follows immediately for the frequentist for the axiomatic approach it is adopted as an axiom, but one imbued with Laplace s commonsense. Any other range could be chosen at the cost of greater difficulty of interpretation. 

A new qualitative quantum-chemical concept of the elementary act of addition polymerization has been proposed as the development of the polymerization theory. An extensive set of various data on the kinetics and the mechanism of polymer structure controlling has been found to have a new explanation from an uniform viewpoint. This concept is developed in the framework of the axiomatic approach to the general polymerization theory and is based on five postulates, namely the principle of the intermediate, the principle of intermediate cyclicity, the principle of correspond-... 

The limitations of modem experimental physico-chemical methods in the direct and detailed study of the monomer insertion in the polymer chain on the one hand, and the large amount of information about the dependence of the macromolecules primary structure on the reaction conditions on the other hand should be taken into account. Consequently, it seems that the only real path of the search for relationships between the electronic aspects of the reaction and the observed microstructural statistics of polymers is the heuristic method of formulation of definite rules. It is most suitable for carrying out an adequate formulation of these imaginary discoveries in the framework of an axiomatic approach. 

A considerable advantage of the axiomatic approach is its universality (for instance, the author previously succeeded in developing the kinetic statistic theory of molecular weight distributions of polymers formulated according to the model for the axiomatic theory [6]). Moreover, in the initial stage of the... 

The fundamental and promising character of any concept is determined not only by its form (although it is doubtless very important in itself) but rather, by its content. In our case the form is caused by the axiomatic approach, whereas the content is based on the ideology of symmetry. Hence, it seems desirable to characterize briefly the sources, development and historical prospects of the ideas determining the objective regularity of applying the proposed approach to the investigation of the polymerization mechanism. 

The formal presentation of the problem was an attempt to develop a qualitative quantum-chemical model of the elementary polymerization act - adequately describing the complex of available experimental facts taking into account the above properties of propagation, the theoretical achievements of organic chemistry and the known principles of invariance in the framework of the axiomatic approach on the ideological basis of symmetry concepts. It should be emphasized that we do not mean to achieve in the present work a global interpretation of the entire experimental material accumulated in the literature. The aim of the author is mainly to demonstrate the promising character of this approach as a whole. 

The axiomatic approach to the development of the theory of addition polymerization presented here is the first consequent attempt to interpret as many various experimental facts as possible on the basis of as few assumptions as possible. It should be noted that at present this concept has neither a wide recognition nor has been the subject of profound criticism.. However, the author hopes that this is due to the fact that these concepts are not very well known. In this connection, the author should like to draw the attention of the reader to the following points ... 

Secondly, the possibilities of using the axiomatic approach are not exhausted by the present paper. From the methodological standpoint, this is the most... 

We emphasize that we did not discover a new or more basically justified axiom of thermodynamics. Namely, the minimum principle of energy is derived from the maximum principle of entropy, and the stability theorems are derived from the minimum principles of energy enthalpy, free energy, and free enthalpy. Thus, we cannot construct an axiomatical approach to thermodynamics by simply adding the stability theorem to the axioms of thermodynamics. 

In the axiomatic approach to quantum mechanics these equations are obtained by substituting differential operators for the classical variables of momentum and energy... 

This chapter reviews basic ideas of probability, random variables, probability distributions, and sampling distributions of a statistic. The axiomatic approach is followed to define probabilities of events in a sample space, which is the collection of all possible outcomes of an experiment. When the outcomes are equally likely, the probability of an event is just the ratio of the number of outcomes in the event to the number of outcomes in the sample space. 

The axiomatic approach is credited to Kolmogorov. The relative frequency approach to the definition of probability states that the probability P(A) of an event A is the limit ... 

Thus, today, it is understandable that it would be desirable to have axiomatic approaches, which are so characteristic of mathematics, also in other sciences that use mathematics, which includes chemistry. Hilbert himself made significant and fundamental contributions to theoretical phyacs, which was at that time the nearest candidate for more rigorous formulatim. Chemistry has been moving gradually in that direction. In the mid-1950s, chemistry was still viewed as an empirical science. This is not currently the case. In those earlier days, there were rnmors among physicists that chemistry was... 

The Molecular Theory of Solutions , by Prigogine, includes thermodynamic considerations, and Zemansky s book is a later edition of the book mentioned earlier. Wilson s volume is of a fairly advanced level and requires a fairly high standard of mathematical ability. It is mainly designed for the use of physicists. Chapters include accounts of partial differentiation, and the book approaches entropy through Carnot cycles but also describes Caratheodory s axiomatic approach. Superconductivity and solutions are considered thermodynamically. Caldin s introduction is designed for chemistry undergraduates. A student who has mastered the text should be well prepared to go on to more advanced work. Chisholm and de Borde s book develops the equations for Bose-Einstein, Fermi-Dirac, and classical statistics by unusual routes and then applies the... 

HB Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd edition, Wiley, New York, 1985. The classic text on the axiomatic approach to thermodynamics. The logic of thermodynamics is explained with great clarity. ... 

Ayers, P.W. Atoms in molecules, an axiomatic approach. I. Maximum transferability. J. Chem. Phys. 2000,113, 10886. 

In this paper we present an axiomatic approach to the elasto-plastic rate-independent behaviour of the metals. The approach is based on mechanical and physical considerations given by Lee, Liu [1], [2], Teodosiu, Kroner, Sidoroff [3]-[6], Mandel [7], Kratochvil [81, [9], llalphen [loj. 

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