Behavior Study Of Affine Transformation Logic Metric Space And Several Types Of Special Formula And Its Applications

In Quantitative Logic, truth degree of formulas and similarity degree between formulas are systematically introduced through establishing graded versions of basic logical notions. Meanwhile the pseudo-distance between formulas is constructed and theory of the logic metric space (LMS for short) is established correspondingly. Based on these, one can further discuss the divergency degree and consistency degree of a logical theory Γ and several types of approximate reasoning can be studied in LMS. Up to the present, the theory of LMS has been generalized from classical two-valued propositional logic to many kinds of n-valued propositional logics (n>2).It is worth noting that the theory of the structure of LMS, as a metric space, has not been thoroughly developed. However, some discussion about reflexive trans-formations in classical LMS has been found recently and seems to be a new beginning of research on the structure of LMS although not mature. To furtherly develop this idea, the present paper studies the affine transformations and obtains that the con-cepts of truth degree as well as similarity degree keep unchanged under the affine transformations. Meanwhile, the theory of reflexive transformations is generalized into (?)*-Lindenbaum algebra.On the other hand, since the theory of Boolean functions sets a foundation for the concept of truth degree in classical LMS and also plays an important part in cryptog-raphy, one can see that the relationship between Quantitative Logic and cryptography seems very close in the sense. Based on this thought, the concepts of linear logic formu-las, symmetric logic formulas as well as avalanche logic formulas of LMS are proposed in this paper one after another, and the sparse degree of these formulas is obtained through their distribution in the whole space. More importantly, these results can be fed back to cryptography and offer useful informations for cryptographists during code transferring.Furthermore, the Shannon expansion of Boolean functions is applied to n-valued Lukasiewicz logic system Ln. The methods of expressing McNaughton functions are given and counting problem of m-ary n-valued McNaughton functions is solved.The present paper is divided into5chapters.The first chapter overviews some basic knowledge about Quantitative Logic and Boolean function in cryptography, which are preparatory for the latter chapters.Chapter2firstly gives a reflexive transformation in logic system (?)*. And the properties of reflexive transformation are studied in (?)*logic metric space. Then the concept of affine transformation is introduced into the classical logic system. The definition of affine transformation Φ on F(S) is proposed, which is proved to be an automorphism of F(S). It is obtained that the concepts of truth degree, similarity degree as well as pseudo-distance keep unchanged under the affine transformation Φ. In the classical logic system, the affine transformation is a new transformation with reflexive transformation being its special case.Based on the concept of linear Boolean functions in cryptology, Chapter3gives the concept of linear logic formulas in classical logic metric space. And methods of construction of n-ary linear logic formulas are given. In classical logic metric space, the properties of linear logic formulas under reflexive transformation are studied. It is proved that the truth degrees of all linear logic formulae are equal to1/2. However there are2n+1kinds of truth degrees to all Boolean functions which contain n variables. This shows that the distribution of linear logic formulas in all logic formulas is sparse. This also verifies from the theory of Quantitative Logic that the structure of linear Boolean function is relatively simple. Then a kind of Boolean functions of degree equaling to k can be obtained by multiplying k linear Boolean functions. It is also proved that the truth degree of logic formulas corresponding to this kind of Boolean functions is equal to1/2k. This shows that the distribution of logic formulas corresponding to this kind of Boolean functions of degree equaling to k in all logic formulas is sparse. Hence this kind of Boolean functions can be used in cryptography.Chapter4generalizes Shannon expansion in symbolic computation tree logic. In n-valued Lukasiewicz logic system, the expansion of n-valued McNaughton func-tions which are induced by logical formulas are studied. The quasi disjunctive normal form and quasi conjunctive normal form of m-ary n-valued McNaughton functions are given. Then the counting problems of m-ary n-valued McNaughton functions are solved. In n-valued Lukasiewicz logic system, the construction of formulas and counting problems of their logic equivalence class are solved.As the method of construction of many-valued McNaughton functions is given, we propose the concept of symmetric three-valued McNaughton functions. Then the concepts of symmetric logic formulas and pseudo-symmetric logic formulas are given in three valued Lukasiewicz logic system L3. The properties of symmetric logic formulas under logically equivalence are studied. The relationship of symmetric logic formulas in L3and classical logical system L, and the number of them are given. It is proved that the ratio of the number of symmetric formulas with n atoms over the number of all formulas with n atoms converges to zero when n tends to infinite. It is also proved that the set of truth degrees of symmetric logic formulas is dense in [0,1]. On the other hand, the set consisting of all symmetric logic formulas is a nowhere-dense set in the logic metric space. Finally, the construction of symmetric logic formulas in L3is given.Chapter5introduces the concept of Boolean functions satisfying strict avalanche criterion in cryptology into Quantitative Logic. The concept of avalanche logic formu-las is proposed. The truth degree of avalanche logic formulas and its properties are studied. It is proved that the truth degree τ(A) of avalanche logic formula A satisfies following equation1/4≤τ(A)≤3/4. In particularly, it is proved that the set of truth degrees of avalanche logic formulas which contain at least three atom formulas is H1={k/2n-1|2n-3≤k≤3×2n-3;n-3,4,…}, Using cryptography terms, the set of Hamming weight of n-ary (n≥3) Boolean functions satisfying strict avalanche criterion is w(n)={ω(f(x))|2n…2≤ω(f(x))≤3×2n-2, and ω(f(x)) is even number.}It can simplify the counting problem of avalanche Boolean functions through excluding the number of Boolean functions which do not satisfy this condition. Then, a formula for calculating the total number of n-ary (n≥3) avalanche Boolean functions is established by means of a newly introduced function ζ, and a method of construction of avalanche logic formulas are given. Finally, the properties of avalanche logic formulas of order k under reflexive transformation are studied. The upper and lower bounds of the number of Boolean functions satisfying strict avalanche criterion are obtained.