Extra Concepts and Domains of Truth Values in Interpreting Logical Languages in Artificial Intelligence Systems

So-called extra concepts introduced to represent structurally defined objects and structures with unlimited complexity in their traditional understanding are suggested. The concepts of extra-word, extra-regular expression, context-free extra-grammar, and context-free extra-language are extensions of the well-known concepts used in the theory of formal languages. Extra words are a special case of symbol sequences; however, the set of all extra words in any alphabet is countable, whereas the set of all symbol sequences is not countable. The periodic codes of rational number representations in some positional numeration system are in fact extra words in this terminology. The concept of an extra-tuple is a generalization of the tuple concept, which implies the possibility of interpreting extra-tuples both as finite and as the indicated type of infinite sequences of elements of an arbitrary, not more than countable set, and it should be noted that the set of all possible sequences of such sort remains countable. By using the introduced concepts, a countable family of the domains of truth values has been specified for multivalued and countable-valued logics, each of which is a bounded lattice of finite or countable power with the traditional definition of basic logical operations of negation, conjunction, and disjunction. The hierarchical construction of the proposed truth domains makes it possible to introduce new logical operations in consideration that do not have analogues in the classical logic.

UNIVERSAL SUBGROUPS OF POLISH GROUPS

Given a class${\cal C}$of subgroups of a topological groupG, we say that a subgroup$H \in {\cal C}$is auniversal${\cal C}$subgroupofGif every subgroup$K \in {\cal C}$is a continuous homomorphic preimage ofH. Such subgroups may be regarded as complete members of${\cal C}$with respect to a natural preorder on the set of subgroups ofG. We show that for any locally compact Polish groupG, the countable powerGωhas a universalKσsubgroup and a universal compactly generated subgroup. We prove a weaker version of this in the nonlocally compact case and provide an example showing that this result cannot readily be improved. Additionally, we show that many standard Banach spaces (viewed as additive topological groups) have universalKσand compactly generated subgroups. As an aside, we explore the relationship between the classes ofKσand compactly generated subgroups and give conditions under which the two coincide.

STOCHASTIC EQUATIONS AND QUASI-INVARIANCE ON INFINITE PRODUCT GROUPS

A stochastic differential equation on an infinite-dimensional Lie group G constructed as the countable power of a compact Lie group G is considered. The existence and uniqueness of the solutions and quasi-invariance of their distribution are proved.