User:Leonid2/sandbox

Dynamic logic (DL) has been created by Leonid Perlovsky to overcome computational complexity encountered since the 1960s in artificial intelligence, modeling the mind, and in many areas of applied and engineering mathematics,.

Combinatorial Complexity
The combinatorial complexity (CC) is often encountered due to a need to consider combinations of multiple elements. For example, in recognition, detection, or in modeling perception abilities of the mind, an object of interest should be found among many other objects. To find an object usually its observed pattern (on the screen or visual cortex) should be matched to a prototype or "mental representation" of this object. In this process a prototype or "mental representation" should be modified in many ways to match the observed pattern. This might require many computations. But even worse complexity is due to the fact that the process needs to identify pixels making up the object, and those belonging to other objects. In other words, surrounding objects have to be identified in parallel. Thus many prototypes (or representations) have to be matched to many patterns. This requires considering combinations of many patterns and prototypes. 100 objects in the field of view is not too many. But combinations] of 100 objects and prototypes is a very large number, 100100. This number is larger than all interactions of all [[elementary particles in the Universe in its entire lifetime. Thus this relatively simple problem seems unsolvable.

CC and the Gödelian difficulties of logic
If Gödelian arguments leading to fundamental difficulties of logic (incompleteness) are applied to a finite system (a computer, or mind), the arguments do not lead to fundamental difficulty, but to CC. Practically, from an algorithmic point of view CC is as bad as fundamental difficulty of logic. Or in other way, CC is as fundamental as Gödelian incompleteness. For this reason, any algorithm using classical logic (or simply logic for shortness) encounters CC. Moreover, algorithmic methods created specifically to overcome limitations of logic, such as neural networks or fuzzy systems, still make logical steps, such as during training, or during setting degree of fuzziness, and thus encounter CC.

DL overcomes CC
For the above reasons creating a mathematical technique capable of solving complex problems without using logic has been a fundamental problem. DL accomplishes this by considering not logical states. Whereas classical logic operates with static logical states (e.g. "this is a chair"), DL operates with processes. DL processes proceed "from vague to crisp." From vague states ( or representations to crisp ones. Crisp, approximately logical states (prototypes, representations") are achieved at the end of a DL process. Initial DL states are highly vague, so that any prototype matches all representation and there is no need to consider combinations of representations. In this way DL overcomes CC.

DL vs. Fuzzy Logic (FL)
DL can be considered an extension of FL toward dynamic fuzziness.

DL engineering applications
DL algorithms solved many engineering problems unsolvable for decades, and problems from emerging areas, which could not have even be formulated mathematically. These include detection of objects below noise, tracking of objects below noise ("track before detect"), fusion of objects from diverse sources, swarm intelligence, learning situations, integration of language and cognition, emotionality of languages, hierarchical intelligence, cognitive models of conscious and unconscious, evolution of cultures, symbol processes, perceptual symbol system, grounding, binding, and others.

DL, the mind and cognitive science
DL has been experimentally demonstrated to be an adequate model of perception and cognition ;.

DL cognition, language, and aesthetic emotions
DL is a mathematical foundation for cognitive models describing interaction between cognition and language ;. These models made a number of unique observable predictions, some of them were confirmed, none were disconfirmed.

DL is a mathematical foundation for cognitive models of aesthetic emotions. According to DL, aesthetic emotions are related to knowledge and include emotions of the beautiful, sublime, musical emotions, emotions of prosody, and emotions of cognitive dissonances. This is a previously unsolved problem in cognition, psychology, and philosophy; in particular, musical emotions Aristotle considered an unsolved problem, and Darwin considered the greatest mystery. DL predicts certain difficulties in measuring aesthetic emotions. These models have made a number of unique, experimentally verifiable predictions, in particular that emotions of the beautiful are related to the meaning of life (which is mostly unconscious), and musical emotions are related to overcoming cognitive dissonances and keeping in mind contradictory knowledge; the later one has been experimentally confirmed,.

Aristotle's theory of the mind and DL
Aristotle was specific in emphasizing that logic is not a fundamental mechanism of the mind and suggested a theory of the mind similar to DL: forms of the mind (representations) are initially not logical, they become logical after "the mind meets matter" (in interaction of top-down with bottom-up signals).