Intuitively, task A is reducible to task B if the algorithm for solving problem B (if it exists) can also be used as a subroutine to efficiently solve problem A. When this is true, solution A cannot be more complicated than solving problem B. Higher complexity means Higher appreciation of required computing resources in this context. For example, high time costs, high memory requirements, expensive need for additional hardware processor cores.
The mathematical structure generated on many problems by reductions of a certain type usually generates a preorder, the equivalence classes of which can be used to determine the degrees of unsolvability and complexity classes.
Mathematical definition
In mathematics, reduction is a rewriting of a process into a simpler form. For example, the process of rewriting the fractional part to the one with the lowest denominator of an integer (while keeping the numerator as an integer) is called "fractional reduction". Rewriting a radical (or “root”) example with the smallest possible integer and radical is called “radical reduction.” This also includes various forms of reduction of numbers.
Types of Mathematical Reduction
As described in the above example, there are two main types of reductions used in complex calculations, multiple abbreviations, and Turing abbreviations. Multiple reduction displays instances of one problem in case another occurs. Turing abbreviations make it possible to calculate a solution to one problem, assuming that the other problem will also be easily solved. Multiple reduction is a stronger type of Turing reduction and more effectively divides problems into separate complexity classes. However, increasing restrictions on multiple reductions makes it difficult to find them, and quantitative reduction often comes to the rescue.
Difficulty classes
The task is completed for one complexity class, if each problem in the class is reduced to this task, and it is also in it. Any solution to a problem can be used in conjunction with abbreviations to solve every problem in the class.
Reduction problem
However, cuts should be light. For example, it is entirely possible to reduce a complex task, such as the problem of logical feasibility, to a completely trivial one. For example, to determine whether a number is equal to zero, due to the fact that the reduction machine solves the problem in exponential time and outputs zero only if there is a solution. However, this is not enough, because although we can solve the new problem, the reduction is as difficult as the solution to the old problem. Similarly, an abbreviation that calculates an uncountable function can reduce an unsolvable problem to a solvable level. As Michael Sipser notes in Introduction to Computing Theory: “Reduction should be simple compared to the complexity of typical problems in a class. If the reduction itself was difficult to calculate, then it will not necessarily give an easy solution to the problems associated with the task. "
Optimization tasks
In the case of optimization problems (maximization or minimization), mathematics comes down to the fact that reduction is what helps to display the simplest solutions. This technique is regularly used to solve such problems of varying degrees of complexity.
Vowel Reduction
In phonetics, this word refers to any change in the acoustic quality of vowels associated with changes in voltage, sonority, duration, volume, articulation, or position in a word, and which is perceived as “attenuation” by ear. Reduction is what makes vowels shorter.
Such vowels are often called reduced or weak. In contrast, unreduced vowels can be described as full or strong.
Reduction in the language
Phonetic reduction is most often associated with the centralization of vowels, i.e., a decrease in the number of movements of the tongue when they are pronounced, as with a characteristic change in many unstressed vowels at the ends of English words to something closer to schwa. A well-studied example of vowel reduction is the neutralization of acoustic differences in unstressed vowels, which occurs in many languages. The most common example of this phenomenon is schwa sound.
Common features
The duration of sound is a common factor in reduction: in fast speech, vowels are reduced due to the physical limitations of articulatory organs, for example, the tongue cannot move to the prototype position quickly or completely to get a full vowel sound (compare with clipping). Different languages have different types of vowel reduction, and this is one of the difficulties in mastering the language. Learning the vowels of a second language is a science.
Vowel reduction associated with stress is a major factor in the development of Indo-European Ablaut, as well as other changes restored by historical linguistics.
Languages without reduction
Some languages, such as Finnish, Hindi, and Classic Spanish, are said to lack vowel reduction. They are often called syllable languages. At the other end of the spectrum, Mexican Spanish is characterized by a reduction or loss of unstressed vowels, mainly when they are in contact with the sound “s”.
Reduction in terms of biology and biochemistry
Reduction is sometimes called correction of a fracture, dislocation or hernia. Also, reduction in biology is an act of reducing any organ as a result of evolutionary or physiological processes. Any process in which electrons are added to an atom or ion (as by removing oxygen or adding hydrogen) and accompanied by oxidation is called reduction. Do not forget about the reduction of chromosomes.
Reduction in philosophy
Reduction (reductionism) covers several related philosophical topics. At least three types can be distinguished: ontological, methodological and epistemic. Although the arguments for and against reductionism often include a combination of positions related to all three types of reductions, these differences are significant because there is no unity between the different types.
Ontology
Ontological reduction is the idea that each specific biological system (for example, an organism) consists only of molecules and their interactions. In metaphysics, this idea is often called physicalism (or materialism), and it assumes in a biological context that biological properties control physical properties and each specific biological process (or token) is metaphysically identical to a specific physicochemical process. This last principle is sometimes called token reduction, in contrast to the stronger principle, according to which each type of biological process is identical to the type of physico-chemical process.
Ontological reduction in this weaker sense is today the mainstream position among philosophers and biologists, although philosophical details remain controversial (for example, are there really emerging properties?). Different concepts of physicalism can have different consequences for ontological reduction in biology. The denial of physicalism by vitalism - the view that biological systems are controlled by non-physicochemical forces is largely of historical interest. (Vitalism also allows for various concepts, especially with regard to how non-physicochemical forces are understood) Some authors have vigorously stated the importance of metaphysical concepts in discussions of reductionism in biology.
Methodology
Methodological reduction is the idea that biological systems are most effectively studied at the lowest possible level, and that experimental studies should be aimed at identifying the molecular and biochemical causes of everything. A common example of this type of strategy is the decomposition of a complex system into parts: a biologist can examine the cellular parts of an organism to understand its behavior, or examine the biochemical components of a cell to understand its features. Although methodological reductionism is often motivated by the presumption of ontological contraction, this procedural recommendation does not follow directly from it. In fact, unlike token reduction, methodological reductionism can be quite controversial. It is argued that exclusively reductionist research strategies exhibit systematic biases that overlook relevant biological features and that for some issues a more fruitful methodology is to integrate the discovery of molecular causes with the study of higher-level functions.
Episteme
Epistical contraction is the idea that knowledge of one scientific field (usually about processes of a higher level) can be reduced to another body of scientific knowledge (usually of a relatively lower or more fundamental level). Although the endorsement of some form of epistemic reduction may be motivated by ontological decline in combination with methodological reductionism (for example, the past success of reductionist research in biology), the possibility of epistemic reduction does not follow directly from their relationship. Indeed, the debate about reduction in philosophy, biology (and the philosophy of science in general) has focused on this third type of reduction as the most controversial of all. Before evaluating any reduction of one aggregate knowledge to another, one should study the concept of these knowledge organs and what this will mean for their “reduction”. A number of different reduction models are proposed. Thus, the discussion about the reduction of biology not only revolved around how epistemic reduction is possible, but also those concepts that play a role in real scientific research and discussion. Two main categories can be distinguished:
- reduction models of the theory that claim that one theory can be inferred from another theory;
- explanatory reduction models that focus on whether higher level functions can be explained by lower characteristics.
General conclusion
The definitions of reduction from various sciences mentioned in this article are far from the limit, because in fact there are many more. Despite all the differences in the definition of reduction, they all have something in common. First of all, reduction is perceived as reduction, reduction, simplification and reduction of something more complex, cumbersome and systemic, to something simpler, understandable and easily explainable. This is a key idea explaining the popularity of the term “reduction” in so many unrelated sciences. Qualitative reduction wanders from science to science, making each of them simpler and more understandable to both professional scientists and ordinary people.