Intuitively, problem A is reducible to problem 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, solving A cannot be more difficult than solving problem B • Higher complexity means a higher estimate of the required computational resources in a given context. For example, high time costs, large memory requirements, expensive need for additional hardware processor cores.
A mathematical structure generated on a set of problems by reductions of a certain type usually forms a preorder whose equivalence classes can be used to determine degrees of unsolvability and complexity classes.
Mathematical definition
In mathematics, reduction is the rewriting of a process into a simpler form. For example, the process of rewriting a fractional part into one with the smallestthe denominator of an integer (while keeping the numerator integer) is called "reduction of the share". Rewriting the radical (or "radical") example with the smallest possible integer and radical is called "radical reduction". This also includes various forms of number reduction.
Types of mathematical reduction
As described in the example above, there are two main types of reductions used in complex calculations, multiple reductions and Turing reductions. Multiple reduction maps instances of one problem in case another occurs. Turing contractions allow one to calculate a solution to one problem, assuming that another problem will also be easily solved. Multiple reduction is a stronger type of Turing reduction and separates problems more efficiently into distinct complexity classes. However, the increase in restrictions on multiple reduction makes it difficult to find them, and here quantitative reduction often comes to the rescue.
Classes of difficulty
A problem is complete for one difficulty class if every problem in the class reduces to this problem and it is also in it. Any problem solution can be combined 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 problem such as the logical satisfiability problem to something quite trivial. For example, to determine whether a number is equal to zero, due to the fact that the reduction machine decidesproblem 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, doing the reduction is just as difficult as solving the old problem. Similarly, a reduction that computes an uncomputable function can reduce an undecidable problem to a solvable one. As Michael Sipser points out in An Introduction to the Theory of Computation: “The reduction should be simple, compared to the complexity of typical problems in the classroom. If the reduction itself were intractable, then it would not necessarily provide an easy solution to the problems associated with the problem.”
Optimization problems
In the case of optimization problems (maximization or minimization), mathematics boils down to the fact that reduction is what helps to display the simplest possible solutions. This technique is regularly used to solve similar 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 tension, sonority, duration, volume, articulation or position in the word, and which is perceived by the ear as "weakening". 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 language
Phonetic reduction is most often associated with the centralization of vowels, i.e., a decrease in the number of language movements during their pronunciation, as with a characteristicchanging many unstressed vowels at the ends of English words to something approaching 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 the sound schwa.
Common features
Sound length is a common factor in reduction: in fast speech, vowels are shortened due to physical limitations of the articulatory organs, e.g. the tongue cannot move into prototypical position quickly or completely to produce a full vowel (compare with clipping). Different languages have different types of vowel reduction, and this is one of the difficulties in language acquisition. Learning the vowels of a second language is a whole science.
Stress-related vowel contraction is a major factor in the development of Indo-European ablaut, as well as other changes reconstructed by historical linguistics.
Languages without reduction
Some languages such as Finnish, Hindi and Classical Spanish are said to lack vowel reduction. They are often called syllabic languages. At the other end of the spectrum, Mexican Spanish is characterized by the reduction or loss of unstressed vowels, mainly when they are in contact with the "s" sound.
Reduction in terms of biology and biochemistry
Reduction is sometimes called the correction of a fracture, dislocationor hernia. Also, reduction in biology is the act of reducing an 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 themes. At least three types can be distinguished: ontological, methodological and epistemic. Although arguments for and against reductionism often involve a combination of positions associated with 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 suggests in a biological context that biological properties control physical properties and that each specific biological process (or token) is metaphysically identical to any specific physical-chemical process. This last principle is sometimes referred to as token reduction, as opposed to the stronger principle that each type of biological process is identical to a type of physico-chemical process.
Ontological reduction in this weaker sense today ismainstream position among philosophers and biologists, although the philosophical details remain debatable (for example, are there really emergent properties?). Different conceptions of physicalism can have different implications for ontological reduction in biology. Vitalism's rejection of physicalism, the view that biological systems are governed by forces other than physical-chemical forces, is largely of historical interest. (Vitalism also allows for different conceptions, especially with regard to how non-physico-chemical forces are understood) Some writers have vigorously asserted 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 research should be aimed at revealing the molecular and biochemical causes of everything that exists. A common example of this type of strategy is breaking down a complex system into parts: a biologist might 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 reduction, this procedural recommendation does not follow directly from it. In fact, unlike token reduction, methodological reductionism can be quite controversial. It is argued that purely reductionist research strategies exhibit systematic biases that missrelevant biological features and that, for some questions, a more fruitful methodology is to integrate the discovery of molecular causes with the study of higher-level functions.
Epistema
Epistic reduction is the idea that knowledge about one scientific area (usually about higher level processes) can be reduced to another body of scientific knowledge (usually at a relatively lower or more fundamental level). While the endorsement of some form of epistemic reduction may be motivated by ontological reduction coupled with methodological reductionism (e.g., 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 from one body of knowledge to another, the concept of these bodies of knowledge and what this would mean for their "reduction" should be examined. A number of different reduction models have been proposed. Thus, the discussion about the reduction of biology has not only revolved around the extent to which epistemic reduction is possible, but also about those concepts of it that play a role in real scientific research and discussion. Two main categories can be distinguished:
- theory reduction models that state that one theory can be logically derived from anothertheory;
- models of explanatory reduction that focus on whether higher-level features can be explained by lower features.
General conclusion
Definitions of reduction from various sciences mentioned in this article are far from the limit, because in fact there are many more of them. Despite all the differences in the definition of reduction, they all have something in common. First of all, reduction is perceived as a reduction, reduction, simplification and reduction of something more complex, cumbersome and systemic, to something simpler, understandable and easily explainable. This is the key idea behind 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 for both professional scientists and ordinary people.
