Essence and universal significance of recursive methods of work Accumulation

In terms of methodology, recursive methods are a continuous solution to problems from simple to complex, from low to advanced. Each step can be operated, and each step is continuously converted. Recursive definition describes, constructs, and describes a complex whole with simple and self-evident elements. The recursive method solves complicated problems by solving simple problems. There is a general recursive mechanism in people's thinking process. Recursive Method is an exquisite technique for solving problems and an effective method for solving problems. It is of great significance to study recursive methods from the perspective of philosophical methodology.
Recursive definitions and recursive methods are commonly used definitions and methods in logic, mathematics, and computer science (programming). They have precise meanings and are especially effective in solving certain difficult problems, it seems to be an interesting and sophisticated processor that can solve the problem perfectly. There are also common recursive phenomena and recursive mechanisms in human thinking activities. Some problems can only be solved by recursive methods. For some problems, recursive methods are more effective than other methods. In computer programs, recursion can constitute an automatic reasoning mechanism. One of the main problems of artificial intelligence is to enable computers to have the function of automatic reasoning. After a "recursive mechanism" is established, the computer can perform automatic reasoning once the initial value (prerequisite) is input to obtain the result (conclusion ). Recursive Method is an exquisite technique for solving problems and an effective method for solving problems.

1. Recursive Concept
The so-called recursion (from the Latin recurso, the original intention is to run back, recall, etc.) refers to the use of "regression" and the unknown is attributed to known. The so-called recursive function is a number theory function. That is to say, the defined and value fields of this function are all natural numbers, and the calculation of unknown values is often obtained by returning to known values. Recursion is a loop structure that recursively sums the computing of "more complex" situations into the calculation of "more simple" situations and always comes down to the calculation of "simplest" situations, until the calculation result is obtained. This is the essence of recursion. The definition is recursive, the data structure is recursive, and the solution to the problem is recursive. The recursive method can be used for processing.

Definition of "progressive": an object is partially composed of itself, or defined by itself, which is called recursion.

According to this definition, recursive classes can be divided into the following types:

(1) recursive functions:
Definition: a function is recursive. If another application of the definition exists in its definition.

For example, the factorial function N! Is defined:

I 0! = 1 (N = 0)
Ⅱ N! = (N-1 )! × N (N> 0)

Here:
I is a non-recursive initial value. It is a condition for determining recursive termination. It is called the exit of recursion.
Ⅱ (N-1 )! Is to use a small independent variable to represent a larger independent variable N in recursion! Recursive definition, that is, N! Is defined by its own part (N-1 )! .
The procedure for defining a recursive function is:
First, we will give some easily-defined recursive functions as basic functions, which are zero functions, successor functions, and projection functions. Then we will give some basic rules starting from the basic functions, use these rules (repeated) to generate all recursive functions.

(2) recursive process,

Definition:
A process is recursive. If a process P contains calls to itself, P is called direct recursion. If P contains calls to another process Q, Q calls P, which is called indirect recursion.

Recursion is a method for calculating function values. Each processing stage contains all subsequent tags, that is, the first stage cannot be completed before the end of all other stages. A recursive function is a function that uses an iteration formula to export its value from some natural numbers to a natural number, and this function is an operand in an iteration formula. Recursive Computing refers to the process of self-calling processes with some test functions and using the values obtained from the previous recursive results. In general, a recursive process refers to a process in which each step uses the results of previous steps.

(3) recursive data structure:

A Data Structure is recursive if its structure definition contains its own definition. The stack in the data structure is a recursive data structure.

(4) recursive routine subroutine

A recursive routine subroutine is a recursive program that can be used as a routine subroutine. It can be called directly or by another user. For example, in the calculation expression N = n! = N × (n-1) × (n-2 )...... In × 2 × 1 (that is, n factorial), n is the program transmission parameter. If n = 1 or 0, the program returns 1. Otherwise, the program uses (n-1) value, multiply n by the call result, and return the product. This program calls itself repeatedly until the passed parameter value is equal to 1. When the program value is 1, the program value is obtained by successive calls, and finally the program value is N = n !. The most common method of calling a recursive program is to use a stack to push the most recent return address and any other status information during each call to the stack exit. The address at the top of the stack is popped up as the return address. Therefore, recursive routines allow you to call your own subroutines. In each subsequent call, the state of a recursive routine subroutine must be stored, indicating that the State data is often stored in the stack, before the subroutine is transferred out, you can call the subroutine itself directly or indirectly. A recursive program is composed of an initial value and a recursive relationship. The recursion method is to first find the recursion relationship, and solve the obtained recursive equation under the conditions of known initial values, and find the same formula to obtain the expected results.

Ii. Recursive Method

1. Progressive Analysis
It is also called "recursive function theory" and "executable theory ".
It is mainly used in mathematics to study the discipline of "constructive" or "competent process.

The construction of recursive functions is also an important aspect of its research. The main content of the progressive theory includes the original recursive function, General recursive function, partial recursive function, recursive enumerative function, determination problem, recursive undefinable theory, α-progressive theory, and pedigree theory. The main method of progressive theory is to thoroughly reveal the essence of the process by studying number theory functions, so as to effectively solve many important mathematical problems.

The progressive discussion discusses the intuitive idea of describing the enforceability of an operation or process in form, that is, in principle, they can mechanically produce a definite result. The concept of "doing something" involves specific implementations, effective, and effective. The concept of recursion is not difficult to understand. It means that the previous results can be used to obtain the subsequent results. The number theory functions studied by the progressive theory have precise mathematical definitions. For example, use recursive definitions to define the "Fibonacci function" as follows:

Initial provisions:
F (0) = 0,
F (1) = 1,
Recursive operation relationship:
F (n) = f (n-1) + f (n-2 ).

It is easy to see that given a natural number N, F (n) constant can be obtained gradually using the recursive definition above.

Throughout history, there have been many precise definitions of the vague concept of "Process of operation. For example, General recursion, Turing Machine, regular algorithm, finite automatic machine, etc. Later, we found that these definitions are equivalent to each other. Therefore, it is critical to find out how the process works. Among those who have made significant contributions in this field are Hilbert, geder, Qiu Qi, Turing, klinny, and Paster. Today, the progressive theory has achieved fruitful results. It not only has an extremely important application in the basic mathematical theory, but also in other emerging disciplines, especially in the electronic computer science, it has become increasingly important.

2. Recursive Operation Sequence

A sequence of items. Each item after the first item is determined by some of its previous items or all of its operations as the operands, it is a recursive operation sequence. If the first item is n [, 0] and the operation is f, the following operations are performed:

(1) N [, 0] (initial state)
N [, 1] = f (N [, 0])
N [, 2] = f (N [, 1])
......
N [, I] = f (N [, I-1])
(2) n [, 0] (initial value)
N [, 1] = F (n [, 0])
N [, 2] = F (n [, 0], n [, 1])
......
N [, I] = F (n [, 0], n [, 1]… N [, I-1])

This is a continuous operation process. In a general sense, it is to start from an initial state of a thing and perform continuous operations on the thing to make a series of changes in the state of the thing. The difference is that the operation can always be the same, or there can be different operations in different links. States are represented by S (State), and operations are represented by O (Operate:

(1) S [, 0] (initial state)
S [, 1] = O (S [, 0])
S [, 2] = O (S [, 1])
......
S [, I] = O (S [, I-1])
The same operation O is performed throughout the process.
(2) S [, 0] (initial state)
S [, 1] = O [, 1] (S [, 0])
S [, 2] = O [, 2] (S [, 1])
......
S [, I] = O [, 1] (S [, I-1])
Different operations are performed in the entire operation process: O [, 1], O [, 2]…, O [, I].

3. Recursive Identification

This method is also called mathematical induction, Which is concise and powerful. For example, we can use recursive proofs to derive all elementary number theory.
The most common form of recursive evidence is to prove that proposition A (n) is true for any natural number n. We only need to prove the following two points:
(1) verify that A (0) is correct. This step is called the foundation.
(2) Suppose A (n) is correct (this is called induction hypothesis), and try to prove that A (n + 1) is also correct. This step is called induction.
There are other forms of recursive evidence, some of which are equivalent to the form of the above recursive evidence, but there are also some more powerful or weaker proofs. It should be pointed out that in the set theory, a branch of mathematics, when we discuss ultra-poor numbers, we often promote the Recursive Identification Method to the ultra-poor number to obtain the super-poor recursive identification method. This certification method is also an important tool for discussing ultra-poor data. The recursive evidence law is closely related to the progressive theory. Recursive Identification is a very important and powerful proof method. It plays an important role in the development of progressive analysis and other mathematical theories.

4. Recursive Definition

Such a definition may seem surprising: when defining an object, can the same object not fully defined? Such a definition is called a recursive definition.
Logically, they are completely correct and understandable.
Recursive definition is a method defined by a concept using recursive methods.
For example, in logic, a recursive definition method is used to define a "formula. The formula is defined as follows:

(1) proposition variables A and B are well-formed formulas;
(2) If variable A is A combination formula, then ~ A is A combination formula;
(3) If the variable A and B are the combination formulas, then A between B, A between B, A → B, A between → B is the combination formula;
(4) The sequence of symbols that are composed of the preceding (1), (2), and (3) rules is a combination formula.

This is the definition under the "formula" by recursive method, that is, the recursive definition. Here we use the "formula" to define the "formula", that is, "define yourself ".
Is recursive definition a self-loop? Is it "loop definition" or "simultaneous repetition? From the perspective of general logic, "Define Yourself" is a circular definition, which violates the defined rules. There is a fundamental difference between the two. Recursive definitions are often used to represent (define) A transfer relation.
The essence of recursive definition reflects the limitation and initiation of human thinking. At a certain level and at a certain stage, there is always a starting point for a specific thinking process. This starting point is often a self-evident preset and can be recognized directly, it is no longer interpreted and defined as the starting point of the thinking process. Human thinking cannot be traced forward infinitely. Otherwise, any thinking process cannot be carried out or completed.
The general meaning of recursive definition: A Starting object has a certain nature. After some Transformation (operation) is performed on it, a new object is obtained. This new object also has this nature. Or, a composite object (whole) has a certain nature, which is determined by its composition unit object (part. For example, A between B is A combination formula, which is determined by A and B.


Iii. essence of Recursion

Recursion includes restoration.
The so-called "restoration" is to find the most basic initial elements and the most basic operation steps. Recursive methods are related to operability issues.
Recursive, that is, recoverable, that is, operable.
Some problems are solved in the process of starting from beginning to end and closely linking the rings. To get the final result, you must start from the first step, and no step is required. Any two adjacent steps, the last step can be completed only after the previous step is completed. This problem must be solved using recursive methods.

In a general sense, recursive methods are a continuous solution to problems from simple to complex, from low to advanced. Each step can be operated, and each step is continuously converted. Recursive definition describes, constructs, and describes a complex whole with simple and self-evident elements. The recursive method solves complicated problems by solving simple problems. There is a recursive mechanism in people's thinking process. Certain problems must be defined or solved using recursive methods.

Recursive methods research problems from the perspective of link and continuous transformation of things. The state of a series of objects or objects studied by recursive methods is not isolated, but associated and capable of transformation, one object generates another object, and one State of the object generates (converts to) another state. For example:
N [, 1], n [, 2] = f (n [, 1]), n [, 3] = f (n [, 2]),…
N [, 2] is produced by n [, I], n [, 3] is produced by n [, 2 ,...... These objects are interrelated sequences. Human Thinking contains the "energy-shaped mechanism ". This is a basic nature of thinking. Otherwise, thinking cannot proceed. Logical thinking is shaped. Intuition and Epiphany are also shaped, but they have not been discovered yet. "Controllability" is "progressive ". "Enforceability" is the premise for the thinking process to proceed. Without "enforceability", the various stages of thinking activities cannot be converted. In any process, there is an internal "enforceability" mechanism.
In the progressive theory, "calculability" is to calculate another number from a number. In general, this "calculability" is operable and convertible, that is to say, an object is generated from one object, and another state is generated from one state. Logically speaking, recursion is also a type of reasoning. Recursive methods play an important role in the study of the theory of enforceability (determination problem, incomprehensibility, etc. The recursive method is also called the progressive method, which is an important method for finding mathematical patterns and solving ideas.

Algorithms are an important concept in progressive theory.

The so-called algorithms are generally regarded as uniform processing rules for a category of problems. At this time, "unified processing" means that such a rule is not an independent solution to each specific problem, but applicable to a type of problem. An algorithm is composed of a few statements. When any problem in a class of questions is given, these rules can be executed successfully and the answer is obtained after a few steps. An algorithm is defined as a clear instruction set that can be used as a computing program (not necessarily a numerical value, it can be used to find answers to any of the given questions. Intuitively, the method of doing is mechanical, and the method of doing it is mechanical, without any randomness. A closer description refers to the fact that every step of the process is clearly defined by a given rule (Computing Program), that is, the rule specifies how to do the first step, it also specifies how to proceed after a certain step, and requires that this process be completed in a poor step. Features of a recursive algorithm: a recursive algorithm uses its own algorithm in definition. Recursion is an important concept. With it, you can use a function or process to define the function or process yourself.

Recursive definition, in terms of specific rules, is simple and mechanical, but the overall functions they converge are beyond the imagination of people. The function of recursion is to automatically generate the next generation from the previous generation in a fixed manner, and then generate the next generation from the next generation. Although the "Generation" method remains the same, the input "raw materials" are different, and the "Descendant" generated is generally different. Recursion is defined as a "processor ". A recursive definition is a set of rules that determine a class of objects. The first rule specifies that some objects belong to the set, and the other rules stipulate that if some objects belong to the set, what other objects also belong to the set.
Recursive definition has the following characteristics: To prove that each of a class of objects determined by the inductive definition has a certain nature, only the objects that meet the rules stated in the definition have this property. For example, to prove that every certain theory of a formal system has a property P, you only need to prove that the formula satisfying the rule for defining the theorem has a property P. In other words, you only need to prove:
(1) Every right of the formal system has a property P;
(2) If all the premises of a reasoning rule have properties P, the conclusion of the rule also has properties P (the reasoning rule maintains the properties P ). This is also called "the inheritance of reasoning rules ".
The proof made by this method is the inductive proof of the theorem. (1) It is called the Foundation; (2) it is called the induction step. (2) hypothesis: all premises of a reasoning rule have properties P, which is called inductive hypothesis. Inductive proof is an important and widely used proof method.
The basic content of the progressive theory is to discuss the function classes of "algorithm Computability" or "algorithm Computability. Computing is precisely related to algorithm concepts. There are two ideal computational forms in computer science theory to construct a system. One is the Thue system, which is purely a form system. In this system, the symbolic sequence can be interpreted as other Symbolic Sequences by inference of certain rules. In this way, after an input sequence is given, these rules allow conversion to an output sequence. This is a form of information processing system. The other is the Turing machine, which deeply describes the Computing Features of algorithm concepts. Turing machine is an ideal computer that is precisely defined by mathematical methods and can reflect the concept of computing programs. The Turing Machine converts all the elements used in computing into a basic program, sums computing into the most basic elements, and finally expresses them in the form of a program.

"Recursion" is a common thinking mechanism, but it is manifested in some special phenomena and special methods in different scientific fields. In methodology research, although we cannot find a universal method to solve all problems, in fact this method does not exist, but we can classify problems in different scientific fields, transition from abstract problems to specific problems, so that the definition and description of problems are accurate, so as to seek a precise method (algorithm) to solve a certain type of problems ). In a variety of scientific fields, as well as in social structures and people's various operational behaviors, there is a common class of problems with recursive structures. We refer to these problems as "recursive problems ". The recursive method is an accurate method to solve this type of "recursive problem ".