The relationship between duality is usually reflected in form, that is, two structures or expressions can be transformed into each other in some way, but they are not necessarily identical in semantics. For example, in group theory, the concept of duality is embodied in the symmetry of group automorphism and fixed automorphism.

For a group G, its automorphism group Aut(G) is a subgroup of symmetric group S_n, so there is a dual relationship between Aut(G) and S _ N. In addition, for a ring or domain, its dual concept is embodied in the symmetry of its automorphism and fixed automorphism.

In geometry, the concept of duality is mainly embodied in vector space and affine space. For example, in computer science, duality is embodied in compiler design and programming language design.

In programming language design, duality is reflected in the duality of language type system and language implementation, that is, language type system defines the semantics and behavior of language, and language implementation realizes language type system.

In a word, duality is a concept in mathematics, logic and computer science, which means that there is a symmetrical relationship between two structures or expressions. The concept of duality has applications in many fields, including algebra, geometry, probability theory and statistics. Through the understanding and application of the dual concept, we can better understand and solve various mathematical and computer science problems.

In addition to the duality mentioned above, for example, there is a duality relationship between resistors and power supply elements, and there is a duality relationship between series circuits and parallel circuits.

For example, Fourier transform and Laplace transform have dual relationship, and time domain convolution and frequency domain product have dual relationship. Through the understanding and application of the concept of duality, we can better understand and solve various problems and promote the exchange and intersection between different disciplines.

In addition to the duality mentioned above. For example, the shortest path problem and longest common subsequence problem are both classic problems in graph theory, which can be solved by greedy algorithm or dynamic programming. In the shortest path problem, the greedy algorithm chooses the shortest edge to expand the path, while in the longest common subsequence problem, the dynamic programming algorithm calculates the maximum length of all subsequences.