Coinduction

Coinduction is a method used in mathematics and computer science, which allows for the definition and proof of properties of infinite data structures and non-terminating computations. It is a dual to the more familiar method of induction, and is used in a similar way to prove properties of certain types of mathematical objects.

Definition

Coinduction is defined in terms of a bisimulation, a relation between two objects that shows they behave in the same way in the context of some process. An object is defined to be coinductive if it is bisimilar to some process or data structure.

In the context of computer science, coinduction is often used to reason about infinite data structures and non-terminating computations. It is a fundamental tool in the theory of process calculi, and is used to define and reason about the semantics of programming languages.

Use in Mathematics

In mathematics, coinduction is used to define and reason about infinite data structures. For example, the set of all natural numbers can be defined coinductively as the smallest set containing zero and closed under the successor function. This definition allows for the proof of properties of the natural numbers using coinduction.

Use in Computer Science

In computer science, coinduction is used to reason about non-terminating computations and infinite data structures. For example, a stream (an infinite sequence of data) can be defined coinductively as a pair consisting of a value and a function that produces the rest of the stream. This definition allows for the proof of properties of streams using coinduction.

See Also

• Induction (mathematics)
• Bisimulation
• Process calculus
• Stream (computing)