# Coinduction

In computer science, **coinduction** is a technique for defining and proving properties of systems of concurrent interacting objects.

Coinduction is the mathematical dual to structural induction. Coinductively defined types are known as **codata** and are typically infinite data structures, such as streams.

As a definition or specification, coinduction describes how an object may be "observed", "broken down" or "destructed" into simpler objects. As a proof technique, it may be used to show that an equation is satisfied by all possible implementations of such a specification.

To generate and manipulate codata, one typically uses corecursive functions, in conjunction with lazy evaluation. Informally, rather than defining a function by pattern-matching on each of the inductive constructors, one defines each of the "destructors" or "observers" over the function result.

In programming, co-logic programming (co-LP for brevity) "is a natural generalization of logic programming and coinductive logic programming, which in turn generalizes other extensions of logic programming, such as infinite trees, lazy predicates, and concurrent communicating predicates. Co-LP has applications to rational trees, verifying infinitary properties, lazy evaluation, concurrent logic programming, model checking, bisimilarity proofs, etc."[1] Experimental implementations of co-LP are available from The University of Texas at Dallas [2] and in Logtalk (for examples see [3]) and SWI-Prolog.

## References

## Further reading

- Textbooks

- Davide Sangiorgi (2012).
*Introduction to Bisimulation and Coinduction*. Cambridge University Press. - Davide Sangiorgi and Jan Rutten (2011).
*Advanced Topics in Bisimulation and Coinduction*. Cambridge University Press.

- Introductory texts

- Andrew D. Gordon (1994). "A Tutorial on Co-induction and Functional Programming". CiteSeerX 10.1.1.37.3914. Cite journal requires
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(help) — mathematically oriented description - Bart Jacobs and Jan Rutten (1997). A Tutorial on (Co)Algebras and (Co)Induction (alternate link) — describes induction and coinduction simultaneously
- Eduardo Giménez and Pierre Castéran (2007). "A Tutorial on [Co-]Inductive Types in Coq"
- Coinduction — short introduction

- History

- Miscellaneous

- Co-Logic Programming: Extending Logic Programming with Coinduction — describes the co-logic programming paradigm