Coinduction

In computer science, coinduction is a technique for defining and proving properties of systems of concurrent interacting objects. Coinduction is the mathematical 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." Experimental implementations of co-LP are available from the University of Texas at Dallas and in Logtalk (for examples see ) and SWI-Prolog. In a concise statement is given of both the principle of induction and the principle of coinduction. While this article is not primarily concerned with induction, it is useful to consider their somewhat generalized forms at once. In order to state the principles, a few preliminaries are required. Let be a set and be a monotone function , that is: Unless otherwise stated, will be assumed to be monotone. X is F-closed if X is F-consistent if X is a fixed point if These terms can be intuitively understood in the following way. Suppose that is a set of assertions, and is the operation which takes the implications of .