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An example is the set of natural numbers N, where 0 is the first element, and the others are produced by adding 1 successively. [ 2 ] Roitman considers the same construction in a more concrete form: the elements are sets, the empty set ∅ {\displaystyle \emptyset } among them, and the successor of every element y {\displaystyle y} is the set y ...
Another example is a similar singly linked type in Java: class List < E > { E value ; List < E > next ; } This indicates that non-empty list of type E contains a data member of type E, and a reference to another List object for the rest of the list (or a null reference to indicate that this is the end of the list).
Most recursive definitions have two foundations: a base case (basis) and an inductive clause. The difference between a circular definition and a recursive definition is that a recursive definition must always have base cases, cases that satisfy the definition without being defined in terms of the definition itself, and that all other instances in the inductive clauses must be "smaller" in some ...
Structural recursion is usually proved correct by structural induction; in particularly easy cases, the inductive step is often left out. The length and ++ functions in the example below are structurally recursive. For example, if the structures are lists, one usually introduces the partial order "<", in which L < M whenever list L is the tail ...
Inductive programming (IP) is a special area of automatic programming, covering research from artificial intelligence and programming, which addresses learning of typically declarative (logic or functional) and often recursive programs from incomplete specifications, such as input/output examples or constraints.
A set for which membership can be decided by a recursive procedure or algorithm, also known as a decidable or computable set. recursively enumerable set A set for which there exists a Turing machine that will list all members of the set, possibly without halting if the set is infinite; also called "semi-decidable set" or "Turing recognizable set".
HoTT differs from ITT by its identity type (equality). Higher inductive types not only define a new type with constants and functions that create elements of the type, but also new instances of the identity type that relate them. A simple example is the circle type, which is defined with two constructors, a basepoint; base : circle. and a loop;
Inductive logic programming (ILP) is an approach to machine learning that induces logic programs as hypothetical generalisations of positive and negative examples. Given a logic program representing background knowledge and positive examples together with constraints representing negative examples, an ILP system induces a logic program that ...