This special issue collects selected articles from the Third International Workshop on Linearity (LINEARITY 2014), which was held in Vienna, on July 13th, 2014. The workshop was a one-day satellite event of FLoC 2014, the sixth Federated Logic Conference, which was held as part of the 2014 Vienna Summer of Logic.

In this article we discuss three different notions of linearity: syntactical, operational and denotational. We briefly define each notion of linearity, pointing out some of the main results in the area, and describe applications of linear languages and type systems.

We show that the full PCF language can be encoded in L_rec, a syntactically linear λ-calculus extended with numbers, pairs, and an unbounded recursor that preserves the syntactic linearity of the calculus. We give call-by-name and call-by-value evaluation strategies and discuss implementation techniques for L_rec, exploiting its linearity.

The linear lambda calculus, where variables are restricted to occur in terms exactly once, has a very weak expressive power: in particular, all functions terminate in linear time. In this paper we consider a simple extension with natural numbers and a restricted iterator: only closed linear functions can be iterated. We show properties of this linear version of Godel's T using a closed reduction strategy, and study the class of functions that can be represented. Surprisingly, this linear calculus offers a huge increase in expressive power over previous linear versions of T, which are 'closed at construction' rather than 'closed at reduction'. We show that a linear T with closed reduction is as powerful as T.