Non-linearity as the Metric Completion of Linearity

We summarize some recent results showing how the lambdacalculus may be obtained by considering the metric completion (with respect to a suitable notion of distance) of a space of affine lambda-terms, i.e., lambda-terms in which abstractions bind variables appearing at most once. This formalizes the intuitive idea that multiplicative additive linear logic is “dense” in full linear logic (in fact, a proof-theoretic version of the above-mentioned construction is also possible). We argue that thinking of non-linearity as the “limit” of linearity gives an interesting point of view on well-known properties of the lambda-calculus and its relationship to computational complexity (through lambda-calculi whose normalization is time-bounded). 1 Linearity and Approximations The concept of linearity in logic and computer science, introduced over two decades ago [12], has now entered firmly into the “toolbox” of proof theorists and functional programming language theorists. It is present, in one way or another, in a broad range of contexts, such as: denotational semantics [11], games semantics [22] and categorical semantics [8]; computational interpretations of classical logic [18,9]; optimal implementation of functional programming languages [3,19]; the theory of explicit substitutions [2]; higher-order languages for probabilistic [10] and quantum computation [24]; typing systems for polynomialtime [4], non-size-increasing [14] and resource-aware computation [17]; and even concurrency theory [6,15]. Technically, linearity imposes a severe restriction on the behavior of programs: data must be accessed exactly once. Its cousin affinity, which is more relevant for the purposes of this text, slightly relaxes the constraint: although data may be discarded, it may nevertheless be accessed at most once. In any case, linearity and affinity forbid re-use, forcing the programmer to explicitly keep track of how many copies of a given piece of information are needed in order to perform a computation. How can general, non-linear computation be performed in an affine setting? In other words, how can a persistent memory be simulated by a volatile memory? The intuitive answer is clear: one persistent memory cell, accessible arbitrarily many times, may be perfectly simulated by infinitely many volatile memory cells, each accessible only once. Of course, if only a finite memory is available, then M. Hasegawa (Ed.): TLCA 2013, LNCS 7941, pp. 3–14, 2013. c © Springer-Verlag Berlin Heidelberg 2013