Semantically linear programming languages

Abstract

We propose a paradigmatic programming language (called S PCF) which is linear in a semantic sense. S PCF is not syntactically linear, namely its programs can contain more than one occurrencies of the same variable. We give an interpretation of S PCF into a model of linear coherence spaces and we show that such semantics is fully abstract with respect to our language. Furthermore, we discuss the independence of new syntactical operators and we address the universality problem.

A Process-Model for Linear Programs

“…. We remark that SℓPCF − is a Turing-complete syntactical restriction of PCF [21] and it is fully abstract with respect to the model of linear function between coherence spaces considered in [20] (as noted just in page 104 of [20], just before the Theorem 4). Hence SℓPCF − is denotationally linear.…”

“…Ground and stable variables belong to distinct kinds only for sake of simplicity. Their free use implies that SℓPCF − is not syntactically linear (in the sense of [20]). …”

“…Accordingly, we tackle the construction of a process-model for SℓPCF − , namely a simple programming language conceived in order to program "linear" functions between coherence spaces. The least full sub-category of coherence spaces, including the infinite flat domain (representing natural numbers) and the coherence spaces representing linear functions between domains in the model itself (by avoiding the use of exponential domain constructors) forms a fully abstract model for such programming language [20]. In order to build a process model for SℓPCF − , we introduce ℓinSolos, namely a process calculus based on a typed calculus of Solos [14].…”

Linearity and recursion in a typed Lambda-calculus

“…This approach has been used to extend linear functional calculi (see, e.g., [12,14,41,45]), however, it relies on the existence of a non-linear conditional which throws away a possibly infinite computation in one of the branches. Instead, in this paper, we obtain a Turing-complete linear λ-calculus through the use of an unbounded recursor with a built-in test on pairs, which allows the encoding of both finite iteration and minimisation.…”

“…Denotational linearity is achieved when only linear functions can be defined in the language (see, e.g., [45]). The language defined in [45] is a linear version of PCF in a denotational sense: it has a linear model (linear coherence spaces) but its terms can contain more than one occurrence of the same variable. Finally, syntactical linearity, requires a linear use of variables in terms [34].…”