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Added by
Arnaud Spiwack
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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A linear argument must be consumed exactly once in the body of its function. A linear type system can verify the correct usage of resources such as file handles and manually managed memory. But this verification requires bureaucracy. This paper presents linear constraints, a front-end feature for linear typing that decreases the bureaucracy of working with linear types. Linear constraints are implicit linear arguments that are to be filled in automatically by the compiler. Linear constraints are presented as a qualified type system,together with an inference algorithm which extends GHCs existing constraint solver algorithm. Soundness of linear constraints is ensured by the fact that they desugar into Linear Haskell.
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In this paper we show that checking satisfiability of a set of non-linear Horn clauses (also called a non-linear Horn clause program) can be achieved using a solver for linear Horn clauses. We achieve this by interleaving a program transformation with a satisfiability checker for linear Horn clauses (also called a solver for linear Horn clauses). The program transformation is based on the notion of tree dimension, which we apply to a set of non-linear clauses, yielding a set whose derivation trees have bounded dimension. Such a set of clauses can be linearised. The main algorithm then proceeds by applying the linearisation transformation and solver for linear Horn clauses to a sequence of sets of clauses with successively increasing dimension bound. The approach is then further developed by using a solution of clauses of lower dimension to (partially) linearise clauses of higher dimension. We constructed a prototype implementation of this approach and performed some experiments on a set of verification problems, which shows some promise.
Program equivalence in linear contexts, where programs are used or executed exactly once, is an important issue in programming languages. However, existing techniques like those based on bisimulations and logical relations only target at contextual equivalence in the usual (non-linear) functional languages, and fail in capturing non-trivial equivalent programs in linear contexts, particularly when non-determinism is present. We propose the notion of linear contextual equivalence to formally characterize such program equivalence, as well as a novel and general approach to studying it in higher-order languages, based on labeled transition systems specifically designed for functional languages. We show that linear contextual equivalence indeed coincides with trace equivalence - it is sound and complete. We illustrate our technique in both deterministic (a linear version of PCF) and non-deterministic (linear PCF in Moggis framework) functional languages.
Recently, the iterative approach named linear tabling has received considerable attention because of its simplicity, ease of implementation, and good space efficiency. Linear tabling is a framework from which different methods can be derived based on the strategies used in handling looping subgoals. One decision concerns when answers are consumed and returned. This paper describes two strategies, namely, {it lazy} and {it eager} strategies, and compares them both qualitatively and quantitatively. The results indicate that, while the lazy strategy has good locality and is well suited for finding all solutions, the eager strategy is comparable in speed with the lazy strategy and is well suited for programs with cuts. Linear tabling relies on depth-first iterative deepening rather than suspension to compute fixpoints. Each cluster of inter-dependent subgoals as represented by a top-most looping subgoal is iteratively evaluated until no subgoal in it can produce any new answers. Naive re-evaluation of all looping subgoals, albeit simple, may be computationally unacceptable. In this paper, we also introduce semi-naive optimization, an effective technique employed in bottom-up evaluation of logic programs to avoid redundant joins of answers, into linear tabling. We give the conditions for the technique to be safe (i.e. sound and complete) and propose an optimization technique called {it early answer promotion} to enhance its effectiveness. Benchmarking in B-Prolog demonstrates that with this optimization linear tabling compares favorably well in speed with the state-of-the-art implementation of SLG.
A number of domain specific languages, such as circuits or data-science workflows, are best expressed as diagrams of boxes connected by wires. Unfortunately, functional languages have traditionally been ill-equipped to embed this sort of languages. The Arrow abstraction is an approximation, but we argue that it does not capture the right properties. A faithful abstraction is Symmetric Monoidal Categories (SMCs), but,so far,it hasnt been convenient to use. We show how the advent of linear typing in Haskell lets us bridge this gap. We provide a library which lets us program in SMCs with linear functions instead of SMC combinators. This considerably lowers the syntactic overhead of the EDSL to be on par with that of monadic DSLs. A remarkable feature of our library is that, contrary to previously known methods for categories, it does not use any metaprogramming.
Instead of a monolithic programming language trying to cover all features of interest, some programming systems are designed by combining together simpler languages that cooperate to cover the same feature space. This can improve usability by making each part simpler than the whole, but there is a risk of abstraction leaks from one language to another that would break expectations of the users familiar with only one or some of the involved languages. We propose a formal specification for what it means for a given language in a multi-language system to be usable without leaks: it should embed into the multi-language in a fully abstract way, that is, its contextual equivalence should be unchanged in the larger system. To demonstrate our proposed design principle and formal specification criterion, we design a multi-language programming system that combines an ML-like statically typed functional language and another language with linear types and linear state. Our goal is to cover a good part of the expressiveness of languages that mix functional programming and linear state (ownership), at only a fraction of the complexity. We prove that the embedding of ML into the multi-language system is fully abstract: functional programmers should not fear abstraction leaks. We show examples of combined programs demonstrating in-place memory updates and safe resource handling, and an implementation extending OCaml with our linear language.
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