On the Representation of McCarthy's amb in the π-calculus

Math. Struct. Comp. Sci.

2008

Abstract. We present a higher-order call-by-need lambda calculus enriched with constructors, case-expressions, recursive letrec-expressions, a seq-operator for sequential evaluation and a non-deterministic operator amb that is locally bottom-avoiding. We use a small-step operational semantics in form of a single-step rewriting system that defines a (nondeterministic) normal order reduction. This strategy can be made fair by adding resources for bookkeeping. As equational theory we use contextual equivalence, i.e. terms are equal if plugged into any program context their termination behaviour is the same, where we use a combination of may-as well as must-convergence, which is appropriate for non-deterministic computations. We show that we can drop the fairness condition for equational reasoning, since the valid equations w.r.t. normal order reduction are the same as for fair normal order reduction. We evolve different proof tools for proving correctness of program transformations, in particular, a context lemma for may-as well as mustconvergence is proved, which restricts the number of contexts that need to be examined for proving contextual equivalence. In combination with so-called complete sets of commuting and forking diagrams we show that all the deterministic reduction rules and also some additional transformations preserve contextual equivalence. We also prove a standardisation theorem for fair normal order reduction. The structure of the ordering ≤c is also analysed: Ω is not a least element, and ≤c already implies contextual equivalence w.r.t. may-convergence.

Section: Fair Normal Order Reductionmentioning

confidence: 50%

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

A call-by-need lambda calculus with locally bottom-avoiding choice: context lemma and correctness of transformations

Math. Struct. Comp. Sci.

2008

“…Besides observing whether a program can terminate (called may-convergence) our notion of contextual equivalence also observes whether a program never loses the ability to terminate after some reductions (called should-convergence or sometimes mustconvergence, see e.g. [3,15,22,23]). The latter notion slightly differs from the classic notion of must-convergence (e.g.…”

Section: Introductionmentioning

confidence: 99%

Conservative Concurrency in Haskell

2012 27th Annual IEEE Symposium on Logic in Computer Science

2012

Abstract. The calculus CHF models Concurrent Haskell extended by concurrent, implicit futures. It is a process calculus with concurrent threads, monadic concurrent evaluation, and includes a pure functional lambda-calculus which comprises data constructors, case-expressions, letrec-expressions, and Haskell's seq. Futures can be implemented in Concurrent Haskell using the primitive unsafeInterleaveIO, which is available in most implementations of Haskell. Our main result is conservativity of CHF, that is, all equivalences of pure functional expressions are also valid in CHF. This implies that compiler optimizations and transformations from pure Haskell remain valid in Concurrent Haskell even if it is extended by futures. We also show that this is no longer valid if Concurrent Haskell is extended by the arbitrary use of unsafeInterleaveIO.

“…We use a contextual observational semantics for the concurrent lambda calculus with futures, based on operationally-defined forms of may-and must-convergence (De Nicola and Hennessy 1984;Ong 1993;Carayol et al 2005). Our form of must-convergence is similar to the should-testing of Rensink and Vogler (2007).…”

Section: Introductionmentioning

confidence: 99%

Correctly translating concurrency primitives

Schwinghammer

Proceedings of the 2009 ACM SIGPLAN Workshop on ML

et al. 2009

Motivated by the question of correctness of a specific implementation of concurrent buffers in the lambda calculus with futures underlying Alice ML, we prove that concurrent buffers and handled futures can correctly encode each other. Our translations map waiting on handled futures to queuing of concurrent buffers and vice versa. Correctness of translations means that they preserve and reflect the observations of may-and must-convergence. As a consequence of compositionality, they are also adequate with respect to a contextually defined notion of observational program semantics. We demonstrate that our approach to the correctness of implementations applies uniformly to the whole compilation process from high-level to low-level concurrent languages.