## Technical report Frank / Johann-Wolfgang-Goethe-Universität, Fachbereich Informatik und Mathematik, Institut für Informatik

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#### Year of publication

- 2006 (3) (remove)

#### Keywords

- Alice ML (1)
- Contextual Equivalence (1)
- Futures (1)
- Lambda Calculus (1)
- Lambda-Kalkül (1)
- ML <Programmiersprache> (1)
- Nebenläufigkeit (1)
- Operational Semantics (1)
- Operationale Semantik (1)

- 26
- Program equivalence for a concurrent lambda calculus with futures (2006)
- Reasoning about the correctness of program transformations requires a notion of program equivalence. We present an observational semantics for the concurrent lambda calculus with futures Lambda(fut), which formalizes the operational semantics of the programming language Alice ML. We show that natural program optimizations, as well as partial evaluation with respect to deterministic rules, are correct for Lambda(fut). This relies on a number of fundamental properties that we establish for our observational semantics.

- 24
- A call-by-need lambda-calculus with locally bottom-avoiding choice: context lemma and correctness of transformations (2006)
- 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, which is locally bottom-avoiding. We use a small-step operational semantics in form of a normal order reduction. As equational theory we use contextual equivalence, i.e. terms are equal if plugged into an arbitrary program context their termination behaviour is the same. We use a combination of may- as well as must-convergence, which is appropriate for non-deterministic computations. We evolve different proof tools for proving correctness of program transformations. We provide a context lemma for may- as well as must- convergence 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 keep contextual equivalence. In contrast to other approaches our syntax as well as semantics does not make use of a heap for sharing expressions. Instead we represent these expressions explicitely via letrec-bindings.

- 24 [v.2]
- A call-by-need lambda-calculus with locally bottom-avoiding choice: context lemma and correctness of transformations (2006)
- 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 a is also analysed: Ω is not a least element, and <=c already implies contextual equivalence w.r.t. may-convergence.