Technical report Frank / Johann-Wolfgang-Goethe-Universität, Fachbereich Informatik und Mathematik, Institut für Informatik
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28
Call-by-need lambda calculi with letrec provide a rewritingbased operational semantics for (lazy) call-by-name functional languages. These calculi model the sharing behavior during evaluation more closely than let-based calculi that use a fixpoint combinator. In a previous paper we showed that the copy-transformation is correct for the small calculus LR-Lambda. In this paper we demonstrate that the proof method based on a calculus on infinite trees for showing correctness of instantiation operations can be extended to the calculus LRCC-Lambda with case and constructors, and show that copying at compile-time can be done without restrictions. We also show that the call-by-need and call-by-name strategies are equivalent w.r.t. contextual equivalence. A consequence is correctness of all the transformations like instantiation, inlining, specialization and common subexpression elimination in LRCC-Lambda. We are confident that the method scales up for proving correctness of copy-related transformations in non-deterministic lambda calculi if restricted to "deterministic" subterms.
29
This paper extends the internal frank report 28 as follows: It is shown that for a call-by-need lambda calculus LRCCP-Lambda extending the calculus LRCC-Lambda by por, i.e in a lambda-calculus with letrec, case, constructors, seq and por, copying can be done without restrictions, and also that call-by-need and call-by-name strategies are equivalent w.r.t. contextual equivalence.
24 [v.3]
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.
24 [v.2]
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.
47 [v.2]
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.
47
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.
44
In this paper we analyze the semantics of a higher-order functional language with concurrent threads, monadic IO and synchronizing variables as in Concurrent Haskell. To assure declarativeness of concurrent programming we extend the language by implicit, monadic, and concurrent futures. As semantic model we introduce and analyze the process calculus CHF, which represents a typed core language of Concurrent Haskell extended by concurrent futures. Evaluation in CHF is defined by a small-step reduction relation. Using contextual equivalence based on may- and should-convergence as program equivalence, we show that various transformations preserve program equivalence. We establish a context lemma easing those correctness proofs. An important result is that call-by-need and call-by-name evaluation are equivalent in CHF, since they induce the same program equivalence. Finally we show that the monad laws hold in CHF under mild restrictions on Haskell’s seq-operator, which for instance justifies the use of the do-notation.
53
The pi-calculus is a well-analyzed model for mobile processes and mobile computations.
While a lot of other process and lambda calculi that are core languages of higher-order concurrent and/or functional programming languages use a contextual semantics observing the termination behavior of programs in all program contexts, traditional program equivalences in the pi-calculus are bisimulations and barbed testing equivalences, which observe the communication capabilities of processes under reduction and in contexts.
There is a distance between these two approaches to program equivalence which makes it hard to compare the pi-calculus with other languages. In this paper we contribute to bridging this gap by investigating a contextual semantics of the synchronous pi-calculus with replication and without sums.
To transfer contextual equivalence to the pi-calculus we add a process Stop as constant which indicates success and is used as the base to define and analyze the contextual equivalence which observes may- and should-convergence of processes.
We show as a main result that contextual equivalence in the pi-calculus with Stop conservatively extends barbed testing equivalence in the (Stop-free) pi-calculus. This implies that results on contextual equivalence can be directly transferred to the (Stop-free) pi-calculus with barbed testing equivalence.
We analyze the contextual ordering, prove some nontrivial process equivalences, and provide proof tools for showing contextual equivalences. Among them are a context lemma, and new notions of sound applicative similarities for may- and should-convergence.