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The paper proposes a variation of simulation for checking and proving contextual equivalence in a non-deterministic call-by-need lambda-calculus with constructors, case, seq, and a letrec with cyclic dependencies. It also proposes a novel method to prove its correctness. The calculus' semantics is based on a small-step rewrite semantics and on may-convergence. The cyclic nature of letrec bindings, as well as non-determinism, makes known approaches to prove that simulation implies contextual equivalence, such as Howe's proof technique, inapplicable in this setting. The basic technique for the simulation as well as the correctness proof is called pre-evaluation, which computes a set of answers for every closed expression. If simulation succeeds in finite computation depth, then it is guaranteed to show contextual preorder of expressions.
The goal of this report is to prove correctness of a considerable subset of transformations w.r.t. contextual equivalence in a an extended lambda-calculus with case, constructors, seq, let, and choice, with a simple set of reduction rules. Unfortunately, a direct proof appears to be impossible. The correctness proof is by defining another calculus comprising the complex variants of copy, case-reduction and seq-reductions that use variablebinding chains. This complex calculus has well-behaved diagrams and allows a proof that of correctness of transformations, and also that the simple calculus defines an equivalent contextual order.
We model sequential synchronous circuits on the logical level by signal-processing programs in an extended lambda calculus Lpor with letrec, constructors, case and parallel or (por) employing contextual equivalence. The model describes gates as (parallel) boolean operators, memory using a delay, which in turn is modeled as a shift of the list of signals, and permits also constructive cycles due to the parallel or. It opens the possibility of a large set of program transformations that correctly transform the expressions and thus the represented circuits and provides basic tools for equivalence testing and optimizing circuits. A further application is the correct manipulation by transformations of software components combined with circuits. The main part of our work are proof methods for correct transformations of expressions in the lambda calculus Lpor, and to propose the appropriate program transformations.
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.
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.
This paper proves several generic variants of context lemmas and thus contributes to improving the tools to develop observational semantics that is based on a reduction semantics for a language. The context lemmas are provided for may- as well as two variants of mustconvergence and a wide class of extended lambda calculi, which satisfy certain abstract conditions. The calculi must have a form of node sharing, e.g. plain beta reduction is not permitted. There are two variants, weakly sharing calculi, where the beta-reduction is only permitted for arguments that are variables, and strongly sharing calculi, which roughly correspond to call-by-need calculi, where beta-reduction is completely replaced by a sharing variant. The calculi must obey three abstract assumptions, which are in general easily recognizable given the syntax and the reduction rules. The generic context lemmas have as instances several context lemmas already proved in the literature for specific lambda calculi with sharing. The scope of the generic context lemmas comprises not only call-by-need calculi, but also call-by-value calculi with a form of built-in sharing. Investigations in other, new variants of extended lambda-calculi with sharing, where the language or the reduction rules and/or strategy varies, will be simplified by our result, since specific context lemmas are immediately derivable from the generic context lemma, provided our abstract conditions are met.
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.
We develop a proof method to show that in a (deterministic) lambda calculus with letrec and equipped with contextual equivalence the call-by-name and the call-by-need evaluation are equivalent, and also that the unrestricted copy-operation is correct. Given a let-binding x = t, the copy-operation replaces an occurrence of the variable x by the expression t, regardless of the form of t. This gives an answer to unresolved problems in several papers, it adds a strong method to the tool set for reasoning about contextual equivalence in higher-order calculi with letrec, and it enables a class of transformations that can be used as optimizations. The method can be used in different kind of lambda calculi with cyclic sharing. Probably it can also be used in non-deterministic lambda calculi if the variable x is "deterministic", i.e., has no interference with non-deterministic executions. The main technical idea is to use a restricted variant of the infinitary lambda-calculus, whose objects are the expressions that are unrolled w.r.t. let, to define the infinite developments as a reduction calculus on the infinite trees and showing a standardization theorem.
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.
Static analysis of different non-strict functional programming languages makes use of set constants like Top, Inf, and Bot denoting all expressions, all lists without a last Nil as tail, and all non-terminating programs, respectively. We use a set language that permits union, constructors and recursive definition of set constants with a greatest fixpoint semantics. This paper proves decidability, in particular EXPTIMEcompleteness, of subset relationship of co-inductively defined sets by using algorithms and results from tree automata. This shows decidability of the test for set inclusion, which is required by certain strictness analysis algorithms in lazy functional programming languages.
Extending the method of Howe, we establish a large class of untyped higher-order calculi, in particular such with call-by-need evaluation, where similarity, also called applicative simulation, can be used as a proof tool for showing contextual preorder. The paper also demonstrates that Mann’s approach using an intermediate “approximation” calculus scales up well from a basic call-by-need non-deterministic lambdacalculus to more expressive lambda calculi. I.e., it is demonstrated, that after transferring the contextual preorder of a non-deterministic call-byneed lambda calculus to its corresponding approximation calculus, it is possible to apply Howe’s method to show that similarity is a precongruence. The transfer is not treated in this paper. The paper also proposes an optimization of the similarity-test by cutting off redundant computations. Our results also applies to deterministic or non-deterministic call-by-value lambda-calculi, and improves upon previous work insofar as it is proved that only closed values are required as arguments for similaritytesting instead of all closed expressions.
Sharing of substructures like subterms and subcontexts in terms is a common method for space-efficient representation of terms, which allows for example to represent exponentially large terms in polynomial space, or to represent terms with iterated substructures in a compact form. We present singleton tree grammars as a general formalism for the treatment of sharing in terms. Singleton tree grammars (STG) are recursion-free context-free tree grammars without alternatives for non-terminals and at most unary second-order nonterminals. STGs generalize Plandowski's singleton context free grammars to terms (trees). We show that the test, whether two different nonterminals in an STG generate the same term can be done in polynomial time, which implies that the equality test of terms with shared terms and contexts, where composition of contexts is permitted, can be done in polynomial time in the size of the representation. This will allow polynomial-time algorithms for terms exploiting sharing. We hope that this technique will lead to improved upper complexity bounds for variants of second order unification algorithms, in particular for variants of context unification and bounded second order unification.
This paper proves correctness of Nöcker's method of strictness analysis, implemented in the Clean compiler, which is an effective way for strictness analysis in lazy functional languages based on their operational semantics. We improve upon the work of Clark, Hankin and Hunt did on the correctness of the abstract reduction rules. Our method fully considers the cycle detection rules, which are the main strength of Nöcker's strictness analysis. Our algorithm SAL is a reformulation of Nöcker's strictness analysis algorithm in a higher-order call-by-need lambda-calculus with case, constructors, letrec, and seq, extended by set constants like Top or Inf, denoting sets of expressions. It is also possible to define new set constants by recursive equations with a greatest fixpoint semantics. The operational semantics is a small-step semantics. Equality of expressions is defined by a contextual semantics that observes termination of expressions. Basically, SAL is a non-termination checker. The proof of its correctness and hence of Nöcker's strictness analysis is based mainly on an exact analysis of the lengths of normal order reduction sequences. The main measure being the number of 'essential' reductions in a normal order reduction sequence. Our tools and results provide new insights into call-by-need lambda-calculi, the role of sharing in functional programming languages, and into strictness analysis in general. The correctness result provides a foundation for Nöcker's strictness analysis in Clean, and also for its use in Haskell.
This paper proves correctness of Nocker s method of strictness analysis, implemented for Clean, which is an e ective way for strictness analysis in lazy functional languages based on their operational semantics. We improve upon the work of Clark, Hankin and Hunt, which addresses correctness of the abstract reduction rules. Our method also addresses the cycle detection rules, which are the main strength of Nocker s strictness analysis. We reformulate Nocker s strictness analysis algorithm in a higherorder lambda-calculus with case, constructors, letrec, and a nondeterministic choice operator used as a union operator. Furthermore, the calculus is expressive enough to represent abstract constants like Top or Inf. The operational semantics is a small-step semantics and equality of expressions is defined by a contextual semantics that observes termination of expressions. The correctness of several reductions is proved using a context lemma and complete sets of forking and commuting diagrams. The proof is based mainly on an exact analysis of the lengths of normal order reductions. However, there remains a small gap: Currently, the proof for correctness of strictness analysis requires the conjecture that our behavioral preorder is contained in the contextual preorder. The proof is valid without referring to the conjecture, if no abstract constants are used in the analysis.
Work on proving congruence of bisimulation in functional programming languages often refers to [How89,How96], where Howe gave a highly general account on this topic in terms of so-called lazy computation systems . Particularly in implementations of lazy functional languages, sharing plays an eminent role. In this paper we will show how the original work of Howe can be extended to cope with sharing. Moreover, we will demonstrate the application of our approach to the call-by-need lambda-calculus lambda-ND which provides an erratic non-deterministic operator pick and a non-recursive let. A definition of a bisimulation is given, which has to be based on a further calculus named lambda-~, since the na1ve bisimulation definition is useless. The main result is that this bisimulation is a congruence and contained in the contextual equivalence. This might be a step towards defining useful bisimulation relations and proving them to be congruences in calculi that extend the lambda-ND-calculus.
In this paper we demonstrate how to relate the semantics given by the nondeterministic call-by-need calculus FUNDIO [SS03] to Haskell. After introducing new correct program transformations for FUNDIO, we translate the core language used in the Glasgow Haskell Compiler into the FUNDIO language, where the IO construct of FUNDIO corresponds to direct-call IO-actions in Haskell. We sketch the investigations of [Sab03b] where a lot of program transformations performed by the compiler have been shown to be correct w.r.t. the FUNDIO semantics. This enabled us to achieve a FUNDIO-compatible Haskell-compiler, by turning o not yet investigated transformations and the small set of incompatible transformations. With this compiler, Haskell programs which use the extension unsafePerformIO in arbitrary contexts, can be compiled in a "safe" manner.
This paper proposes a non-standard way to combine lazy functional languages with I/O. In order to demonstrate the usefulness of the approach, a tiny lazy functional core language FUNDIO , which is also a call-by-need lambda calculus, is investigated. The syntax of FUNDIO has case, letrec, constructors and an IO-interface: its operational semantics is described by small-step reductions. A contextual approximation and equivalence depending on the input-output behavior of normal order reduction sequences is defined and a context lemma is proved. This enables to study a semantics of FUNDIO and its semantic properties. The paper demonstrates that the technique of complete reduction diagrams enables to show a considerable set of program transformations to be correct. Several optimizations of evaluation are given, including strictness optimizations and an abstract machine, and shown to be correct w.r.t. contextual equivalence. Correctness of strictness optimizations also justifies correctness of parallel evaluation. Thus this calculus has a potential to integrate non-strict functional programming with a non-deterministic approach to input-output and also to provide a useful semantics for this combination. It is argued that monadic IO and unsafePerformIO can be combined in Haskell, and that the result is reliable, if all reductions and transformations are correct w.r.t. to the FUNDIO-semantics. Of course, we do not address the typing problems the are involved in the usage of Haskell s unsafePerformIO. The semantics can also be used as a novel semantics for strict functional languages with IO, where the sequence of IOs is not fixed.
Context unification is a variant of second order unification. It can also be seen as a generalization of string unification to tree unification. Currently it is not known whether context unification is decidable. A specialization of context unification is stratified context unification, which is decidable. However, the previous algorithm has a very bad worst case complexity. Recently it turned out that stratified context unification is equivalent to satisfiability of one-step rewrite constraints. This paper contains an optimized algorithm for strati ed context unification exploiting sharing and power expressions. We prove that the complexity is determined mainly by the maximal depth of SO-cycles. Two observations are used: i. For every ambiguous SO-cycle, there is a context variable that can be instantiated with a ground context of main depth O(c*d), where c is the number of context variables and d is the depth of the SO-cycle. ii. the exponent of periodicity is O(2 pi ), which means it has an O(n)sized representation. From a practical point of view, these observations allow us to conclude that the unification algorithm is well-behaved, if the maximal depth of SO-cycles does not grow too large.