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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.
This paper describes a method to treat contextual equivalence in polymorphically typed lambda-calculi, and also how to transfer equivalences from the untyped versions of lambda-calculi to their typed variant, where our specific calculus has letrec, recursive types and is nondeterministic. An addition of a type label to every subexpression is all that is needed, together with some natural constraints for the consistency of the type labels and well-scopedness of expressions. One result is that an elementary but typed notion of program transformation is obtained and that untyped contextual equivalences also hold in the typed calculus as long as the expressions are well-typed. In order to have a nice interaction between reduction and typing, some reduction rules have to be accompanied with a type modification by generalizing or instantiating types.
Context unification is a variant of second-order unification and also a generalization of string unification. Currently it is not known whether context uni cation is decidable. An expressive fragment of context unification is stratified context unification. Recently, it turned out that stratified context unification and one-step rewrite constraints are equivalent. This paper contains a description of a decision algorithm SCU for stratified context unification together with a proof of its correctness, which shows decidability of stratified context unification as well as of satisfiability of one-step rewrite constraints.
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.
We introduce a new method for representing and solving a general class of non-preemptive resource-constrained project scheduling problems. The new approach is to represent scheduling problems as de- scriptions (activity terms) in a language called RSV, which allows nested expressions using pll, seq, and xor. The activity-terms of RSV are similar to concepts in a description logic. The language RSV generalizes previous approaches to scheduling with variants insofar as it permits xor's not only of atomic activities but also of arbitrary activity terms. A specific semantics that assigns their set of active schedules to activity terms shows correctness of a calculus normalizing activity terms RSV similar to propositional DNF-computation. Based on RSV, this paper describes a diagram-based algorithm for the RSV-problem which uses a scan-line principle. The scan-line principle is used for determining and resolving the occurring resource conflicts and leads to a nonredundant generation of all active schedules and thus to a computation of the optimal schedule.
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.
Towards correctness of program transformations through unification and critical pair computation
(2011)
Correctness of program transformations in extended lambda calculi with a contextual semantics is usually based on reasoning about the operational semantics which is a rewrite semantics. A successful approach to proving correctness is the combination of a context lemma with the computation of overlaps between program transformations and the reduction rules, and then of so-called complete sets of diagrams. The method is similar to the computation of critical pairs for the completion of term rewriting systems.We explore cases where the computation of these overlaps can be done in a first order way by variants of critical pair computation that use unification algorithms. As a case study we apply the method to a lambda calculus with recursive let-expressions and describe an effective unification algorithm to determine all overlaps of a set of transformations with all reduction rules. The unification algorithm employs many-sorted terms, the equational theory of left-commutativity modelling multi-sets, context variables of different kinds and a mechanism for compactly representing binding chains in recursive let-expressions.
It is well known that first order uni cation is decidable, whereas second order and higher order unification is undecidable. Bounded second order unification (BSOU) is second order unification under the restriction that only a bounded number of holes in the instantiating terms for second order variables is permitted, however, the size of the instantiation is not restricted. In this paper, a decision algorithm for bounded second order unification is described. This is the fist non-trivial decidability result for second order unification, where the (finite) signature is not restricted and there are no restrictions on the occurrences of variables. We show that the monadic second order unification (MSOU), a specialization of BSOU is in sum p s. Since MSOU is related to word unification, this is compares favourably to the best known upper bound NEXPTIME (and also to the announced upper bound PSPACE) for word unification. This supports the claim that bounded second order unification is easier than context unification, whose decidability is currently an open question.
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.
This paper describes context analysis, an extension to strictness analysis for lazy functional languages. In particular it extends Wadler's four point domain and permits in nitely many abstract values. A calculus is presented based on abstract reduction which given the abstract values for the result automatically finds the abstract values for the arguments. The results of the analysis are useful for veri fication purposes and can also be used in compilers which require strictness information.
The extraction of strictness information marks an indispensable element of an efficient compilation of lazy functional languages like Haskell. Based on the method of abstract reduction we have developed an e cient strictness analyser for a core language of Haskell. It is completely written in Haskell and compares favourably with known implementations. The implementation is based on the G#-machine, which is an extension of the G-machine that has been adapted to the needs of abstract reduction.
This paper shows equivalence of applicative similarity and contextual approximation, and hence also of bisimilarity and contextual equivalence, in LR, the deterministic call-by-need lambda calculus with letrec extended by data constructors, case-expressions and Haskell's seqoperator. LR models an untyped version of the core language of Haskell. Bisimilarity simplifies equivalence proofs in the calculus and opens a way for more convenient correctness proofs for program transformations.
The proof is by a fully abstract and surjective transfer of the contextual approximation into a call-by-name calculus, which is an extension of Abramsky's lazy lambda calculus. In the latter calculus equivalence of similarity and contextual approximation can be shown by Howe's method. Using an equivalent but inductive definition of behavioral preorder we then transfer similarity back to the calculus LR.
The translation from the call-by-need letrec calculus into the extended call-by-name lambda calculus is the composition of two translations. The first translation replaces the call-by-need strategy by a call-by-name strategy and its correctness is shown by exploiting infinite tress, which emerge by unfolding the letrec expressions. The second translation encodes letrec-expressions by using multi-fixpoint combinators and its correctness is shown syntactically by comparing reductions of both calculi. A further result of this paper is an isomorphism between the mentioned calculi, and also with a call-by-need letrec calculus with a less complex definition of reduction than LR.
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.
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. Correctness means that our encodings preserve and reflect the observations of may- and must-convergence, and as a consequence also yields soundness of the encodings with respect to a contextually defined notion of program equivalence. While these translations encode blocking into queuing and waiting, we also describe an adequate encoding of buffers in a calculus without handles, which is more low-level and uses busy-waiting instead of blocking. Furthermore we demonstrate that our correctness concept applies to the whole compilation process from high-level to low-level concurrent languages, by translating the calculus with buffers, handled futures and data constructors into a small core language without those constructs.
The goal of this report is to prove correctness of a considerable subset of transformations w.r.t. contextual equivalence in an extended lambda-calculus LS with case, constructors, seq, let, and choice, with a simple set of reduction rules; and to argue that an approximation calculus LA is equivalent to LS w.r.t. the contextual preorder, which enables the proof tool of simulation. Unfortunately, a direct proof appears to be impossible.
The correctness proof is by defining another calculus L comprising the complex variants of copy, case-reduction and seq-reductions that use variable-binding chains. This complex calculus has well-behaved diagrams and allows a proof of correctness of transformations, and that the simple calculus LS, the calculus L, and the calculus LA all have an equivalent contextual preorder.
The calculus LRP is a polymorphically typed call-by-need lambda calculus extended by data constructors, case-expressions, seq-expressions and type abstraction and type application. This report is devoted to the extension LRPw of LRP by scoped sharing decorations. The extension cannot be properly encoded into LRP if improvements are defined w.r.t. the number of lbeta, case, and seq-reductions, which makes it necessary to reconsider the claims and proofs of properties. We show correctness of improvement properties of reduction and transformation rules and also of computation rules for decorations in the extended calculus LRPw. We conjecture that conservativity of the embedding of LRP in LRPw holds.
The calculus LRP is a polymorphically typed call-by-need lambda calculus extended by data constructors, case-expressions, seq-expressions and type abstraction and type application. This report is devoted to the extension LRPw of LRP by scoped sharing decorations. The extension cannot be properly encoded into LRP if improvements are defined w.r.t. the number of lbeta, case, and seq-reductions, which makes it necessary to reconsider the claims and proofs of properties. We show correctness of improvement properties of reduction and transformation rules and also of computation rules for decorations in the extended calculus LRPw. We conjecture that conservativity of the embedding of LRP in LRPw holds.