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

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- 41
- Towards correctness of program transformations through unification and critical pair computation (2010)
- 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 is the combination of a context lemma with the computation of overlaps between program transformations and the reduction rules, which results in 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 of an application we describe a finitary and decidable unification algorithm for the combination of the equational theory of left-commutativity modelling multi-sets, context variables and many-sorted unification. Sets of equations are restricted to be almost linear, i.e. every variable and context variable occurs at most once, where we allow one exception: variables of a sort without ground terms may occur several times. Every context variable must have an argument-sort in the free part of the signature. We also extend the unification algorithm by the treatment of binding-chains in let- and letrec-environments and by context-classes. This results in a unification algorithm that can be applied to all overlaps of normal-order reductions and transformations in an extended lambda calculus with letrec that we use as a case study.

- 40
- Simulation in the call-by-need lambda-calculus with letrec (2010)
- This paper shows the equivalence of applicative similarity and contextual approximation, and hence also of bisimilarity and contextual equivalence, in the deterministic call-by-need lambda calculus with letrec. Bisimilarity simplifies equivalence proofs in the calculus and opens a way for more convenient correctness proofs for program transformations. Although this property may be a natural one to expect, to the best of our knowledge, this paper is the first one providing a proof. The proof technique is to transfer the contextual approximation into Abramsky's lazy lambda calculus by a fully abstract and surjective translation. This also shows that the natural embedding of Abramsky's lazy lambda calculus into the call-by-need lambda calculus with letrec is an isomorphism between the respective term-models.We show that the equivalence property proven in this paper transfers to a call-by-need letrec calculus developed by Ariola and Felleisen.

- 39
- Reconstruction a logic for inductive proofs of properties of functional programs (2010)
- A logical framework consisting of a polymorphic call-by-value functional language and a first-order logic on the values is presented, which is a reconstruction of the logic of the verification system VeriFun. The reconstruction uses contextual semantics to define the logical value of equations. It equates undefinedness and non-termination, which is a standard semantical approach. The main results of this paper are: Meta-theorems about the globality of several classes of theorems in the logic, and proofs of global correctness of transformations and deduction rules. The deduction rules of VeriFun are globally correct if rules depending on termination are appropriately formulated. The reconstruction also gives hints on generalizations of the VeriFun framework: reasoning on nonterminating expressions and functions, mutual recursive functions and abstractions in the data values, and formulas with arbitrary quantifier prefix could be allowed.

- 38
- Counterexamples to simulation in non-deterministic call-by-need lambda-calculi with letrec (2009)
- This note shows that in non-deterministic extended lambda calculi with letrec, the tool of applicative (bi)simulation is in general not usable for contextual equivalence, by giving a counterexample adapted from data flow analysis. It also shown that there is a flaw in a lemma and a theorem concerning finite simulation in a conference paper by the first two authors.

- 37
- On correctness of buffer implementations in a concurrent lambda calculus with futures (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. Correctness means that our encodings preserve and reflect the observations of may- and must-convergence. This also shows correctness wrt. program semantics, since the encodings are adequate translations wrt. contextual semantics. While these translations encode blocking into queuing and waiting, we also provide 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.

- 36
- Contextual equivalence in lambda-calculi extended with letrec and with a parametric polymorphic type system (2009)
- 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.

- 35
- Closures of may and must convergence for contextual equivalence (2008)
- We show on an abstract level that contextual equivalence in non-deterministic program calculi defined by may- and must-convergence is maximal in the following sense. Using also all the test predicates generated by the Boolean, forall- and existential closure of may- and must-convergence does not change the contextual equivalence. The situation is different if may- and total must-convergence is used, where an expression totally must-converges if all reductions are finite and terminate with a value: There is an infinite sequence of test-predicates generated by the Boolean, forall- and existential closure of may- and total must-convergence, which also leads to an infinite sequence of different contextual equalities.

- 34
- On proving the equivalence of concurrency primitives (2008)
- Various concurrency primitives have been added to sequential programming languages, in order to turn them concurrent. Prominent examples are concurrent buffers for Haskell, channels in Concurrent ML, joins in JoCaml, and handled futures in Alice ML. Even though one might conjecture that all these primitives provide the same expressiveness, proving this equivalence is an open challenge in the area of program semantics. In this paper, we establish a first instance of this conjecture. We show that concurrent buffers can be encoded in the lambda calculus with futures underlying Alice ML. Our correctness proof results from a systematic method, based on observational semantics with respect to may and must convergence.

- 33
- Adequacy of compositional translations for observational semantics (2008)
- We investigate methods and tools for analysing translations between programming languages with respect to observational semantics. The behaviour of programs is observed in terms of may- and must-convergence in arbitrary contexts, and adequacy of translations, i.e., the reﬂection of program equivalence, is taken to be the fundamental correctness condition. For compositional translations we propose a notion of convergence equivalence as a means for proving adequacy. This technique avoids explicit reasoning about contexts, and is able to deal with the subtle role of typing in implementations of language extension.

- 32
- A finite simulation method in a non-deterministic call-by-need calculus with letrec, constructors and case (2008)
- 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.