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

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#### Year of publication

- 2007 (8) (remove)

#### Keywords

- functional programming languages (2)
- Lambda-Kalkül (1)
- Programmtransformation (1)
- call-by-name (1)
- call-by-need (1)
- context lemma (1)
- context lemma (1)
- infinitary lambda calculus (1)
- lambda calculus (1)
- lambda calculus (1)

- 31
- On equivalences and standardization in a non-deterministic call-by-need lambda calculus (2007)
- 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.

- 31 [v.2]
- On equivalences and standardization in a non-deterministic call-by-need lambda calculus (2007)
- 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.

- 30
- Program transformation for functional circuit descriptions (2007)
- 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.

- 29
- Correctness of copy in calculi with letrec, case, constructors and por (2007)
- 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.

- 28
- Correctness of copy in calculi with letrec, case and constructors (2007)
- 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.

- 27
- On generic context lemmas for lambda calculi with sharing (2007)
- 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.

- 27 [v.2]
- On generic context lemmas for lambda calculi with sharing (2007)
- 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.

- 25 [v.2]
- Equivalence of call-by-name and call-by-need for lambda-calculi with letrec (2007)
- 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.