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**Published in:** LIPIcs, Volume 211, 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021)

By adapting Salomaa’s complete proof system for equality of regular expressions under the language semantics, Milner (1984) formulated a sound proof system for bisimilarity of regular expressions under the process interpretation he introduced. He asked whether this system is complete. Proof-theoretic arguments attempting to show completeness of this equational system are complicated by the presence of a non-algebraic rule for solving fixed-point equations by using star iteration.
We characterize the derivational power that the fixed-point rule adds to the purely equational part Mil- of Milner’s system Mil: it corresponds to the power of coinductive proofs over Mil- that have the form of finite process graphs with the loop existence and elimination property LEE. We define a variant system cMil by replacing the fixed-point rule in Mil with a rule that permits LEE-shaped circular derivations in Mil- from previously derived equations as a premise. With this rule alone we also define the variant system CLC for combining LEE-shaped coinductive proofs over Mil-. We show that both cMil and CLC have proof interpretations in Mil, and vice versa. As this correspondence links, in both directions, derivability in Mil with derivation trees of process graphs, it widens the space for graph-based approaches to finding a completeness proof of Milner’s system.

Clemens Grabmayer. A Coinductive Version of Milner’s Proof System for Regular Expressions Modulo Bisimilarity. In 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 211, pp. 16:1-16:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{grabmayer:LIPIcs.CALCO.2021.16, author = {Grabmayer, Clemens}, title = {{A Coinductive Version of Milner’s Proof System for Regular Expressions Modulo Bisimilarity}}, booktitle = {9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021)}, pages = {16:1--16:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-212-9}, ISSN = {1868-8969}, year = {2021}, volume = {211}, editor = {Gadducci, Fabio and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2021.16}, URN = {urn:nbn:de:0030-drops-153712}, doi = {10.4230/LIPIcs.CALCO.2021.16}, annote = {Keywords: regular expressions, process theory, bisimilarity, coinduction, proof theory} }

Document

**Published in:** LIPIcs, Volume 21, 24th International Conference on Rewriting Techniques and Applications (RTA 2013)

We address a problem connected to the unfolding semantics of functional programming languages: give a useful characterization of those infinite lambda-terms that are lambda-letrec-expressible in the sense that they arise as infinite unfoldings of terms in lambda-letrec, the lambda-calculus with letrec. We provide two characterizations, using concepts we introduce for infinite lambda-terms: regularity, strong regularity, and binding–capturing chains.
It turns out that lambda-letrec-expressible infinite lambda-terms
form a proper subclass of the regular infinite lambda-terms.
In this paper we establish these characterizations only for
expressibility in lambda-mu, the lambda-calculus with explicit mu-recursion. We show that for all infinite lambda-terms T the following are equivalent: (i): T is lambda-mu-expressible; (ii): T is strongly regular; (iii): T is regular, and it only has finite binding–capturing chains.
We define regularity and strong regularity for infinite lambda-terms as two different generalizations of regularity for infinite first-order terms: as the existence of only finitely many subterms that are defined as the reducts of two rewrite systems for decomposing lambda-terms. These rewrite systems act on infinite lambda-terms furnished with a bracketed prefix of abstractions for collecting decomposed lambda-abstractions and keeping the terms closed under decomposition. They differ in which vacuous abstractions in the prefix are removed.

Clemens Grabmayer and Jan Rochel. Expressibility in the Lambda Calculus with Mu. In 24th International Conference on Rewriting Techniques and Applications (RTA 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 21, pp. 206-222, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{grabmayer_et_al:LIPIcs.RTA.2013.206, author = {Grabmayer, Clemens and Rochel, Jan}, title = {{Expressibility in the Lambda Calculus with Mu}}, booktitle = {24th International Conference on Rewriting Techniques and Applications (RTA 2013)}, pages = {206--222}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-53-8}, ISSN = {1868-8969}, year = {2013}, volume = {21}, editor = {van Raamsdonk, Femke}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2013.206}, URN = {urn:nbn:de:0030-drops-40635}, doi = {10.4230/LIPIcs.RTA.2013.206}, annote = {Keywords: lambda-calculus, lambda-calculus with letrec, unfolding semantics, regularity for infinite lambda-terms, binding-capturing chain} }

Document

**Published in:** LIPIcs, Volume 6, Proceedings of the 21st International Conference on Rewriting Techniques and Applications (2010)

We present some contributions to the theory of infinitary rewriting
for weakly orthogonal term rewrite systems, in which critical pairs
may occur provided they are trivial.
We show that the infinitary unique normal form property (UNinf)
fails by a simple example of a weakly orthogonal TRS with two
collapsing rules. By translating this example, we show that UNinf
also fails for the infinitary lambda-beta-eta-calculus.
As positive results we obtain the following: Infinitary confluence,
and hence UNinf, holds for weakly orthogonal TRSs that do not contain
collapsing rules. To this end we refine the compression lemma.
Furthermore, we consider the triangle and diamond properties
for infinitary developments in weakly orthogonal TRSs,
by refining an earlier cluster-analysis for the finite case.

Joerg Endrullis, Clemens Grabmayer, Dimitri Hendriks, Jan Willem Klop, and Vincent van Oostrom. Unique Normal Forms in Infinitary Weakly Orthogonal Rewriting. In Proceedings of the 21st International Conference on Rewriting Techniques and Applications. Leibniz International Proceedings in Informatics (LIPIcs), Volume 6, pp. 85-102, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{endrullis_et_al:LIPIcs.RTA.2010.85, author = {Endrullis, Joerg and Grabmayer, Clemens and Hendriks, Dimitri and Klop, Jan Willem and van Oostrom, Vincent}, title = {{Unique Normal Forms in Infinitary Weakly Orthogonal Rewriting}}, booktitle = {Proceedings of the 21st International Conference on Rewriting Techniques and Applications}, pages = {85--102}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-18-7}, ISSN = {1868-8969}, year = {2010}, volume = {6}, editor = {Lynch, Christopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2010.85}, URN = {urn:nbn:de:0030-drops-26469}, doi = {10.4230/LIPIcs.RTA.2010.85}, annote = {Keywords: Weakly orthogonal term rewrite systems, unique normal form property, infinitary rewriting, infinitary lambda-beta-eta-calculus,} }

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