# informatics

### Publications of Fer-Jan de Vries

#### Refereed Publications

1. F.J. de Vries.
On Undefined and Meaningless in Lambda Definability
in proceedings of 1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016)
pdf, DOI: 10.4230/LIPIcs.FSCD.2016.18, URL: http://drops.dagstuhl.de/opus/volltexte/2016/5978/

AbstractWe distinguish between undefined terms as used in lambda definability of partial recursive functions and meaningless terms as used in infinite lambda calculus for the infinitary terms models that generalise the Bohm model. While there are uncountable many known sets of meaningless terms, there are four known sets of undefined terms. Two of these four are sets of meaningless terms. In this paper we first present set of sufficient conditions for a set of lambda terms to serve as set of undefined terms in lambda definability of partial functions. The four known sets of undefined terms satisfy these conditions. Next we locate the smallest set of meaningless terms satisfying these conditions. This set sits very low in the lattice of all sets of meaningless terms. Any larger set of meaningless terms than this smallest set is a set of undefined terms. Thus we find uncountably many new sets of undefined terms. As an unexpected bonus of our careful analysis of lambda definability we obtain a natural modification, strict lambda-definability, which allows for a Barendregt style of proof in which the representation of composition is truly the composition of representations.
2. A. Kurz, A. Pardo, D. Petrisan, P. Severi and F.J. de Vries.
Approximation of Nested Fixpoints – A Coalgebraic View of Parametric Dataypes
in proceedings of 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015)
pdf, DOI: 10.4230/LIPIcs.CALCO.2015.205, URL: http://drops.dagstuhl.de/opus/volltexte/2015/5535/

Abstract The question addressed in this paper is how to correctly approximate infinite data given by systems of simultaneous corecursive definitions. We devise a categorical framework for reasoning about regular datatypes, that is, datatypes closed under products, coproducts and fixpoints. We argue that the right methodology is on one hand coalgebraic (to deal with possible non- termination and infinite data) and on the other hand 2-categorical (to deal with parameters in a disciplined manner). We prove a coalgebraic version of Bekič lemma that allows us to reduce simultaneous fixpoints to a single fix point. Thus a possibly infinite object of interest is regarded as a final coalgebra of a many-sorted polynomial functor and can be seen as a limit of finite approximants. As an application, we prove correctness of a generic function that calculates the approximants on a large class of data types.
3. P. Severi and F.J. de Vries.
The Infinitary Lambda Calculus of the Infinite Eta Bohm Trees
in Mathematical Structures in Computer Science (FirstView Article: 17 August 2015): Computing with lambda-terms: A special issue dedicated to Corrado Böhm for his 90th birthday
pdf DOI: http://dx.doi.org/10.1017/S096012951500033X

Abstract In this paper we introduce a strong form of eta reduction called etabang that we use to construct a confluent and normalising infinitary lambda calculus, of which the normal forms correspond to Barendregt’s infinite eta Bohm trees. This new infinitary perspective on the set of infinite eta Bohm trees allows us to prove that the set of infinite eta B ̈hm trees is a model of the lambda calculus. The model is of interest because it has the same local structure as Scott’s D-infinity-models, i.e. two finite lambda terms are equal in the infinite eta B ̈hm model if and only if they have the same interpretation in Scott’s D-infinity-models
4. A. Kurz, D. Petrisan, P. Severi and F.J. de Vries.
Nominal Coalgebraic Data Types with Applications to Lambda Calculus
in LMCS 2013, volume 9(4), paper 20, pages 1-52, 2013.
pdf

Abstract We investigate final coalgebras in nominal sets. This allows us to define types of infinite data with binding for which all constructions automatically respect alpha equivalence. We give applications to the infinitary lambda calculus.
5. P. Severi and F.J. de Vries.
Completeness of Conversion between Reactive Programs for Ultrametric Models
in Proceedings of the Typed Lambda Calculi and Applications, 11th International Conference, TLCA 2013, Eindhoven, The Netherlands, June 26-28, 2013. SLNCS volume 7941, pages 221-235, 2013. .
pdf

Abstract In 1970 Friedman proved completeness of beta eta conversion in the simply-typed lambda calculus for the set-theoretical model. Re- cently Krishnaswami and Benton have captured the essence of Hudak’s reactive programs in an extension of simply typed lambda calculus with causal streams and a temporal modality and provided this typed lambda calculus for reactive programs with a sound ultrametric semantics. We show that beta eta conversion in the typed lambda calculus of reac- tive programs is complete for the ultrametric model.
6. P. Severi and F.J. de Vries.
Pure Type Systems with Corecursion on Streams
in Proceedings of the 17th ACM SIGPLAN International Conference on Functional Programming, ICFP 2012.
pdf

Abstract In this paper, we use types for ensuring that programs involving streams are well-behaved. We extend pure type systems with a type constructor for streams, a modal operator next and a fixed point operator for expressing corecursion. This extension is called Pure Type Systems with Corecursion (CoPTS). The typed lambda calcu- lus for reactive programs defined by Krishnaswami and Benton can be obtained as a CoPTS. CoPTS’s allow us to study a wide range of typed lambda calculi extended with corecursion using only one framework. In particular, we study this extension for the calculus of constructions which is the underlying formal language of Coq. We use the machinery of infinitary rewriting and formalize the idea of well-behaved programs using the concept of infinitary normalization. We study the properties of infinitary weak and strong normalization for CoPTS’s. The set of finite and infinite terms is defined as a metric completion. We shed new light on the meaning of the modal operator by connecting the modality with the depth used to define the metric. This connection is the key to the proofs of infinitary weak and strong normalization
7. P. Severi and F.J. de Vries.
Meaningless Sets in Infinitary Combinatory Logic
in proceedings of RTA 2012, the 23rd International Conference on Rewriting Techniques and Applications. Editor Ashish Tiwari. Nagoya, May 28 -June 2, 2012. Leibniz International Proceedings in Informatics (LIPIcs) volume 15. Pages 288--304.
pdf

Abstract In this paper we study meaningless sets in infinitary combinatory logic. So far only a handful of meaningless sets were known. We show that there are uncountably many meaningless sets. As an application to the semantics of finite combinatory logics, we show that there exist uncountably many combinatory algebras that are not a lambda algebra. We also study ways of weakening the axioms of meaningless sets to get, not only sufficient, but also necessary conditions for having confluence and normalisation.
8. P. Severi and F.J. de Vries.
Weakening the Axiom of Overlap in Infinitary Lambda Calculus
in proceedings of RTA2011: the 22nd International Conference on Rewriting Techniques and Applications. Editor Manfred Schmidt-Schauß. Novi Sad. May 30 - June 1, 2011. Leibniz International Proceedings in Informatics (LIPIcs) volume 10. Pages 313--328.
pdf

Abstract In this paper we present a set of necessary and sufficient conditions on a set of lambda terms to serve as the set of meaningless terms in an infinitary bottom extension of lambda calculus. So far only a set of sufficient conditions was known for choosing a suitable set of meaningless terms to make this construction produce confluent extensions. The conditions covered the three main known examples of sets of meaningless terms. However, the much later construction of many more examples of sets of meaningless terms satisfying the sufficient conditions renewed the interest in the necessity question and led us to reconsider the old conditions. The key idea in this paper is an alternative solution for solving the overlap between beta reduction and bottom reduction. This allows us to reformulate the Axiom of Overlap, which now determines together with the other conditions a larger class of sets of meaningless terms. We show that the reformulated conditions are not only sufficient but also necessary for obtaining a confluent and normalizing infinitary lambda beta bottom calculus. As an interesting consequence of the necessity proof we obtain for infinitary lambda calculus with beta and bot reduction that confluence implies normalization.
9. A. Kurz, D. Petrisan, P. Severi and F.J. de Vries.
An Alpha-Corecursion Principle for the Infinitary Lambda Calculus
in postproceedings of CMCS 2012: the 11th International Workshop on Coalgebraic Methods in Computer Science. 31 March - 1 April 2012, Tallinn, Estonia. SLNCS volume 7399, pages 130-149, 2012.
pdf

Abstract Gabbay and Pitts proved that lambda-terms up to alpha-equivalence constitute an initial algebra for a certain endofunctor on the category of nominal sets. We show that the terms of the infinitary lambda-calculus form the final coalgebra for the same functor. This allows us to give a corecursion principle for alpha-equivalence classes of finite and infinite terms. As an application, we give corecursive definitions of substitution and of infinite normal forms (Bohm, Levy-Longo and Berarducci trees).
10. P. Severi and F.J. de Vries.
Decomposing the Lattice of Meaningless Sets in the Infinitary Lambda Calculus
in proceedings of WoLLIC 2011, Logic, Language, Information and Computation - 18th International Workshop, Philadelphia, PA, USA, May 18-20, 2011. Editors Lev D. Beklemishev and Ruy de Queiroz. SLNAI 6642, pages 210-227, 2011.
pdf

Abstract The notion of a meaningless set has been defined for infinitary lambda calculus axiomatically. Standard examples of meaningless sets are sets of terms that have no head normal form, the set of terms without weak head normal form and the set of rootactive terms. The collection of meaningless sets is a lattice. In this paper, we study the way this lattices decompose as union of more elementary key intervals. We also analyse the distribution of the sets of meaningless terms in the lattice by selecting some sets as key vertices and study the cardinality in the intervals between key vertices. As an application, we prove that the lattice of meaningless sets is neither distributive nor modular. Interest- ingly, the example translates into a counterexample that the lattice of lambda theories is not modular.
11. J.R. Kennaway, P. Severi, M.R. Sleep, and F.J. de Vries.
Infinite rewriting: from syntax to semantics,
in proceedings of Processes, Terms and Cycles: Steps on the Road to Infinity: Essays dedicated to Jan Willem Klop on the Occasion of His 60th Birthday, editors: A. Middeldorp, V. van Oostrom, F. van Raamsdonk and R. de Vrijer. SLNCS 3838, pages 148-172, 2005.
pdf

Part one of this paper is a brief summary of the theory of infinitary rewriting; part two presents as application recent work presented at CSL05 and TLCA05 plus some new results by Severi and de Vries on the lattice of tree semantics for lambda calculus. Each tree semantics consist of the set of normal forms of a confluent and normalising infinitary extension of lambda calculus build from a set of "meaningless" terms. Using their syntactic descriptions these sets of meaningless terms can be ordered into a rich lattice, providing some insight in the even richer lattice of lambda models.
12. P. Severi and F.J. de Vries.
Separability of infinite lambda terms
in proceedings of the LCTTNL Workshop: The second workshop on Lambda Calculus, Type Theory and Natural Language, King's College London, 12th September 2005. Page 1-12.
pdf

Abstract Infinite lambda calculi extend finite lambda calculus with infinite terms and transfinite reduction. In this paper we extend some classical results of finite lambda calculus to infinite terms. First we extend to infinite terms is B\"ohm Theorem which states the separability of two finite $\beta \eta$-normal forms. Next we extend to infinite terms is the equivalence of the prefix relation up to infinite eta expansions and the contextual preorder that observes head normal forms. Finally we prove that the theory given by equality of $\infty \eta$-B\"ohm trees is the largest theory induced by the confluent and normalising infinitary lambda calculi extending the calculus of B\"ohm trees.
13. P. Severi and F.J. de Vries.
Order Structures on Böhm-like Models
Conference paper in Luke Ong, editor, Computer Science Logic: 19th International Workshop, CSL 2005, 14th Annual Conference of the EACSL, Oxford, UK. August 22-25, 2005. (Lecture Notes in Computer Science 3634), Springer-Verlag. Page 103-116. 2005.
dvi, ps and pdf

Abstract In [Kennaway-de Vries 2003] we presented a general construction of tree semantics for lambda calculus parametrised by a set of "meaningless" terms. Here we give many new examples of meaningless sets. We study whether the resulting tree semantics are orderable. The six principal models (including the Böhm and the Lévy-Longo model) are orderable by the prefix relation. The Berarducci model is also orderable. However, orderability is not the norm. We give a syntactic proof that at least 2^c such tree semantics are unorderable, where c is the cardinality of the continuum. Salibra's Theorem is a corollary.
14. P. Severi and F.J. de Vries.
Continuity and Discontinuity in Lambda Calculus
Conference paper in Pawel Urzyczyn, editor, Typed Lambda Calculus and Applications, Proceedings of the 7th International Conference, TLCA 2005, Nara Japan, April 21-23 2005. (Lecture Notes in Computer Science 3461), Springer-Verlag. Page 369-385. 2005.
dvi, ps and pdf

Abstract In [Kennaway-de Vries 2003] we presented a general construction of tree semantics from a set of "meaningless" terms. Properties of these semantics have not been studied before. Observing that Barendregt's Böhm model construction for lambda calculus depends heavily on continuity, in contrast to our general construction, and that the Berarducci model is not continuous, it is a natural question to ask when a tree model of the lambda calculus is continuous. We prove that the only continuous trees semantics are the Böhm and the Levy-Longo model and that only these two models have an approximation theorem and continuous context operators.
15. J.R. Kennaway and F.J. de Vries.
Infinitary Rewriting.
Chapter 12 in Terese, editor, Term Rewriting Systems (Cambridge Tracts in Theoretical Computer Science 55), Cambridge University Press. Page 668-711. 2003. Erratum.
dvi, ps and pdf

Abstract In this chapter we develop a general theory of infinite orthogonal rewriting unifying our earlier work on infinite term rewriting and infinite lambda calculus. Confluence in the infinitary setting is significantly harder than in the finitary case. The technique of transfinite tiling diagrams in the confluence proof is new. Infinitary extensions of confluent finite systems may not be confluent, unless one adds a rule that identifies "meaningless" terms by reducing them to bottom. Then the extensions become confluent and normalising; the sets of their normal forms can give new denotational semantics of the original finite systems.
16. M. Dezani-Ciancaglini, P. Severi and F.J. de Vries.
Infinitary Lambda Calculus and Discrimination of Berarducci trees.
Theoretical Computer Science 298(2):275 - 302, 2003.
dvi, ps and pdf

AbstractTree semantics of lambda calculus induce equality relations on lambda terms. For the common tree semantics various authors have given alternative characterisations in the form of observational equivalence of suitable extensions of lambda calculus. Berarducci trees are a recent tree semantics. The induced equality relation is included in all equality relations induced by a tree semantics. In this paper we give such a characterisation for Berarducci trees. To do so we make an essential use of our recently developed techniques for infinite rewriting.
17. P. Severi and F.J. de Vries.
An Extensional Böhm Model.
Conference paper in Sophie Tison, editor, Rewiting Techniques and Applications, Proceedings of the 13th International Conference, RTA 2002, Copenhagen Denmark, July 2002. (Lecture Notes in Computer Science 2378), Springer-Verlag. Page 159-173. 2002.
dvi, ps and pdf (15 pages)

Abstract Here we present the first confluent and normalising infinitary extension of extensional lambda calculus using transfinite tiling diagrams from [Kennaway-de Vries 2003]. This requires new infinitary commutation and postponement results for eta reduction and beta-bottom reduction. Application: the set of normal forms of the extension is a simple (i.e.,\ not using Barendregt's continuity techniques) description of Barendregt and Dezani's extensional eta-Böhm tree model and gives us a simple and direct (no detour via a D-infinity semantics!) syntax-based proof that two lambda terms are equal in this model if and only if they are observationally equivalent wrt to eta-beta normal forms.
18. S. van Bakel, F. Barbanera, M. Dezani-Ciancaglini, F.J. de Vries.
Intersection Types for Lambda Trees.
Theoretical Computer Science 272(1-2): 3-40, 2002.
ps and pdf (42 pages)

Abstract Trees have been used to give syntax-flavoured semantics to lambda calculus. As such they play an important role in the understanding of lambda calculus. In this paper we study five families of trees and we present an intersection type assignment system parametric with respect to these five families. Then we prove, in an (almost) uniform way, that each type assignment system fully describes the observational equivalence induced by the corresponding tree representation of lambda terms. For two families of extensional trees this gives new results and, for the other families of trees, this unifies and improves earlier results.
19. M. Dezani-Ciancaglini, P. Severi and F.J. de Vries.
Böhm's theorem for Berarducci trees.
In Proceedings CATS 2000 Computing: the Australasian Theory Symposium, Canberra, Australia, February 1-2, 2000. Electronic Notes in Theoretical Computer Science, volume 31, 24 pages, 2000.
dvi (24 pages)

20. J.R. Kennaway, V. van Oostrom, F.J. de Vries.
Meaningless terms in rewriting.
Journal of Logic and Functional Programming. The MIT Press. Article 1, 35 pages. Volume 1999.
dvi, ps and pdf (35 pages)

Abstract We present an axiomatic approach to the concept of meaninglessness in finite and transfinite term rewriting and lambda calculus. We justify our axioms in several ways. They can be intuitively justified from the viewpoint of rewriting as computation. They are shown to imply important properties of meaninglessness: genericity of the class of meaningless terms, confluence modulo equality of meaningless terms, the consistency of equating all meaningless terms, and the construction of Böhm trees and models of rewrite systems. Finally, we show that they can be easily verified for many existing notions of meaninglessness, and easily refuted for some notions that are known not to be good characterisations of meaninglessness.
21. F. Barbanera, M. Dezani-Ciancaglini, F.J. de Vries.
Types for trees.
In Proceedings IFIP Working Conference on Programming Concepts and Methods (PROCOMET '98), Shelter Island, New York, Chapman and Hall, pages 6-29, 1998.

22. J.R. Kennaway, J.W. Klop, M.R. Sleep, F.J. de Vries.
Infinitary lambda calculus.
Theoretical Computer Science 175(1):93-125, 1997.
dvi, ps and pdf (38 pages)

Abstract In a previous paper we have established the theory of transfinite reduction for orthogonal term rewriting systems. In this paper we perform the same task for the lambda calculus. From the viewpoint of infinitary rewriting, the Böhm model of the lambda calculus can be seen as an infinitary term model. In contrast to term rewriting, there are several different possible notions of infinite term, which give rise to different Böhm-like models, which embody different notions of lazy or eager computation.
23. F.J. de Vries.
Böhm trees, bisimulations and observations in lambda calculus.
In Proceedings of the Second Fuji International Workshop on Functional and Logic Programming Workshop Shonan Village Center, Japan 1-4 November 1996. Editors: T. Ida, A. Ohori and M. Takeichi. World Scientific, Singapore, pages 230-245, 1997.

24. J.R. Kennaway, J.W. Klop, M.R. Sleep and F.J. de Vries.
Comparing curried and uncurried rewrite systems.
Journal of Symbolic Computation, 21(1):15-39, 1996.
dvi, ps and pdf (25 pages)

25. J.R. Kennaway, V. van Oostrom, F.J. de Vries.
Meaningless terms in rewriting.
In Proceedings of the Fifth International Conference on Algebraic and Logic Programming Aachen (Germany), September 25-27, 1996. Series: Lecture Notes in Computer Science 1139, Springer-Verlag, pages 254-268, 1996.

26. J.R. Kennaway, J.W. Klop, M.R. Sleep, F.J. de Vries.
Infinite lambda calculus and Böhm models.
In Proceedings Rewriting Techniques and Applications, Kaiserslautern, 1995. Series: Lecture Notes in Computer Science 914, Springer-Verlag, pages 257-270, 1995.

27. J.R. Kennaway, J.W. Klop, M.R. Sleep, F.J. de Vries.
From Finite to Infinite Lambda Calculi.
Bulletin of the section of Logic, University of Lodz, Department of Logic (Special issue dedicated to the Workshop on Non-standard Logics and Logical Aspects of Computer Science, Kanazawa, Japan, Dec. 5-8. 1994, editor Hiroakira Ono), 24(1)13-20, 1995.
pdf (8 pages)

Abstract In a previous paper we have established the theory of transfinite reduction for orthogonal term rewriting systems. In this paper we perform the same task for the lambda calculus. This results in several new Bohm models of the lambda calculus and new unifying descriptions of existing models.
28. D.J.N van Eijck and F.J. de Vries.
Journal of Philosophical Logic, 24(1):19-47, 1995.
dvi, ps and pdf (19 pages)

29. E. Horita and F.J. de Vries.
A fully abstract denotational model for communicating processes with label-passing.

In Proceedings of Concurrency Theory and Its Application, RIMS Kokyuroku 902, page 26-48, 1995.

30. J.R. Kennaway, J.W. Klop, M.R. Sleep and F.J. de Vries.
Transfinite Reductions in Orthogonal Term Rewriting Systems.
Information and Computation, 119(1):18-38, 1995.
ps and pdf (36 pages)

Abstract We define the notion of transfinite term rewriting: rewriting in which terms may be infinitely large and rewrite sequences may be of any ordinal length. For orthogonal rewrite systems, some fundamental properties known in the finite case are extended to the transfinite case. Among these are the Parallel Moves lemma and the Unique Normal Form property. The transfinite Church-Rosser property fails in general, even for orthogonal systems, including such well-known systems as Combinatory Logic. Syntactic characterisations are given of some classes of orthogonal TRSs which do satisfy the transfinite Church-Rosser property. We also prove a weakening of the transfinite Church-Rosser property for all orthogonal systems, in which the property is only required to hold up to a certain equivalence relation on terms. Finally, we extend the theory of needed reduction from the finite to the transfinite case. The reduction strategy of needed reduction is normalising in the finite case, but not in the transfinite case. To obtain a normalising strategy, it is necessary and sufficient to add a requirement of fairness. Parallel outermost reduction is such a strategy.
31. Z. Ariola, J.R. Kennaway, J.W. Klop, M.R. Sleep and F.J. de Vries.
On defining the undefined.
In the proceedings: TACS'94 (Theoretical Aspects of Computer Software) International conference in Sendai. Series: Lecture Notes in Computer Science 789, Springer-Verlag, pages 543-554, 1994.

32. J.R. Kennaway, J.W. Klop, M.R. Sleep and F.J. de Vries.
On the Adequacy of Graph Rewriting for Simulating Term Rewriting.
Transactions on Programming Languages and Systems, 16(3):493-523, 1994.
ps (29 pages), pdf

33. F.J. de Vries and J. Yamada.
On termination of rewriting with real numbers.
In proceedings: Functional Programming II, JSSST'94. Editor Masato Takeichi. Series: Lecture Notes on Software Gaku 10. Publisher: Kindai-kagaku-sya, Tokyo. Pages 233-247, 1994.
ps (18 pages), pdf (18 pages) and Addendum (Oct 13, 2004)

34. J.R. Kennaway, J.W. Klop, M.R. Sleep and F.J. de Vries.
An Introduction to Term Graph Rewriting.
Chapter 1 in M.R. Sleep, M.J. Plasmeijer and M.C. van Eekelen, editors, Term Graph Rewriting: Theory and Practice, John Wiley ™Sons Ltd, pages 1-13, 1993.

35. J.R. Kennaway, J.W. Klop, M.R. Sleep and F.J. de Vries.
An infinitary Church-Rosser property for non-collapsing orthogonal term rewriting systems.
Chapter 4 in M.R. Sleep, M.J. Plasmeijer and M.C. van Eekelen, editors, Term Graph Rewriting: Theory and Practice, John Wiley ™ Sons Ltd, pages 47-59, 1993.

36. J.R. Kennaway, J.W. Klop, M.R. Sleep and F.J. de Vries.
Event Structures and Orthogonal Term Graph Rewriting.
Chapter 11 in M.R. Sleep, M.J. Plasmeijer and M.C. van Eekelen, editors, Term Graph Rewriting: Theory and Practice, John Wiley ™Sons Ltd, pages 141-155, 1993.

37. J.R. Kennaway, J.W. Klop, M.R. Sleep and F.J. de Vries.
On the adequacy of graph rewriting for simulating term rewriting.
Chapter 12 in M.R. Sleep, M.J. Plasmeijer and M.C. van Eekelen, editors, Term Graph Rewriting: Theory and Practice, John Wiley ™ Sons Ltd, pages 157-169, 1993.

38. D.J.N van Eijck and F.J. de Vries.
Dynamic interpretation and Hoare deduction.
Journal of Logic, Language and Information, 1(1):1-44, 1992.
dvi, ps and pdf (43 pages)

39. D.J.N van Eijck and F.J. de Vries.
Dynamic interpretation and Hoare deduction (Extended Abstract).
In S. Moore and A.Z. Wyner, editors, Proceedings of Semantics and Linguistic Theory, SALT I, CLC Publications, Cornell University, Ithaca, N.Y., 10:65-85, 1991.

40. J.R. Kennaway, J.W. Klop, M.R. Sleep and F.J. de Vries.
Transfinite reductions in orthogonal term rewriting systems.
In R.V. Book, editor, Proceedings of Rewriting Techniques and Applications, Como, Lecture Notes in computer Science 488, pages 1-12, 1991.
pdf (12 pages)

41. F.J. de Vries.
Type theoretical topics in topos theory.
PhD Thesis, University of Utrecht, April 13, 1989. Supervisor Prof D. van Dalen.

42. D. van Dalen and F.J. de Vries.
Intuitionistic Free Abelian Groups.
Zeitschrift für Mathematischen Logik und Grundlagen der Mathematik, 34(1):3-12, 1988.

43. F.J. de Vries.
A functional program for the Fast Fourier Transform.
SigmaPlan Notices, 23(1):67-74, 1988.

#### Papers in Unrefereed Proceedings

1. J. Giesl, J.R, V. van Oostrom, F.J. de Vries. Strong convergence of term rewriting using strong dependency pairs. Extended abstract. In Proceedings of Termination Workshop, Schloss Dagstuhl, 1999.
pdf (2 pages plus erratum)

2. F.J. de Vries. Projection spaces and recursive domain equations. Information Processing Society of Japan SIG Notes. 95(114):37-38, 1995 .

3. J.R. Kennaway, J.W. Klop, M.R. Sleep and F.J. de Vries. From finite lambda calculus to infinite lambda calculi (abstract) Information Processing Society of Japan, SIG Notes 94-PRG-19-6:43-50, 1994.

4. J.R. Kennaway, J.W. Klop, M.R. Sleep and F.J. de Vries. On comparing curried and uncurried rewrite systems. In H.P. Barendregt, M. Bezem and J.W. Klop, editors, Festschrift dedicated to the sixtieth anniversary of Dirk van Dalen, Quaestiones Infinitae, Logic Series, Department of Philosophy, University of Utrecht, pages 57-78, 1993.

#### Reports

1. P. Severi and F.J. de Vries.
A lambda calculus for D-infinity.
Presented at the Workshop at Domain Theory held at the honour of Dana Scott's 70th birthday.
Technical Report tr-2002-29, University of Leicester, June 2002.
dvi (10 pages), ps (10 pages)

Abstract We define an extension of lambda calculus which is fully abstract for Scott's D-infinity models. We do so by constructing an infinitary lambda calculus which not only has the confluence property, but also is normalising: every term has its inf-eta-Böhm tree as unique normal form. The extension incorporates a strengthened form of eta-reduction besides infinite terms, infinite reductions and a bottom rule allowing to replace terms without head normal form by bottom. The new eta!-reduction is the key idea of this paper. It allows us to capture in a compact and natural way Barendregt's complex infinite eta-operation on Böhm trees. As a corollary we obtain a new congruence proof for Böhm tree equivalence modulo infinite eta-expansion.
2. P. Severi and F.J. de Vries.
An Extensional Böhm Model.
In the Proceedings of RTA'02, Springer Lecture Notes in Computer Science 2378.
Technical Report tr-2002-28, University of Leicester, June 2002.
dvi (15 pages), ps (15 pages)

Abstract We show the existence of an infinitary confluent and normalising extension of the finite extensional lambda calculus with beta and eta. Besides infinite beta reductions also infinite eta reductions are possible in this extension, and terms without head normal form can be reduced to bottom. As corollaries we obtain a simple, syntax based construction of an extensional Böhm model of the finite lambda calculus; and a simple, syntax based proof that two lambda terms have the same semantics in this model if and only if they have the same eta-Böhm tree if and only if they are observationally equivalent wrt to beta normal forms. The confluence proof reduces confluence of beta, bottom and eta via infinitary commutation and postponement arguments to confluence of beta and bottom and confluence of eta. We give counterexamples against confluence of similar extensions based on the identification of the terms without weak head normal form and the terms without top normal form (rootactive terms) respectively.
3. S. Byun, J.R. Kennaway, V. van Oostrom, F.J. de Vries.
Separability and translatability of sequential term rewrite systems into the lambda calculus.
Technical Report tr-2001-16, University of Leicester, April 2001.
dvi (35 pages), ps (35 pages)

Abstract Orthogonal term rewrite systems do not currently have any semantics other than syntactically-based ones such as term models and event structures. For a functional language which combines lambda calculus with term rewriting, a semantics is most easily given by translating the rewrite rules into lambda calculus and then using well-understood semantics for the lambda calculus. We therefore study in this paper the question of which classes of TRSs do or do not have such translations. We demonstrate by construction that forward branching orthogonal term rewrite systems are translatable into the lambda calculus. The translation satisfies some strong properties concerning preservation of equality and of some inequalities. We prove that the forward branching systems are exactly the systems permitting such a translation which is, in a precise sense, uniform in the right-hand sides. Connections are drawn between translatability, sequentiality and separability properties. Simple syntactic proofs are given of the non-translatability of a class of TRSs, including Berry's F and several variants of it.
4. J.R. Kennaway and F.J. de Vries.
Infinitary Rewriting.
Technical Report tr-2001-13, University of Leicester, March 2001.
dvi (48 pages), ps (48 pages)

Abstract In this chapter we will give the basic definitions and properties of infinite terms and infinite reduction sequences, for both term rewrite systems and the lambda calculus. We will then study confluence properties in orthogonal systems, which turns out to be significantly more complicated than in the finitary case. In general, these systems are only confluent up to an identification of a certain class of terms. The breakdown of confluence leads us to consider the concept of a meaningless term, which further suggests a link with the lambda-calculus concept of Böhm reduction, and to denotational semantics for TRSs.
5. M. Dezani-Ciancaglini, P. Severi and F.J. de Vries.
Infinitary Lambda Calculus and Discrimination.
Technical Report tr-2001-06, University of Leicester, February 2001.
dvi (33 pages), ps (33 pages)

Abstract We propose an extension of lambda calculus for which the Berarducci trees equality coincides with observational equivalence, when we observe rootstable or rootactive behavior of terms. In one direction the proof is an adaptation of the classical Böhm out technique. In the other direction the proof is based on confluence for strongly converging reductions in this extension.
6. J.R. Kennaway, V. van Oostrom, F.J. de Vries. Meaningless terms in rewriting, revised version. Technical Report: Utrecht Universiteit, Artificial Intelligence Preprint Series No: 003, May, 1999.

7. F. Barbanera, M. Dezani-Ciancaglini, F.J. de Vries. Types for Trees. Technical Report: Dipartimento di Unformatica, Universita' di Torino, 1997.

8. J.R. Kennaway, V. van Oostrom, F.J. de Vries. Meaningless terms in rewriting Technical Report: Vrije Universiteit, Amsterdam, IR-418, 23 pages, January, 1997.

9. J.R. Kennaway, J.W. Klop, M.R. Sleep, F.J. de Vries. Infinite lambda calculus, Report CS-R9535, CWI, Amsterdam, 1995.

10. E. Horita and F.J. de Vries. A fully abstract denotational model for communicating processes with label-passing. ECL Technical Report. NTT Kyoto, 1994.

11. J.R. Kennaway, J.W. Klop, M.R. Sleep, F.J. de Vries. On comparing curried and uncurried rewrite systems. Revised version of Festschrift version. Report CS-R9350, CWI, Amsterdam, 1993.

12. D.J.N. van Eijck and F.J. de Vries. Reasoning about Update Logic, (Completely revised version of report CS-R9155), Report CS-R9312, CWI, Amsterdam, 1993.

13. Kennaway J.R., Klop J.W., Sleep M.R., Vries F.-J. de, Transfinite reductions in orthogonal term rewriting systems, Report SYS-C93-10, School of Information Systems, Univ. of East Anglia, Norwich, England. Revised version of 17.

14. J.R. Kennaway, J.W. Klop, M.R. Sleep, F.J. de Vries. On the Adequacy of Graph Rewriting for Simulating Term Rewriting, Report CS9204, CWI, Amsterdam, 1992. Similar as 10.

15. J.R. Kennaway, J.W. Klop, M.R. Sleep, F.J. de Vries. On the Adequacy of Graph Rewriting for Simulating Term Rewriting, Report IR-287, Vrije Universiteit Amsterdam.

16. D.J.N. van Eijck and F.J. de Vries. A sound and complete calculus for Update Logic, report CS-R9155, CWI, Amsterdam, 1991.

17. J.R. Kennaway, J.W. Klop, M.R. Sleep, F.J. de Vries. Event structures and orthogonal term graph rewriting, in M.J. Plasmeijer and M.R. Sleep, editors, Sema Graph '91, Part II, Technical Report 91-25, Department of Informatics, University of Nijmegen, 1991.

18. J.R. Kennaway, J.W. Klop, M.R. Sleep, F.J. de Vries. Finite orthogonal graph rewriting is adequate for rational orthogonal term rewriting, in M.J. Plasmeijer and M.R. Sleep, editors, Sema Graph '91, Part I, Technical Report 91-25, Department of Informatics, University of Nijmegen, 1991.

19. D.J.N. van Eijck and F.J. de Vries. Dynamic interpretation and Hoare deduction, report CS-R9115, CWI, Amsterdam, 1991.

20. J.R. Kennaway, J.W. Klop, M.R. Sleep, F.J. de Vries. An infinitary Church-Rosser property for non-collapsing orthogonal term rewriting systems, report CS-R9043, CWI, Amsterdam, 1990. pdf

21. J.R. Kennaway, J.W. Klop, M.R. Sleep, F.J. de Vries. Transfinite reductions in orthogonal term rewriting systems (Extended abstract), report CS-R9042, CWI, Amsterdam, 1990. pdf (somewhat corrupted font)

22. J.R. Kennaway, J.W. Klop, M.R. Sleep, F.J. de Vries. Transfinite reductions in orthogonal term rewriting systems (Full version) , report CS-R9041, CWI, Amsterdam, 1990. .
pdf (47 pages)

23. A. Ponse and F.J. de Vries. Strong completeness for Hoare Logics of Recursive Processes: an infinitary approach, report CS-R8957, CWI, Amsterdam, 1989.

24. F.J. de Vries. Applications of constructive logic to sheaf constructions in toposes, Logic Group Preprint Series No. 25, Department of Philosophy, University of Utrecht, October 1987.

25. F.J. de Vries. A functional program for Gaussian elimination, Logic Group Preprint Series No. 23, Department of Philosophy, University of Utrecht. September 1987.

26. F.J. de Vries. A functional program for the Fast Fourier Transform, Logic Group Preprint Series No. 19, Department of Philosophy, University of Utrecht, April 1987.

27. D. van Dalen and F.J. de Vries. Intuitionistic Free Abelian Groups, Logic Group Preprint Series No. 7, Department of Philosophy, University of Utrecht, April 1986.

28. F.J. de Vries. Type theory with a geometric modality and extensions of the reals, Preprint 340, Mathematical Institute, University of Utrecht, June 1984.

#### Book reviews

1. Review of Algebra in a localic topos, with applications to ring theory by F.Borceux and G. van den Bossche, Nov 1985, Mededelingen van het Wiskundig Genootschap.

2. Review of Antimorphic action by W.H. Cornish, 1988, Mededelingen van het Wiskundig Genootschap.

3. Review of Graph-grammars and their application to computer science by H.Ehrig, e.a. (eds), 1990, Informatie.

4. Review of Cartesian closed categories of domains by A. Jung, 1990, Mededelingen van het Wiskundig Genootschap.

 Author: F.J. de Vries (fdv1 at mcs.le.ac.uk).© University of Leicester 6 August 2003 . Last modified: 1st July 2016, 12:49:28.Informatics Web Maintainer. Any opinions expressed on this page are those of the author.