9 Search Results for "Heijltjes, Willem"

Document
DOI: 10.4230/LIPIcs.CSL.2026.28
Resourceful Traces for Commuting Processes

Authors: Matthew Earnshaw, Chad Nester, and Mario Román

Published in: LIPIcs, Volume 363, 34th EACSL Annual Conference on Computer Science Logic (CSL 2026)

Abstract
We show that, when the actions of a Mazurkiewicz trace are considered not merely as atomic but as transformations from a specified type of inputs to a specified type of outputs, we obtain a novel notion of presentation for effectful categories (also known as generalized Freyd categories), a well-known algebraic structure in the semantics of side-effecting computation. Like the usual representation of traces as graphs, our notion of presentation gives rise to a graphical representation of morphisms in effectful categories. We use our presentations to give a construction of the commuting tensor product of free effectful categories, capturing the combination of systems in which the actions of each must commute with one another, while still permitting exchange of resources.
Cite as

Matthew Earnshaw, Chad Nester, and Mario Román. Resourceful Traces for Commuting Processes. In 34th EACSL Annual Conference on Computer Science Logic (CSL 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 363, pp. 28:1-28:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

Copy BibTex To Clipboard

@InProceedings{earnshaw_et_al:LIPIcs.CSL.2026.28,
  author =	{Earnshaw, Matthew and Nester, Chad and Rom\'{a}n, Mario},
  title =	{{Resourceful Traces for Commuting Processes}},
  booktitle =	{34th EACSL Annual Conference on Computer Science Logic (CSL 2026)},
  pages =	{28:1--28:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-411-6},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{363},
  editor =	{Guerrini, Stefano and K\"{o}nig, Barbara},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2026.28},
  URN =		{urn:nbn:de:0030-drops-254522},
  doi =		{10.4230/LIPIcs.CSL.2026.28},
  annote =	{Keywords: Mazurkiewicz traces, premonoidal categories, monoidal categories, effectful categories}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2025.24
Quantitative Types for the Functional Machine Calculus

Authors: Willem Heijltjes

Published in: LIPIcs, Volume 337, 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)

Abstract
The Functional Machine Calculus (FMC, Heijltjes 2022) extends the lambda-calculus with the computational effects of global mutable store, input/output, and probabilistic choice while maintaining confluent reduction and simply-typed strong normalization. Based in a simple call-by-name stack machine in the style of Krivine, the FMC models effects through additional argument stacks, and introduces sequential composition through a continuation stack to encode call-by-value behaviour, where simple types guarantee termination of the machine. The present paper provides a discipline of quantitative types, also known as non-idempotent intersection types, for the FMC, in two variants. In the weak variant, typeability coincides with termination of the stack machine and with spine normalization, while exactly measuring the transitions in machine evaluation. The strong variant characterizes strong normalization through a notion of perpetual evaluation, while giving an upper bound to the length of reductions. Through the encoding of effects, quantitative typeability coincides with termination for higher-order mutable store, input/output, and probabilistic choice.
Cite as

Willem Heijltjes. Quantitative Types for the Functional Machine Calculus. In 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 337, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

Copy BibTex To Clipboard

@InProceedings{heijltjes:LIPIcs.FSCD.2025.24,
  author =	{Heijltjes, Willem},
  title =	{{Quantitative Types for the Functional Machine Calculus}},
  booktitle =	{10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)},
  pages =	{24:1--24:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-374-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{337},
  editor =	{Fern\'{a}ndez, Maribel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2025.24},
  URN =		{urn:nbn:de:0030-drops-236391},
  doi =		{10.4230/LIPIcs.FSCD.2025.24},
  annote =	{Keywords: lambda-calculus, computational effects, intersection types}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2025.5
The Cost of Skeletal Call-By-Need, Smoothly

Authors: Beniamino Accattoli, Francesco Magliocca, Loïc Peyrot, and Claudio Sacerdoti Coen

Published in: LIPIcs, Volume 337, 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)

Abstract
Skeletal call-by-need is an optimization of call-by-need evaluation also known as "fully lazy sharing": when the duplication of a value has to take place, it is first split into "skeleton", which is then duplicated, and "flesh" which is instead kept shared. Here, we provide two cost analyses of skeletal call-by-need. Firstly, we provide a family of terms showing that skeletal call-by-need can be asymptotically exponentially faster than call-by-need in both time and space; it is the first such evidence, to our knowledge. Secondly, we prove that skeletal call-by-need can be implemented efficiently, that is, with bi-linear overhead. This result is obtained by providing a new smooth presentation of ideas by Shivers and Wand for the reconstruction of skeletons, which is then smoothly plugged into the study of an abstract machine following the distillation technique by Accattoli et al.
Cite as

Beniamino Accattoli, Francesco Magliocca, Loïc Peyrot, and Claudio Sacerdoti Coen. The Cost of Skeletal Call-By-Need, Smoothly. In 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 337, pp. 5:1-5:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

Copy BibTex To Clipboard

@InProceedings{accattoli_et_al:LIPIcs.FSCD.2025.5,
  author =	{Accattoli, Beniamino and Magliocca, Francesco and Peyrot, Lo\"{i}c and Sacerdoti Coen, Claudio},
  title =	{{The Cost of Skeletal Call-By-Need, Smoothly}},
  booktitle =	{10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)},
  pages =	{5:1--5:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-374-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{337},
  editor =	{Fern\'{a}ndez, Maribel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2025.5},
  URN =		{urn:nbn:de:0030-drops-236206},
  doi =		{10.4230/LIPIcs.FSCD.2025.5},
  annote =	{Keywords: \lambda-calculus, abstract machines, call-by-need, cost models}
}
Document
DOI: 10.4230/LIPIcs.CSL.2025.31
Simple Types for Probabilistic Termination

Authors: Willem Heijltjes and Georgina Majury

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract
We present a new typing discipline to guarantee the probability of termination in probabilistic lambda-calculi. The main contribution is a particular naturality and simplicity: our probabilistic types are as simple types, but generated from probabilities as base types, representing a least probability of termination. Simple types are recovered by restricting probabilities to one. Our vehicle is the Probabilistic Event Lambda-Calculus by Dal Lago, Guerrieri, and Heijltjes, which presents a solution to the issue of confluence in probabilistic lambda-calculi. Our probabilistic type system provides an alternative solution to that using counting quantifiers by Antonelli, Dal Lago, and Pistone, for the same calculus. The problem that both type systems address is to give a lower bound on the probability that terms head-normalize. Following the recent Functional Machine Calculus by Heijltjes, our development takes the (simplified) Krivine machine as primary, and proceeds via an extension of the calculus with sequential composition and identity on the machine. Our type system then gives a natural account of termination probability on the Krivine machine, reflected back onto head-normalization for the original calculus. In this way we are able to avoid the use of counting quantifiers, while improving on the termination bounds given by Antonelli, Dal Lago, and Pistone.
Cite as

Willem Heijltjes and Georgina Majury. Simple Types for Probabilistic Termination. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 31:1-31:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

Copy BibTex To Clipboard

@InProceedings{heijltjes_et_al:LIPIcs.CSL.2025.31,
  author =	{Heijltjes, Willem and Majury, Georgina},
  title =	{{Simple Types for Probabilistic Termination}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{31:1--31:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.31},
  URN =		{urn:nbn:de:0030-drops-227885},
  doi =		{10.4230/LIPIcs.CSL.2025.31},
  annote =	{Keywords: lambda-calculus, probabilistic termination, simple types}
}
Document
DOI: 10.4230/LIPIcs.CSL.2023.10
The Functional Machine Calculus II: Semantics

Authors: Chris Barrett, Willem Heijltjes, and Guy McCusker

Published in: LIPIcs, Volume 252, 31st EACSL Annual Conference on Computer Science Logic (CSL 2023)

Abstract
The Functional Machine Calculus (FMC), recently introduced by the second author, is a generalization of the lambda-calculus which may faithfully encode the effects of higher-order mutable store, I/O and probabilistic/non-deterministic input. Significantly, it remains confluent and can be simply typed in the presence of these effects. In this paper, we explore the denotational semantics of the FMC. We have three main contributions: first, we argue that its syntax - in which both effects and lambda-calculus are realised using the same syntactic constructs - is semantically natural, corresponding closely to the structure of a Scott-style domain theoretic semantics. Second, we show that simple types confer strong normalization by extending Gandy’s proof for the lambda-calculus, including a small simplification of the technique. Finally, we show that the typed FMC (without considering the specifics of encoded effects), modulo an appropriate equational theory, is a complete language for Cartesian closed categories.
Cite as

Chris Barrett, Willem Heijltjes, and Guy McCusker. The Functional Machine Calculus II: Semantics. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 10:1-10:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{barrett_et_al:LIPIcs.CSL.2023.10,
  author =	{Barrett, Chris and Heijltjes, Willem and McCusker, Guy},
  title =	{{The Functional Machine Calculus II: Semantics}},
  booktitle =	{31st EACSL Annual Conference on Computer Science Logic (CSL 2023)},
  pages =	{10:1--10:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-264-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{252},
  editor =	{Klin, Bartek and Pimentel, Elaine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2023.10},
  URN =		{urn:nbn:de:0030-drops-174716},
  doi =		{10.4230/LIPIcs.CSL.2023.10},
  annote =	{Keywords: lambda-calculus, computational effects, denotational semantics, strong normalization}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2022.19
Normalization Without Syntax

Authors: Willem B. Heijltjes, Dominic J. D. Hughes, and Lutz Straßburger

Published in: LIPIcs, Volume 228, 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)

Abstract
We present normalization for intuitionistic combinatorial proofs (ICPs) and relate it to the simply-typed lambda-calculus. We prove confluence and strong normalization. Combinatorial proofs, or "proofs without syntax", form a graphical semantics of proof in various logics that is canonical yet complexity-aware: they are a polynomial-sized representation of sequent proofs that factors out exactly the non-duplicating permutations. Our approach to normalization aligns with these characteristics: it is canonical (free of permutations) and generic (readily applied to other logics). Our reduction mechanism is a canonical representation of reduction in sequent calculus with closed cuts (no abstraction is allowed below a cut), and relates to closed reduction in lambda-calculus and supercombinators. While we will use ICPs concretely, the notion of reduction is completely abstract, and can be specialized to give a reduction mechanism for any representation of typed normal forms.
Cite as

Willem B. Heijltjes, Dominic J. D. Hughes, and Lutz Straßburger. Normalization Without Syntax. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 19:1-19:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{heijltjes_et_al:LIPIcs.FSCD.2022.19,
  author =	{Heijltjes, Willem B. and Hughes, Dominic J. D. and Stra{\ss}burger, Lutz},
  title =	{{Normalization Without Syntax}},
  booktitle =	{7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)},
  pages =	{19:1--19:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-233-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{228},
  editor =	{Felty, Amy P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2022.19},
  URN =		{urn:nbn:de:0030-drops-163004},
  doi =		{10.4230/LIPIcs.FSCD.2022.19},
  annote =	{Keywords: combinatorial proofs, intuitionistic logic, lambda-calculus, Curry-Howard, proof nets}
}
Document
DOI: 10.4230/LIPIcs.CSL.2021.24
A Deep Quantitative Type System

Authors: Giulio Guerrieri, Willem B. Heijltjes, and Joseph W.N. Paulus

Published in: LIPIcs, Volume 183, 29th EACSL Annual Conference on Computer Science Logic (CSL 2021)

Abstract
We investigate intersection types and resource lambda-calculus in deep-inference proof theory. We give a unified type system that is parametric in various aspects: it encompasses resource calculi, intersection-typed lambda-calculus, and simply-typed lambda-calculus; it accommodates both idempotence and non-idempotence; it characterizes strong and weak normalization; and it does so while allowing a range of algebraic laws to determine reduction behaviour, for various quantitative effects. We give a parametric resource calculus with explicit sharing, the "collection calculus", as a Curry-Howard interpretation of the type system, that embodies these computational properties.
Cite as

Giulio Guerrieri, Willem B. Heijltjes, and Joseph W.N. Paulus. A Deep Quantitative Type System. In 29th EACSL Annual Conference on Computer Science Logic (CSL 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 183, pp. 24:1-24:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{guerrieri_et_al:LIPIcs.CSL.2021.24,
  author =	{Guerrieri, Giulio and Heijltjes, Willem B. and Paulus, Joseph W.N.},
  title =	{{A Deep Quantitative Type System}},
  booktitle =	{29th EACSL Annual Conference on Computer Science Logic (CSL 2021)},
  pages =	{24:1--24:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-175-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{183},
  editor =	{Baier, Christel and Goubault-Larrecq, Jean},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2021.24},
  URN =		{urn:nbn:de:0030-drops-134586},
  doi =		{10.4230/LIPIcs.CSL.2021.24},
  annote =	{Keywords: Lambda-calculus, Deep inference, Intersection types, Resource calculus}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2019.22
Proof Nets for First-Order Additive Linear Logic

Authors: Willem B. Heijltjes, Dominic J. D. Hughes, and Lutz Straßburger

Published in: LIPIcs, Volume 131, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)

Abstract
We present canonical proof nets for first-order additive linear logic, the fragment of linear logic with sum, product, and first-order universal and existential quantification. We present two versions of our proof nets. One, witness nets, retains explicit witnessing information to existential quantification. For the other, unification nets, this information is absent but can be reconstructed through unification. Unification nets embody a central contribution of the paper: first-order witness information can be left implicit, and reconstructed as needed. Witness nets are canonical for first-order additive sequent calculus. Unification nets in addition factor out any inessential choice for existential witnesses. Both notions of proof net are defined through coalescence, an additive counterpart to multiplicative contractibility, and for witness nets an additional geometric correctness criterion is provided. Both capture sequent calculus cut-elimination as a one-step global composition operation.
Cite as

Willem B. Heijltjes, Dominic J. D. Hughes, and Lutz Straßburger. Proof Nets for First-Order Additive Linear Logic. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 22:1-22:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{heijltjes_et_al:LIPIcs.FSCD.2019.22,
  author =	{Heijltjes, Willem B. and Hughes, Dominic J. D. and Stra{\ss}burger, Lutz},
  title =	{{Proof Nets for First-Order Additive Linear Logic}},
  booktitle =	{4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)},
  pages =	{22:1--22:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-107-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{131},
  editor =	{Geuvers, Herman},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.22},
  URN =		{urn:nbn:de:0030-drops-105297},
  doi =		{10.4230/LIPIcs.FSCD.2019.22},
  annote =	{Keywords: linear logic, first-order logic, proof nets, Herbrand’s theorem}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2018.10
Proof Nets for Bi-Intuitionistic Linear Logic

Authors: Gianluigi Bellin and Willem B. Heijltjes

Published in: LIPIcs, Volume 108, 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)

Abstract
Bi-Intuitionistic Linear Logic (BILL) is an extension of Intuitionistic Linear Logic with a par, dual to the tensor, and subtraction, dual to linear implication. It is the logic of categories with a monoidal closed and a monoidal co-closed structure that are related by linear distributivity, a strength of the tensor over the par. It conservatively extends Full Intuitionistic Linear Logic (FILL), which includes only the par. We give proof nets for the multiplicative, unit-free fragment MBILL-. Correctness is by local rewriting in the style of Danos contractibility, which yields sequentialization into a relational sequent calculus extending the existing one for FILL. We give a second, geometric correctness condition combining Danos-Regnier switching and Lamarche's Essential Net criterion, and demonstrate composition both inductively and as a one-off global operation.
Cite as

Gianluigi Bellin and Willem B. Heijltjes. Proof Nets for Bi-Intuitionistic Linear Logic. In 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 108, pp. 10:1-10:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

Copy BibTex To Clipboard

@InProceedings{bellin_et_al:LIPIcs.FSCD.2018.10,
  author =	{Bellin, Gianluigi and Heijltjes, Willem B.},
  title =	{{Proof Nets for Bi-Intuitionistic Linear Logic}},
  booktitle =	{3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)},
  pages =	{10:1--10:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-077-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{108},
  editor =	{Kirchner, H\'{e}l\`{e}ne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2018.10},
  URN =		{urn:nbn:de:0030-drops-91800},
  doi =		{10.4230/LIPIcs.FSCD.2018.10},
  annote =	{Keywords: proof nets, intuitionistic linear logic, contractibility, linear logic}
}
  • Refine by Type
  • 9 Document/PDF
  • 3 Document/HTML

  • Refine by Publication Year
  • 1 2026
  • 3 2025
  • 1 2023
  • 1 2022
  • 1 2021
  • Show More...

  • Refine by Author
  • 4 Heijltjes, Willem B.
  • 3 Heijltjes, Willem
  • 2 Hughes, Dominic J. D.
  • 2 Straßburger, Lutz
  • 1 Accattoli, Beniamino
  • Show More...

  • Refine by Series/Journal
  • 9 LIPIcs

  • Refine by Classification
  • 5 Theory of computation → Lambda calculus
  • 4 Theory of computation → Proof theory
  • 2 Theory of computation → Linear logic
  • 1 Theory of computation
  • 1 Theory of computation → Categorical semantics
  • Show More...

  • Refine by Keyword
  • 4 lambda-calculus
  • 3 proof nets
  • 2 computational effects
  • 2 linear logic
  • 1 Curry-Howard
  • Show More...

Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact