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**Published in:** LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

We study pseudo-deterministic query complexity - randomized query algorithms that are required to output the same answer with high probability on all inputs. We prove Ω(√n) lower bounds on the pseudo-deterministic complexity of a large family of search problems based on unsatisfiable random CNF instances, and also for the promise problem (FIND1) of finding a 1 in a vector populated with at least half one’s. This gives an exponential separation between randomized query complexity and pseudo-deterministic complexity, which is tight in the quantum setting. As applications we partially solve a related combinatorial coloring problem, and we separate random tree-like Resolution from its pseudo-deterministic version. In contrast to our lower bound, we show, surprisingly, that in the zero-error, average case setting, the three notions (deterministic, randomized, pseudo-deterministic) collapse.

Shafi Goldwasser, Russell Impagliazzo, Toniann Pitassi, and Rahul Santhanam. On the Pseudo-Deterministic Query Complexity of NP Search Problems. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 36:1-36:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{goldwasser_et_al:LIPIcs.CCC.2021.36, author = {Goldwasser, Shafi and Impagliazzo, Russell and Pitassi, Toniann and Santhanam, Rahul}, title = {{On the Pseudo-Deterministic Query Complexity of NP Search Problems}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {36:1--36:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-193-1}, ISSN = {1868-8969}, year = {2021}, volume = {200}, editor = {Kabanets, Valentine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.36}, URN = {urn:nbn:de:0030-drops-143104}, doi = {10.4230/LIPIcs.CCC.2021.36}, annote = {Keywords: Pseudo-determinism, Query complexity, Proof complexity} }

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**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

We consider the following question: using a source of labeled data and interaction with an untrusted prover, what is the complexity of verifying that a given hypothesis is "approximately correct"? We study interactive proof systems for PAC verification, where a verifier that interacts with a prover is required to accept good hypotheses, and reject bad hypotheses. Both the verifier and the prover are efficient and have access to labeled data samples from an unknown distribution. We are interested in cases where the verifier can use significantly less data than is required for (agnostic) PAC learning, or use a substantially cheaper data source (e.g., using only random samples for verification, even though learning requires membership queries). We believe that today, when data and data-driven algorithms are quickly gaining prominence, the question of verifying purported outcomes of data analyses is very well-motivated.
We show three main results. First, we prove that for a specific hypothesis class, verification is significantly cheaper than learning in terms of sample complexity, even if the verifier engages with the prover only in a single-round (NP-like) protocol. Moreover, for this class we prove that single-round verification is also significantly cheaper than testing closeness to the class. Second, for the broad class of Fourier-sparse boolean functions, we show a multi-round (IP-like) verification protocol, where the prover uses membership queries, and the verifier is able to assess the result while only using random samples. Third, we show that verification is not always more efficient. Namely, we show a class of functions where verification requires as many samples as learning does, up to a logarithmic factor.

Shafi Goldwasser, Guy N. Rothblum, Jonathan Shafer, and Amir Yehudayoff. Interactive Proofs for Verifying Machine Learning. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 41:1-41:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{goldwasser_et_al:LIPIcs.ITCS.2021.41, author = {Goldwasser, Shafi and Rothblum, Guy N. and Shafer, Jonathan and Yehudayoff, Amir}, title = {{Interactive Proofs for Verifying Machine Learning}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {41:1--41:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.41}, URN = {urn:nbn:de:0030-drops-135806}, doi = {10.4230/LIPIcs.ITCS.2021.41}, annote = {Keywords: PAC learning, Fourier analysis of boolean functions, Complexity gaps, Complexity lower bounds, Goldreich-Levin algorithm, Kushilevitz-Mansour algorithm, Distribution testing} }

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**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

A pseudo-deterministic algorithm is a (randomized) algorithm which, when run multiple times on the same input, with high probability outputs the same result on all executions. Classic streaming algorithms, such as those for finding heavy hitters, approximate counting, ?_2 approximation, finding a nonzero entry in a vector (for turnstile algorithms) are not pseudo-deterministic. For example, in the instance of finding a nonzero entry in a vector, for any known low-space algorithm A, there exists a stream x so that running A twice on x (using different randomness) would with high probability result in two different entries as the output.
In this work, we study whether it is inherent that these algorithms output different values on different executions. That is, we ask whether these problems have low-memory pseudo-deterministic algorithms. For instance, we show that there is no low-memory pseudo-deterministic algorithm for finding a nonzero entry in a vector (given in a turnstile fashion), and also that there is no low-dimensional pseudo-deterministic sketching algorithm for ?_2 norm estimation. We also exhibit problems which do have low memory pseudo-deterministic algorithms but no low memory deterministic algorithm, such as outputting a nonzero row of a matrix, or outputting a basis for the row-span of a matrix.
We also investigate multi-pseudo-deterministic algorithms: algorithms which with high probability output one of a few options. We show the first lower bounds for such algorithms. This implies that there are streaming problems such that every low space algorithm for the problem must have inputs where there are many valid outputs, all with a significant probability of being outputted.

Shafi Goldwasser, Ofer Grossman, Sidhanth Mohanty, and David P. Woodruff. Pseudo-Deterministic Streaming. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 79:1-79:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{goldwasser_et_al:LIPIcs.ITCS.2020.79, author = {Goldwasser, Shafi and Grossman, Ofer and Mohanty, Sidhanth and Woodruff, David P.}, title = {{Pseudo-Deterministic Streaming}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {79:1--79:25}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.79}, URN = {urn:nbn:de:0030-drops-117644}, doi = {10.4230/LIPIcs.ITCS.2020.79}, annote = {Keywords: streaming, pseudo-deterministic} }

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**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

We introduce pseudo-deterministic interactive proofs (psdIP): interactive proof systems for search problems where the verifier is guaranteed with high probability to output the same output on different executions. As in the case with classical interactive proofs, the verifier is a probabilistic polynomial time algorithm interacting with an untrusted powerful prover.
We view pseudo-deterministic interactive proofs as an extension of the study of pseudo-deterministic randomized polynomial time algorithms: the goal of the latter is to find canonical solutions to search problems whereas the goal of the former is to prove that a solution to a search problem is canonical to a probabilistic polynomial time verifier.
Alternatively, one may think of the powerful prover as aiding the probabilistic polynomial time verifier to find canonical solutions to search problems, with high probability over the randomness of the verifier. The challenge is that pseudo-determinism should hold not only with respect to the randomness, but also with respect to the prover: a malicious prover should not be able to cause the verifier to output a solution other than the unique canonical one.
The IP=PSPACE characterization implies that psdIP = IP. The challenge is to find constant round pseudo-deterministic interactive proofs for hard search problems. We show a constant round pseudo-deterministic interactive proof for the graph isomorphism problem: on any input pair of isomorphic graphs (G_0,G_1), there exist a unique isomorphism phi from G_0 to G_1 (although many isomorphism many exist) which will be output by the verifier with high probability, regardless of any dishonest prover strategy.
In contrast, we show that it is unlikely that psdIP proofs with constant rounds exist for NP-complete problems by showing that if any NP-complete problem has a constant round psdIP protocol, then the polynomial hierarchy collapses.

Shafi Goldwasser, Ofer Grossman, and Dhiraj Holden. Pseudo-Deterministic Proofs. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 17:1-17:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{goldwasser_et_al:LIPIcs.ITCS.2018.17, author = {Goldwasser, Shafi and Grossman, Ofer and Holden, Dhiraj}, title = {{Pseudo-Deterministic Proofs}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {17:1--17:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.17}, URN = {urn:nbn:de:0030-drops-83669}, doi = {10.4230/LIPIcs.ITCS.2018.17}, annote = {Keywords: Pseudo-Deterministic, Interactive Proofs} }

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**Published in:** LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Instances of computational problems do not exist in isolation. Rather, multiple and correlated instances of the same problem arise naturally in the real world. The challenge is how to gain computationally from correlations when they can be found.
[DGH, ITCS 2015] showed that significant computational gains can be made by having access to auxiliary instances which are correlated to the primary problem instance via the solution space. They demonstrate this for constraint satisfaction problems, which are NP-hard in the general worst case form.
Here, we set out to study the impact of having access to correlated instances on the complexity of polynomial time problems. Namely, for a problem P that is conjectured to require time n^c for c>0,
we ask whether access to a few instances of P that are correlated in some natural way can be used to solve P on one of them (the designated "primary instance") faster than the conjectured lower bound of n^c.
We focus our attention on a number of problems: the Longest Common Subsequence (LCS), the minimum Edit Distance between sequences, and Dynamic Time Warping Distance (DTWD) of curves, for all of which the best known algorithms achieve O(n^2/polylog(n)) runtime via dynamic programming. These problems form an interesting case in point to study, as it has been shown that a O(n^(2 - epsilon)) time algorithm for a worst-case instance would imply improved algorithms
for a host of other problems as well as disprove complexity hypotheses such as the Strong Exponential Time Hypothesis.
We show how to use access to a logarithmic number of auxiliary correlated instances, to design novel o(n^2) time algorithms for LCS, EDIT, DTWD, and more generally improved algorithms for computing any tuple-based similarity measure - a generalization which we define within on strings. For the multiple sequence alignment problem on k strings, this yields an O(nk\log n) algorithm
contrasting with classical O(n^k) dynamic programming.
Our results hold for several correlation models between the primary and the auxiliary instances. In the most general correlation model we address, we assume that the primary instance is a worst-case instance and the auxiliary instances are chosen with uniform distribution subject to the constraint that their alignments are
epsilon-correlated with the optimal alignment of the primary instance. We emphasize that optimal solutions for the auxiliary instances will not generally coincide with optimal solutions for the worst case primary instance.
We view our work as pointing out a new avenue for looking for significant improvements for sequence alignment problems and
computing similarity measures, by taking advantage of access to sequences which are correlated through natural generating processes.
In this first work we show how to take advantage of mathematically inspired simple clean models of correlation - the intriguing question, looking forward, is to find correlation models which coincide with evolutionary models and other relationships and for which our approach to multiple sequence alignment gives provable guarantees.

Shafi Goldwasser and Dhiraj Holden. The Complexity of Problems in P Given Correlated Instances. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 13:1-13:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{goldwasser_et_al:LIPIcs.ITCS.2017.13, author = {Goldwasser, Shafi and Holden, Dhiraj}, title = {{The Complexity of Problems in P Given Correlated Instances}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {13:1--13:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.13}, URN = {urn:nbn:de:0030-drops-81753}, doi = {10.4230/LIPIcs.ITCS.2017.13}, annote = {Keywords: Correlated instances, Longest Common Subsequence, Fine-grained complexity} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We present a pseudo-deterministic NC algorithm for finding perfect matchings in bipartite graphs. Specifically, our algorithm is a randomized parallel algorithm which uses poly(n) processors, poly(log n) depth, poly(log n) random bits, and outputs for each bipartite input graph a unique perfect matching with high probability. That is, on the same graph it returns the same matching for almost all choices of randomness. As an immediate consequence we also find a pseudo-deterministic NC algorithm for constructing a depth first search (DFS) tree. We introduce a method for computing the union of all min-weight perfect matchings of a weighted graph in RNC and a novel set of weight assignments which in combination enable isolating a unique matching in a graph.
We then show a way to use pseudo-deterministic algorithms to reduce the number of random bits used by general randomized algorithms. The main idea is that random bits can be reused by successive invocations of pseudo-deterministic randomized algorithms. We use the technique to show an RNC algorithm for constructing a depth first search (DFS) tree using only O(log^2 n) bits whereas the previous best randomized algorithm used O(log^7 n), and a new sequential randomized algorithm for the set-maxima problem which uses fewer random bits than the previous state of the art.
Furthermore, we prove that resolving the decision question NC = RNC, would imply an NC algorithm for finding a bipartite perfect matching and finding a DFS tree in NC. This is not implied by previous randomized NC search algorithms for finding bipartite perfect matching, but is implied by the existence of a pseudo-deterministic NC search algorithm.

Shafi Goldwasser and Ofer Grossman. Bipartite Perfect Matching in Pseudo-Deterministic NC. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 87:1-87:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{goldwasser_et_al:LIPIcs.ICALP.2017.87, author = {Goldwasser, Shafi and Grossman, Ofer}, title = {{Bipartite Perfect Matching in Pseudo-Deterministic NC}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {87:1--87:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.87}, URN = {urn:nbn:de:0030-drops-74824}, doi = {10.4230/LIPIcs.ICALP.2017.87}, annote = {Keywords: Parallel Algorithms, Pseudo-determinism, RNC, Perfect Matching} }

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Invited Talk

**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

In this talk we describe a new type of probabilistic algorithm which we call "Bellagio" Algorithms: a randomized algorithm which is guaranteed to run in expected polynomial time, and to produce a correct and unique solution with high probability. These algorithms are pseudo-deterministic: they can not be distinguished from deterministic algorithms in polynomial time by a probabilistic polynomial time observer with black box access to the algorithm.
We show a necessary and sufficient condition for the existence of a
Bellagio Algorithm for an NP relation R: R has a Bellagio
algorithm if and only if it is deterministically reducible to some
decision problem in BPP. Several examples of Bellagio algorithms, for
well known problems in algebra and graph theory which improve on
deterministic solutions, follow.
The notion of pseudo-deterministic algorithms (or more generally
computations) is interesting beyond just sequential algorithms. In
particular, it has long been known that it is impossible to solve
deterministically tasks such as "consensus" in a faulty
distributed systems, whereas randomized protocols can achieve
consensus in expected constant time. We thus explore the notion of
pseudo-deterministic fault tolerant distributed protocols: randomized
protocols which are polynomial time indistinguishable from
deterministic protocols in presence of faults.

Shafi Goldwasser. Pseudo-deterministic Algorithms (Invited Talk). In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, p. 29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{goldwasser:LIPIcs.STACS.2012.29, author = {Goldwasser, Shafi}, title = {{Pseudo-deterministic Algorithms}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {29--29}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.29}, URN = {urn:nbn:de:0030-drops-34435}, doi = {10.4230/LIPIcs.STACS.2012.29}, annote = {Keywords: randomized algorithms, distributed computing, Monte Carlo, Las Vegas} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 8491, Theoretical Foundations of Practical Information Security (2009)

From 30.11. to 05.12.2008, the Dagstuhl Seminar 08491 ``Theoretical Foundations of Practical Information Security '' was held in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.

Ran Canetti, Shafi Goldwasser, Günter Müller, and Rainer Steinwandt. 08491 Abstracts Collection – Theoretical Foundations of Practical Information Security. In Theoretical Foundations of Practical Information Security. Dagstuhl Seminar Proceedings, Volume 8491, pp. 1-16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{canetti_et_al:DagSemProc.08491.1, author = {Canetti, Ran and Goldwasser, Shafi and M\"{u}ller, G\"{u}nter and Steinwandt, Rainer}, title = {{08491 Abstracts Collection – Theoretical Foundations of Practical Information Security}}, booktitle = {Theoretical Foundations of Practical Information Security}, pages = {1--16}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2009}, volume = {8491}, editor = {Ran Canetti and Shafi Goldwasser and G\"{u}nter M\"{u}ller and Rainer Steinwandt}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08491.1}, URN = {urn:nbn:de:0030-drops-18945}, doi = {10.4230/DagSemProc.08491.1}, annote = {Keywords: Organic computing, self-organisation, design, adaptivity} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 8491, Theoretical Foundations of Practical Information Security (2009)

Designing, building, and operating secure information processing
systems is a complex task, and the only scientific way to address the
diverse challenges arising throughout the life-cycle of security
criticial systems is to consolidate and increase the knowledge of the
theoretical foundations of practical security problems. To this aim,
the mutual exchange of ideas across individual security research
communities can be extraordinary beneficial. Accordingly, the
motivation of this Dagstuhl seminar was the integration of different
research areas with the common goal of providing an integral
theoretical basis that is needed for the design of secure information
processing systems.

Ran Canetti, Shafi Goldwasser, Günter Müller, and Rainer Steinwandt. 08491 Executive Summary – Theoretical Foundations of Practical Information Security. In Theoretical Foundations of Practical Information Security. Dagstuhl Seminar Proceedings, Volume 8491, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{canetti_et_al:DagSemProc.08491.2, author = {Canetti, Ran and Goldwasser, Shafi and M\"{u}ller, G\"{u}nter and Steinwandt, Rainer}, title = {{08491 Executive Summary – Theoretical Foundations of Practical Information Security }}, booktitle = {Theoretical Foundations of Practical Information Security}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2009}, volume = {8491}, editor = {Ran Canetti and Shafi Goldwasser and G\"{u}nter M\"{u}ller and Rainer Steinwandt}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08491.2}, URN = {urn:nbn:de:0030-drops-18938}, doi = {10.4230/DagSemProc.08491.2}, annote = {Keywords: Organic computing, self-organisation, design, adaptivity} }

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