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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} }

Document

**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} }

Document

**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

We consider variants of the Minimum Circuit Size Problem MCSP, where the goal is to minimize the size of oracle circuits computing a given function. When the oracle is QBF, the resulting problem MCSP^QBF
is known to be complete for PSPACE under ZPP reductions. We show that it is not complete under logspace reductions, and indeed it is not even hard for TC under uniform AC^0 reductions. We obtain a variety of consequences that follow if oracle versions of MCSP are hard for various complexity classes under different types of reductions. We also prove analogous results for the problem of determining the resource-bounded Kolmogorov complexity of strings, for certain types of Kolmogorov complexity measures.

Eric Allender, Dhiraj Holden, and Valentine Kabanets. The Minimum Oracle Circuit Size Problem. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 21-33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{allender_et_al:LIPIcs.STACS.2015.21, author = {Allender, Eric and Holden, Dhiraj and Kabanets, Valentine}, title = {{The Minimum Oracle Circuit Size Problem}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {21--33}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.21}, URN = {urn:nbn:de:0030-drops-49013}, doi = {10.4230/LIPIcs.STACS.2015.21}, annote = {Keywords: Kolmogorov complexity, minimum circuit size problem, PSPACE, NP-intermediate sets} }

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