DROPS

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DOI: 10.4230/LIPIcs.CCC.2024.29

Exact Search-To-Decision Reductions for Time-Bounded Kolmogorov Complexity

Authors: Shuichi Hirahara, Valentine Kabanets, Zhenjian Lu, and Igor C. Oliveira

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract

A search-to-decision reduction is a procedure that allows one to find a solution to a problem from the mere ability to decide when a solution exists. The existence of a search-to-decision reduction for time-bounded Kolmogorov complexity, i.e., the problem of checking if a string x can be generated by a t-time bounded program of description length s, is a long-standing open problem that dates back to the 1960s. In this work, we obtain new average-case and worst-case search-to-decision reductions for the complexity measure 𝖪^t and its randomized analogue rK^t: 1) (Conditional Errorless and Error-Prone Reductions for 𝖪^t) Under the assumption that 𝖤 requires exponential size circuits, we design polynomial-time average-case search-to-decision reductions for 𝖪^t in both errorless and error-prone settings. In fact, under the easiness of deciding 𝖪^t under the uniform distribution, we obtain a search algorithm for any given polynomial-time samplable distribution. In the error-prone reduction, the search algorithm works in the more general setting of conditional 𝖪^t complexity, i.e., it finds a minimum length t-time bound program for generating x given a string y. 2) (Unconditional Errorless Reduction for rK^t) We obtain an unconditional polynomial-time average-case search-to-decision reduction for rK^t in the errorless setting. Similarly to the results described above, we obtain a search algorithm for each polynomial-time samplable distribution, assuming the existence of a decision algorithm under the uniform distribution. To our knowledge, this is the first unconditional sub-exponential time search-to-decision reduction among the measures 𝖪^t and rK^t that works with respect to any given polynomial-time samplable distribution. 3) (Worst-Case to Average-Case Reductions) Under the errorless average-case easiness of deciding rK^t, we design a worst-case search algorithm running in time 2^O(n/log n) that produces a minimum length randomized t-time program for every input string x ∈ {0,1}ⁿ, with the caveat that it only succeeds on some explicitly computed sub-exponential time bound t ≤ 2^{n^ε} that depends on x. A similar result holds for 𝖪^t, under the assumption that 𝖤 requires exponential size circuits. In these results, the corresponding search problem is solved exactly, i.e., a successful run of the search algorithm outputs a t-time bounded program for x of minimum length, as opposed to an approximately optimal program of slightly larger description length or running time.

Cite as

Shuichi Hirahara, Valentine Kabanets, Zhenjian Lu, and Igor C. Oliveira. Exact Search-To-Decision Reductions for Time-Bounded Kolmogorov Complexity. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 29:1-29:56, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hirahara_et_al:LIPIcs.CCC.2024.29,
  author =	{Hirahara, Shuichi and Kabanets, Valentine and Lu, Zhenjian and Oliveira, Igor C.},
  title =	{{Exact Search-To-Decision Reductions for Time-Bounded Kolmogorov Complexity}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{29:1--29:56},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.29},
  URN =		{urn:nbn:de:0030-drops-204256},
  doi =		{10.4230/LIPIcs.CCC.2024.29},
  annote =	{Keywords: average-case complexity, Kolmogorov complexity, search-to-decision reductions}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2024.14

Two-Dimensional Kripke Semantics I: Presheaves

Authors: G. A. Kavvos

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

The study of modal logic has witnessed tremendous development following the introduction of Kripke semantics. However, recent developments in programming languages and type theory have led to a second way of studying modalities, namely through their categorical semantics. We show how the two correspond.

Cite as

G. A. Kavvos. Two-Dimensional Kripke Semantics I: Presheaves. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 14:1-14:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kavvos:LIPIcs.FSCD.2024.14,
  author =	{Kavvos, G. A.},
  title =	{{Two-Dimensional Kripke Semantics I: Presheaves}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{14:1--14:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.14},
  URN =		{urn:nbn:de:0030-drops-203438},
  doi =		{10.4230/LIPIcs.FSCD.2024.14},
  annote =	{Keywords: modal logic, categorical semantics, Kripke semantics, duality, open maps}
}

Document

DOI: 10.4230/DagMan.10.1.1

Current and Future Challenges in Knowledge Representation and Reasoning (Dagstuhl Perspectives Workshop 22282)

Authors: James P. Delgrande, Birte Glimm, Thomas Meyer, Miroslaw Truszczynski, and Frank Wolter

Published in: Dagstuhl Manifestos, Volume 10, Issue 1 (2024)

Abstract

Knowledge Representation and Reasoning is a central, longstanding, and active area of Artificial Intelligence. Over the years it has evolved significantly; more recently it has been challenged and complemented by research in areas such as machine learning and reasoning under uncertainty. In July 2022,sser a Dagstuhl Perspectives workshop was held on Knowledge Representation and Reasoning. The goal of the workshop was to describe the state of the art in the field, including its relation with other areas, its shortcomings and strengths, together with recommendations for future progress. We developed this manifesto based on the presentations, panels, working groups, and discussions that took place at the Dagstuhl Workshop. It is a declaration of our views on Knowledge Representation: its origins, goals, milestones, and current foci; its relation to other disciplines, especially to Artificial Intelligence; and on its challenges, along with key priorities for the next decade.

Cite as

James P. Delgrande, Birte Glimm, Thomas Meyer, Miroslaw Truszczynski, and Frank Wolter. Current and Future Challenges in Knowledge Representation and Reasoning (Dagstuhl Perspectives Workshop 22282). In Dagstuhl Manifestos, Volume 10, Issue 1, pp. 1-61, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@Article{delgrande_et_al:DagMan.10.1.1,
  author =	{Delgrande, James P. and Glimm, Birte and Meyer, Thomas and Truszczynski, Miroslaw and Wolter, Frank},
  title =	{{Current and Future Challenges in Knowledge Representation and Reasoning (Dagstuhl Perspectives Workshop 22282)}},
  pages =	{1--61},
  journal =	{Dagstuhl Manifestos},
  ISSN =	{2193-2433},
  year =	{2024},
  volume =	{10},
  number =	{1},
  editor =	{Delgrande, James P. and Glimm, Birte and Meyer, Thomas and Truszczynski, Miroslaw and Wolter, Frank},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagMan.10.1.1},
  URN =		{urn:nbn:de:0030-drops-201403},
  doi =		{10.4230/DagMan.10.1.1},
  annote =	{Keywords: Knowledge representation and reasoning, Applications of logics, Declarative representations, Formal logic}
}

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Survey

DOI: 10.4230/TGDK.2.1.3

Semantic Web: Past, Present, and Future

Authors: Ansgar Scherp, Gerd Groener, Petr Škoda, Katja Hose, and Maria-Esther Vidal

Published in: TGDK, Volume 2, Issue 1 (2024): Special Issue on Trends in Graph Data and Knowledge - Part 2. Transactions on Graph Data and Knowledge, Volume 2, Issue 1

Abstract

Ever since the vision was formulated, the Semantic Web has inspired many generations of innovations. Semantic technologies have been used to share vast amounts of information on the Web, enhance them with semantics to give them meaning, and enable inference and reasoning on them. Throughout the years, semantic technologies, and in particular knowledge graphs, have been used in search engines, data integration, enterprise settings, and machine learning. In this paper, we recap the classical concepts and foundations of the Semantic Web as well as modern and recent concepts and applications, building upon these foundations. The classical topics we cover include knowledge representation, creating and validating knowledge on the Web, reasoning and linking, and distributed querying. We enhance this classical view of the so-called "Semantic Web Layer Cake" with an update of recent concepts that include provenance, security and trust, as well as a discussion of practical impacts from industry-led contributions. We conclude with an outlook on the future directions of the Semantic Web. This is a living document. If you like to contribute, please contact the first author and visit: https://github.com/ascherp/semantic-web-primer

Cite as

Ansgar Scherp, Gerd Groener, Petr Škoda, Katja Hose, and Maria-Esther Vidal. Semantic Web: Past, Present, and Future. In Special Issue on Trends in Graph Data and Knowledge - Part 2. Transactions on Graph Data and Knowledge (TGDK), Volume 2, Issue 1, pp. 3:1-3:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@Article{scherp_et_al:TGDK.2.1.3,
  author =	{Scherp, Ansgar and Groener, Gerd and \v{S}koda, Petr and Hose, Katja and Vidal, Maria-Esther},
  title =	{{Semantic Web: Past, Present, and Future}},
  journal =	{Transactions on Graph Data and Knowledge},
  pages =	{3:1--3:37},
  ISSN =	{2942-7517},
  year =	{2024},
  volume =	{2},
  number =	{1},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/TGDK.2.1.3},
  URN =		{urn:nbn:de:0030-drops-198607},
  doi =		{10.4230/TGDK.2.1.3},
  annote =	{Keywords: Linked Open Data, Semantic Web Graphs, Knowledge Graphs}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2022.119

Polynomial Identity Testing via Evaluation of Rational Functions

Authors: Dieter van Melkebeek and Andrew Morgan

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract

We introduce a hitting set generator for Polynomial Identity Testing based on evaluations of low-degree univariate rational functions at abscissas associated with the variables. In spite of the univariate nature, we establish an equivalence up to rescaling with a generator introduced by Shpilka and Volkovich, which has a similar structure but uses multivariate polynomials in the abscissas. We study the power of the generator by characterizing its vanishing ideal, i.e., the set of polynomials that it fails to hit. Capitalizing on the univariate nature, we develop a small collection of polynomials that jointly produce the vanishing ideal. As corollaries, we obtain tight bounds on the minimum degree, sparseness, and partition size of set-multi-linearity in the vanishing ideal. Inspired by an alternating algebra representation, we develop a structured deterministic membership test for the vanishing ideal. As a proof of concept we rederive known derandomization results based on the generator by Shpilka and Volkovich, and present a new application for read-once oblivious arithmetic branching programs that provably transcends the usual combinatorial techniques.

Cite as

Dieter van Melkebeek and Andrew Morgan. Polynomial Identity Testing via Evaluation of Rational Functions. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 119:1-119:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{vanmelkebeek_et_al:LIPIcs.ITCS.2022.119,
  author =	{van Melkebeek, Dieter and Morgan, Andrew},
  title =	{{Polynomial Identity Testing via Evaluation of Rational Functions}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{119:1--119:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.119},
  URN =		{urn:nbn:de:0030-drops-157158},
  doi =		{10.4230/LIPIcs.ITCS.2022.119},
  annote =	{Keywords: Derandomization, Gr\"{o}bner Basis, Lower Bounds, Polynomial Identity Testing}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2021.54

Cryptographic Hardness Under Projections for Time-Bounded Kolmogorov Complexity

Authors: Eric Allender, John Gouwar, Shuichi Hirahara, and Caleb Robelle

Published in: LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

Abstract

A version of time-bounded Kolmogorov complexity, denoted KT, has received attention in the past several years, due to its close connection to circuit complexity and to the Minimum Circuit Size Problem MCSP. Essentially all results about the complexity of MCSP hold also for MKTP (the problem of computing the KT complexity of a string). Both MKTP and MCSP are hard for SZK (Statistical Zero Knowledge) under BPP-Turing reductions; neither is known to be NP-complete. Recently, some hardness results for MKTP were proved that are not (yet) known to hold for MCSP. In particular, MKTP is hard for DET (a subclass of P) under nonuniform ≤^{NC^0}_m reductions. In this paper, we improve this, to show that the complement of MKTP is hard for the (apparently larger) class NISZK_L under not only ≤^{NC^0}_m reductions but even under projections. Also, the complement of MKTP is hard for NISZK under ≤^{P/poly}_m reductions. Here, NISZK is the class of problems with non-interactive zero-knowledge proofs, and NISZK_L is the non-interactive version of the class SZK_L that was studied by Dvir et al. As an application, we provide several improved worst-case to average-case reductions to problems in NP, and we obtain a new lower bound on MKTP (which is currently not known to hold for MCSP).

Cite as

Eric Allender, John Gouwar, Shuichi Hirahara, and Caleb Robelle. Cryptographic Hardness Under Projections for Time-Bounded Kolmogorov Complexity. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 54:1-54:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{allender_et_al:LIPIcs.ISAAC.2021.54,
  author =	{Allender, Eric and Gouwar, John and Hirahara, Shuichi and Robelle, Caleb},
  title =	{{Cryptographic Hardness Under Projections for Time-Bounded Kolmogorov Complexity}},
  booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
  pages =	{54:1--54:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-214-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{212},
  editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.54},
  URN =		{urn:nbn:de:0030-drops-154875},
  doi =		{10.4230/LIPIcs.ISAAC.2021.54},
  annote =	{Keywords: Kolmogorov Complexity, Interactive Proofs, Minimum Circuit Size Problem, Worst-case to Average-case Reductions}
}

Document

DOI: 10.4230/DagRep.9.9.135

Comparative Theory for Graph Polynomials (Dagstuhl Seminar 19401)

Authors: Jo Ellis-Monaghan, Andrew Goodall, Iain Moffatt, and Kerri Morgan

Published in: Dagstuhl Reports, Volume 9, Issue 9 (2020)

Abstract

This report documents the programme and outcomes of Dagstuhl Seminar 19401 ``Comparative Theory for Graph Polynomials''. The study of graph polynomials has become increasingly active, with new applications and new graph polynomials being discovered each year. The genera of graph polynomials are diverse, and their interconnections are rich. Experts in the field are finding that proof techniques and results established in one area can be successfully extended to others. From this a general theory is emerging that encapsulates the deeper interconnections between families of graph polynomials and the various techniques, computational approaches, and methodologies applied to them. The overarching aim of this Seminar was to exploit commonalities among polynomial invariants of graphs, matroids, and related combinatorial structures. Model-theoretic, computational and other methods were used in order to initiate a comparative theory that collects the current state of knowledge into a more cohesive and powerful framework.

Cite as

Jo Ellis-Monaghan, Andrew Goodall, Iain Moffatt, and Kerri Morgan. Comparative Theory for Graph Polynomials (Dagstuhl Seminar 19401). In Dagstuhl Reports, Volume 9, Issue 9, pp. 135-155, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@Article{ellismonaghan_et_al:DagRep.9.9.135,
  author =	{Ellis-Monaghan, Jo and Goodall, Andrew and Moffatt, Iain and Morgan, Kerri},
  title =	{{Comparative Theory for Graph Polynomials (Dagstuhl Seminar 19401)}},
  pages =	{135--155},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2020},
  volume =	{9},
  number =	{9},
  editor =	{Ellis-Monaghan, Jo and Goodall, Andrew and Moffatt, Iain and Morgan, Kerri},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.9.9.135},
  URN =		{urn:nbn:de:0030-drops-118460},
  doi =		{10.4230/DagRep.9.9.135},
  annote =	{Keywords: graph polynomials, graph and matroid invariants, Tutte polynomial, topological and algebraic graph theory}
}

Document

DOI: 10.4230/LIPIcs.CSL.2020.14

Unifying Cubical Models of Univalent Type Theory

Authors: Evan Cavallo, Anders Mörtberg, and Andrew W Swan

Published in: LIPIcs, Volume 152, 28th EACSL Annual Conference on Computer Science Logic (CSL 2020)

Abstract

We present a new constructive model of univalent type theory based on cubical sets. Unlike prior work on cubical models, ours depends neither on diagonal cofibrations nor connections. This is made possible by weakening the notion of fibration from the cartesian cubical set model, so that it is not necessary to assume that the diagonal on the interval is a cofibration. We have formally verified in Agda that these fibrations are closed under the type formers of cubical type theory and that the model satisfies the univalence axiom. By applying the construction in the presence of diagonal cofibrations or connections and reversals, we recover the existing cartesian and De Morgan cubical set models as special cases. Generalizing earlier work of Sattler for cubical sets with connections, we also obtain a Quillen model structure.

Cite as

Evan Cavallo, Anders Mörtberg, and Andrew W Swan. Unifying Cubical Models of Univalent Type Theory. In 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 152, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cavallo_et_al:LIPIcs.CSL.2020.14,
  author =	{Cavallo, Evan and M\"{o}rtberg, Anders and Swan, Andrew W},
  title =	{{Unifying Cubical Models of Univalent Type Theory}},
  booktitle =	{28th EACSL Annual Conference on Computer Science Logic (CSL 2020)},
  pages =	{14:1--14:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-132-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{152},
  editor =	{Fern\'{a}ndez, Maribel 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.CSL.2020.14},
  URN =		{urn:nbn:de:0030-drops-116578},
  doi =		{10.4230/LIPIcs.CSL.2020.14},
  annote =	{Keywords: Cubical Set Models, Cubical Type Theory, Homotopy Type Theory, Univalent Foundations}
}

Document

DOI: 10.4230/LIPIcs.ITP.2019.27

Verifying That a Compiler Preserves Concurrent Value-Dependent Information-Flow Security

Authors: Robert Sison and Toby Murray

Published in: LIPIcs, Volume 141, 10th International Conference on Interactive Theorem Proving (ITP 2019)

Abstract

It is common to prove by reasoning over source code that programs do not leak sensitive data. But doing so leaves a gap between reasoning and reality that can only be filled by accounting for the behaviour of the compiler. This task is complicated when programs enforce value-dependent information-flow security properties (in which classification of locations can vary depending on values in other locations) and complicated further when programs exploit shared-variable concurrency. Prior work has formally defined a notion of concurrency-aware refinement for preserving value-dependent security properties. However, that notion is considerably more complex than standard refinement definitions typically applied in the verification of semantics preservation by compilers. To date it remains unclear whether it can be applied to a realistic compiler, because there exist no general decomposition principles for separating it into smaller, more familiar, proof obligations. In this work, we provide such a decomposition principle, which we show can almost halve the complexity of proving secure refinement. Further, we demonstrate its applicability to secure compilation, by proving in Isabelle/HOL the preservation of value-dependent security by a proof-of-concept compiler from an imperative While language to a generic RISC-style assembly language, for programs with shared-memory concurrency mediated by locking primitives. Finally, we execute our compiler in Isabelle on a While language model of the Cross Domain Desktop Compositor, demonstrating to our knowledge the first use of a compiler verification result to carry an information-flow security property down to the assembly-level model of a non-trivial concurrent program.

Cite as

Robert Sison and Toby Murray. Verifying That a Compiler Preserves Concurrent Value-Dependent Information-Flow Security. In 10th International Conference on Interactive Theorem Proving (ITP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 141, pp. 27:1-27:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{sison_et_al:LIPIcs.ITP.2019.27,
  author =	{Sison, Robert and Murray, Toby},
  title =	{{Verifying That a Compiler Preserves Concurrent Value-Dependent Information-Flow Security}},
  booktitle =	{10th International Conference on Interactive Theorem Proving (ITP 2019)},
  pages =	{27:1--27:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-122-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{141},
  editor =	{Harrison, John and O'Leary, John and Tolmach, Andrew},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2019.27},
  URN =		{urn:nbn:de:0030-drops-110829},
  doi =		{10.4230/LIPIcs.ITP.2019.27},
  annote =	{Keywords: Secure compilation, Information flow security, Concurrency, Verification}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2018.20

Minimum Circuit Size, Graph Isomorphism, and Related Problems

Authors: Eric Allender, Joshua A. Grochow, Dieter van Melkebeek, Cristopher Moore, and Andrew Morgan

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

Abstract

We study the computational power of deciding whether a given truth-table can be described by a circuit of a given size (the Minimum Circuit Size Problem, or MCSP for short), and of the variant denoted MKTP where circuit size is replaced by a polynomially-related Kolmogorov measure. All prior reductions from supposedly-intractable problems to MCSP / MKTP hinged on the power of MCSP / MKTP to distinguish random distributions from distributions produced by hardness-based pseudorandom generator constructions. We develop a fundamentally different approach inspired by the well-known interactive proof system for the complement of Graph Isomorphism (GI). It yields a randomized reduction with zero-sided error from GI to MKTP. We generalize the result and show that GI can be replaced by any isomorphism problem for which the underlying group satisfies some elementary properties. Instantiations include Linear Code Equivalence, Permutation Group Conjugacy, and Matrix Subspace Conjugacy. Along the way we develop encodings of isomorphism classes that are efficiently decodable and achieve compression that is at or near the information-theoretic optimum; those encodings may be of independent interest.

Cite as

Eric Allender, Joshua A. Grochow, Dieter van Melkebeek, Cristopher Moore, and Andrew Morgan. Minimum Circuit Size, Graph Isomorphism, and Related Problems. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{allender_et_al:LIPIcs.ITCS.2018.20,
  author =	{Allender, Eric and Grochow, Joshua A. and van Melkebeek, Dieter and Moore, Cristopher and Morgan, Andrew},
  title =	{{Minimum Circuit Size, Graph Isomorphism, and Related Problems}},
  booktitle =	{9th Innovations in Theoretical Computer Science Conference (ITCS 2018)},
  pages =	{20:1--20:20},
  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.2018.20},
  URN =		{urn:nbn:de:0030-drops-83455},
  doi =		{10.4230/LIPIcs.ITCS.2018.20},
  annote =	{Keywords: Reductions between NP-intermediate problems, Graph Isomorphism, Minimum Circuit Size Problem, time-bounded Kolmogorov complexity}
}

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