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

Dawar and Wilsenach (ICALP 2020) introduce the model of symmetric arithmetic circuits and show an exponential separation between the sizes of symmetric circuits for computing the determinant and the permanent. The symmetry restriction is that the circuits which take a matrix input are unchanged by a permutation applied simultaneously to the rows and columns of the matrix. Under such restrictions we have polynomial-size circuits for computing the determinant but no subexponential size circuits for the permanent. Here, we consider a more stringent symmetry requirement, namely that the circuits are unchanged by arbitrary even permutations applied separately to rows and columns, and prove an exponential lower bound even for circuits computing the determinant. The result requires substantial new machinery. We develop a general framework for proving lower bounds for symmetric circuits with restricted symmetries, based on a new support theorem and new two-player restricted bijection games. These are applied to the determinant problem with a novel construction of matrices that are bi-adjacency matrices of graphs based on the CFI construction. Our general framework opens the way to exploring a variety of symmetry restrictions and studying trade-offs between symmetry and other resources used by arithmetic circuits.

Anuj Dawar and Gregory Wilsenach. Lower Bounds for Symmetric Circuits for the Determinant. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 52:1-52:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dawar_et_al:LIPIcs.ITCS.2022.52, author = {Dawar, Anuj and Wilsenach, Gregory}, title = {{Lower Bounds for Symmetric Circuits for the Determinant}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {52:1--52:22}, 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.2022.52}, URN = {urn:nbn:de:0030-drops-156480}, doi = {10.4230/LIPIcs.ITCS.2022.52}, annote = {Keywords: arithmetic circuits, symmetric arithmetic circuits, Boolean circuits, symmetric circuits, permanent, determinant, counting width, Weisfeiler-Leman dimension, Cai-F\"{u}rer-Immerman constructions} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

We introduce symmetric arithmetic circuits, i.e. arithmetic circuits with a natural symmetry restriction. In the context of circuits computing polynomials defined on a matrix of variables, such as the determinant or the permanent, the restriction amounts to requiring that the shape of the circuit is invariant under row and column permutations of the matrix. We establish unconditional, nearly exponential, lower bounds on the size of any symmetric circuit for computing the permanent over any field of characteristic other than 2. In contrast, we show that there are polynomial-size symmetric circuits for computing the determinant over fields of characteristic zero.

Anuj Dawar and Gregory Wilsenach. Symmetric Arithmetic Circuits. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{dawar_et_al:LIPIcs.ICALP.2020.36, author = {Dawar, Anuj and Wilsenach, Gregory}, title = {{Symmetric Arithmetic Circuits}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {36:1--36:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.36}, URN = {urn:nbn:de:0030-drops-124430}, doi = {10.4230/LIPIcs.ICALP.2020.36}, annote = {Keywords: arithmetic circuits, symmetric arithmetic circuits, Boolean circuits, symmetric circuits, permanent, determinant, counting width, Weisfeiler-Leman dimension, Cai-F\"{u}rer-Immerman constructions} }

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**Published in:** LIPIcs, Volume 119, 27th EACSL Annual Conference on Computer Science Logic (CSL 2018)

Fixed-point logic with rank (FPR) is an extension of fixed-point logic with counting (FPC) with operators for computing the rank of a matrix over a finite field. The expressive power of FPR properly extends that of FPC and is contained in P, but it is not known if that containment is proper. We give a circuit characterization for FPR in terms of families of symmetric circuits with rank gates, along the lines of that for FPC given by [Anderson and Dawar 2017]. This requires the development of a broad framework of circuits in which the individual gates compute functions that are not symmetric (i.e., invariant under all permutations of their inputs). This framework also necessitates the development of novel techniques to prove the equivalence of circuits and logic. Both the framework and the techniques are of greater generality than the main result.

Anuj Dawar and Gregory Wilsenach. Symmetric Circuits for Rank Logic. In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 20:1-20:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dawar_et_al:LIPIcs.CSL.2018.20, author = {Dawar, Anuj and Wilsenach, Gregory}, title = {{Symmetric Circuits for Rank Logic}}, booktitle = {27th EACSL Annual Conference on Computer Science Logic (CSL 2018)}, pages = {20:1--20:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-088-0}, ISSN = {1868-8969}, year = {2018}, volume = {119}, editor = {Ghica, Dan R. and Jung, Achim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2018.20}, URN = {urn:nbn:de:0030-drops-96870}, doi = {10.4230/LIPIcs.CSL.2018.20}, annote = {Keywords: fixed-point logic with rank, circuits, symmetric circuits, uniform families of circuits, circuit characterization, circuit framework, finite model theory, descriptive complexity} }

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