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**Published in:** LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)

Abramsky, Dawar, and Wang (2017) introduced the pebbling comonad for k-variable counting logic and thereby initiated a line of work that imports category theoretic machinery to finite model theory. Such game comonads have been developed for various logics, yielding characterisations of logical equivalences in terms of isomorphisms in the associated co-Kleisli category. We show a first limitation of this approach by studying linear-algebraic logic, which is strictly more expressive than first-order counting logic and whose k-variable logical equivalence relations are known as invertible-map equivalences (IM). We show that there exists no finite-rank comonad on the category of graphs whose co-Kleisli isomorphisms characterise IM-equivalence, answering a question of Ó Conghaile and Dawar (CSL 2021). We obtain this result by ruling out a characterisation of IM-equivalence in terms of homomorphism indistinguishability and employing the Lovász-type theorem for game comonads established by Reggio (2022). Two graphs are homomorphism indistinguishable over a graph class if they admit the same number of homomorphisms from every graph in the class. The IM-equivalences cannot be characterised in this way, neither when counting homomorphisms in the natural numbers, nor in any finite prime field.

Moritz Lichter, Benedikt Pago, and Tim Seppelt. Limitations of Game Comonads for Invertible-Map Equivalence via Homomorphism Indistinguishability. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 36:1-36:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{lichter_et_al:LIPIcs.CSL.2024.36, author = {Lichter, Moritz and Pago, Benedikt and Seppelt, Tim}, title = {{Limitations of Game Comonads for Invertible-Map Equivalence via Homomorphism Indistinguishability}}, booktitle = {32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)}, pages = {36:1--36:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-310-2}, ISSN = {1868-8969}, year = {2024}, volume = {288}, editor = {Murano, Aniello and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.36}, URN = {urn:nbn:de:0030-drops-196799}, doi = {10.4230/LIPIcs.CSL.2024.36}, annote = {Keywords: finite model theory, graph isomorphism, linear-algebraic logic, homomorphism indistinguishability, game comonads, invertible-map equivalence} }

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**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Choiceless Polynomial Time (CPT) is one of the few remaining candidate logics for capturing Ptime. In this paper, we make progress towards separating CPT from polynomial time by firstly establishing a connection between the expressive power of CPT and the existence of certain symmetric circuit families, and secondly, proving lower bounds against these circuits. We focus on the isomorphism problem of unordered Cai-Fürer-Immerman-graphs (the CFI-query) as a potential candidate for separating CPT from Ptime. Results by Dawar, Richerby and Rossman, and subsequently by Pakusa, Schalthöfer and Selman show that the CFI-query is CPT-definable on linearly ordered and preordered base graphs with small colour classes. We define a class of CPT-algorithms, that we call "CFI-symmetric algorithms", which generalises all the known ones, and show that such algorithms can only define the CFI-query on a given class of base graphs if there exists a family of symmetric XOR-circuits with certain properties. These properties include that the circuits have the same symmetries as the base graphs, are of polynomial size, and satisfy certain fan-in restrictions. Then we prove that such circuits with slightly strengthened requirements (i.e. stronger symmetry and fan-in and fan-out restrictions) do not exist for the n-dimensional hypercubes as base graphs. This almost separates the CFI-symmetric algorithms from Ptime - up to the gap that remains between the circuits whose existence we can currently disprove and the circuits whose existence is necessary for the definability of the CFI-query by a CFI-symmetric algorithm.

Benedikt Pago. Lower Bounds for Choiceless Polynomial Time via Symmetric XOR-Circuits. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 73:1-73:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{pago:LIPIcs.MFCS.2023.73, author = {Pago, Benedikt}, title = {{Lower Bounds for Choiceless Polynomial Time via Symmetric XOR-Circuits}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {73:1--73:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.73}, URN = {urn:nbn:de:0030-drops-186077}, doi = {10.4230/LIPIcs.MFCS.2023.73}, annote = {Keywords: logic in computer science, finite model theory, descriptive complexity, symmetric computation, symmetric circuits, graph isomorphism} }

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

This paper extends prior work on the connections between logics from finite model theory and propositional/algebraic proof systems. We show that if all non-isomorphic graphs in a given graph class can be distinguished in the logic Choiceless Polynomial Time with counting (CPT), then they can also be distinguished in the bounded-degree extended polynomial calculus (EPC), and the refutations have roughly the same size as the resource consumption of the CPT-sentence. This allows to transfer lower bounds for EPC to CPT and thus constitutes a new potential approach towards better understanding the limits of CPT. A super-polynomial EPC lower bound for a Ptime-instance of the graph isomorphism problem would separate CPT from Ptime and thus solve a major open question in finite model theory. Further, using our result, we provide a model theoretic proof for the separation of bounded-degree polynomial calculus and bounded-degree extended polynomial calculus.

Benedikt Pago. Finite Model Theory and Proof Complexity Revisited: Distinguishing Graphs in Choiceless Polynomial Time and the Extended Polynomial Calculus. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 31:1-31:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{pago:LIPIcs.CSL.2023.31, author = {Pago, Benedikt}, title = {{Finite Model Theory and Proof Complexity Revisited: Distinguishing Graphs in Choiceless Polynomial Time and the Extended Polynomial Calculus}}, booktitle = {31st EACSL Annual Conference on Computer Science Logic (CSL 2023)}, pages = {31:1--31:19}, 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.2023.31}, URN = {urn:nbn:de:0030-drops-174923}, doi = {10.4230/LIPIcs.CSL.2023.31}, annote = {Keywords: finite model theory, proof complexity, graph isomorphism} }

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

The search for a logic capturing PTIME is a long standing open problem in finite model theory. One of the most promising candidate logics for this is Choiceless Polynomial Time with counting (CPT). Abstractly speaking, CPT is an isomorphism-invariant computation model working with hereditarily finite sets as data structures. While it is easy to check that the evaluation of CPT-sentences is possible in polynomial time, the converse has been open for more than 20 years: Can every PTIME-decidable property of finite structures be expressed in CPT? We attempt to make progress towards a negative answer and show that Choiceless Polynomial Time cannot compute a preorder with colour classes of logarithmic size in every hypercube. The reason is that such preorders have super-polynomially many automorphic images, which makes it impossible for CPT to define them. While the computation of such a preorder is not a decision problem that would immediately separate P and CPT, it is significant for the following reason: The so-called Cai-Fürer-Immerman (CFI) problem is one of the standard "benchmarks" for logics and maybe best known for separating fixed-point logic with counting (FPC) from P. Hence, it is natural to consider this also a potential candidate for the separation of CPT and P. The strongest known positive result in this regard says that CPT is able to solve CFI if a preorder with logarithmically sized colour classes is present in the input structure. Our result implies that this approach cannot be generalised to unordered inputs. In other words, CFI on unordered hypercubes is a PTIME-problem which provably cannot be tackled with the state-of-the-art choiceless algorithmic techniques.

Benedikt Pago. Choiceless Computation and Symmetry: Limitations of Definability. In 29th EACSL Annual Conference on Computer Science Logic (CSL 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 183, pp. 33:1-33:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{pago:LIPIcs.CSL.2021.33, author = {Pago, Benedikt}, title = {{Choiceless Computation and Symmetry: Limitations of Definability}}, booktitle = {29th EACSL Annual Conference on Computer Science Logic (CSL 2021)}, pages = {33:1--33:21}, 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.2021.33}, URN = {urn:nbn:de:0030-drops-134673}, doi = {10.4230/LIPIcs.CSL.2021.33}, annote = {Keywords: finite model theory, descriptive complexity, choiceless computation, symmetries of combinatorial objects} }

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**Published in:** LIPIcs, Volume 82, 26th EACSL Annual Conference on Computer Science Logic (CSL 2017)

We establish new, and surprisingly tight, connections between propositional proof complexity and finite model theory.
Specifically, we show that the power of several propositional proof systems, such as Horn resolution, bounded width resolution, and the polynomial calculus of bounded degree, can be characterised in a precise sense by variants of fixed-point logics that are of fundamental importance in descriptive complexity theory.
Our main results are that Horn resolution has the same expressive power as least fixed-point logic, that bounded width resolution captures existential least fixed-point logic, and that the (monomial restriction of the) polynomial calculus of bounded degree solves precisely the problems definable in fixed-point logic with counting.

Erich Grädel, Benedikt Pago, and Wied Pakusa. The Model-Theoretic Expressiveness of Propositional Proof Systems. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 82, pp. 27:1-27:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{gradel_et_al:LIPIcs.CSL.2017.27, author = {Gr\"{a}del, Erich and Pago, Benedikt and Pakusa, Wied}, title = {{The Model-Theoretic Expressiveness of Propositional Proof Systems}}, booktitle = {26th EACSL Annual Conference on Computer Science Logic (CSL 2017)}, pages = {27:1--27:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-045-3}, ISSN = {1868-8969}, year = {2017}, volume = {82}, editor = {Goranko, Valentin and Dam, Mads}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2017.27}, URN = {urn:nbn:de:0030-drops-76897}, doi = {10.4230/LIPIcs.CSL.2017.27}, annote = {Keywords: Propositional proof systems, fixed-point logics, resolution, polynomial calculus, generalized quantifiers} }

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