## Abstract

We establish an exactly tight relation between reversible
pebblings of graphs and Nullstellensatz refutations of pebbling formulas,
showing that a graph *G* can be reversibly pebbled in time *t* and space *s* if and only if there is a Nullstellensatz refutation of the pebbling formula
over *G* in size *t* + 1 and degree *s* (independently of the field in which
the Nullstellensatz refutation is made). We use this correspondence
to prove a number of strong size-degree trade-offs for Nullstellensatz,
which to the best of our knowledge are the first such results for this
proof system.

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## References

Michael Alekhnovich, Eli Ben-Sasson, Alexander A. Razborov & Avi Wigderson (2002). Space Complexity in Propositional Calculus. SIAM Journal on Computing 31(4), 1184–1211. Preliminary version in STOC '00

Joël Alwen & Vladimir Serbinenko (2015). High Parallel Complexity Graphs and Memory-Hard Functions. In Proceedings of the 47th Annual ACM Symposium on Theory of Computing (STOC '15), 595–603

Joël Alwen, Susanna F. de Rezende, Jakob Nordström & Marc Vinyals (2017). Cumulative Space in Black-White Pebbling and Resolution. In Proceedings of the 8th Innovations in Theoretical Computer Science Conference (ITCS '17), volume 67 of Leibniz International Proceedings in Informatics (LIPIcs), 38:1–38:21

Albert Atserias & Tuomas Hakoniemi (2019). Size-Degree Trade- Offs for Sums-of-Squares and Positivstellensatz Proofs. In Proceedings of the 34th Annual Computational Complexity Conference (CCC '19), volume 137 of Leibniz International Proceedings in Informatics (LIPIcs), 24:1–24:20

Albert Atserias, Massimo Lauria & Jakob Nordström (2016). Narrow Proofs May Be Maximally Long. ACM Transactions on Computational Logic 17(3), 19:1–19:30. Preliminary version in CCC '14

Albert Atserias & Joanna Ochremiak (2019). Proof Complexity Meets Algebra. ACM Transactions on Computational Logic 20, 1:1–1:46. Preliminary version in ICALP '17

Paul Beame, Stephen A. Cook, Jeff Edmonds, Russell Impagliazzo & Toniann Pitassi (1998). The Relative Complexity of NP Search Problems. Journal of Computer and System Sciences 57(1), 3–19. Preliminary version in STOC '95

Paul Beame, Russell Impagliazzo, Jan Krajíček, Toniann Pitassi & Pavel Pudlák (1994). Lower Bounds on Hilbert's Nullstellensatz and Propositional Proofs. In Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science (FOCS '94), 794–806

Chris Beck, Jakob Nordström & Bangsheng Tang (2013). Some Trade-off Results for Polynomial Calculus. In Proceedings of the 45th Annual ACM Symposium on Theory of Computing (STOC '13), 813–822

Eli Ben-Sasson (2009). Size-Space Tradeoffs for Resolution. SIAM Journal on Computing 38(6), 2511–2525. Preliminary version in STOC '02

Eli Ben-Sasson & Jakob Nordström (2008). Short Proofs May Be Spacious: An Optimal Separation of Space and Length in Resolution. In Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS '08), 709–718

Eli Ben-Sasson & Jakob Nordström (2011). Understanding Space in Proof Complexity: Separations and Trade-offs via Substitutions. In Proceedings of the 2nd Symposium on Innovations in Computer Science (ICS '11), 401–416

Eli Ben-Sasson & Avi Wigderson (2001). Short Proofs are Narrow—Resolution Made Simple. Journal of the ACM 48(2), 149–169. Preliminary version in STOC '99

Bennett, Charles H.: Logical Reversibility of Computation. IBM Journal of Research and Development

**17**(6), 525–532 (1973)Bennett, Charles H.: Time/Space Trade-offs for Reversible Computation. SIAM Journal on Computing

**18**(4), 766–776 (1989)Christoph Berkholz (2018). The Relation between Polynomial Calculus, Sherali-Adams, and Sum-of-Squares Proofs. In Proceedings of the 35th Symposium on Theoretical Aspects of Computer Science (STACS '18), volume 96 of Leibniz International Proceedings in Informatics (LIPIcs), 11:1–11:14

Archie Blake (1937). Canonical Expressions in Boolean Algebra. Ph.D. thesis, University of Chicago

Harry Buhrman, John Tromp & Paul Vitányi (2001). Time and Space Bounds for Reversible Simulation. Journal of Physics A: Mathematical and general 34, 6821–6830. Preliminary version in ICALP '01

Joshua Buresh-Oppenheim, Matthew Clegg, Russell Impagliazzo & Toniann Pitassi (2002). Homogenization and the Polynomial Calculus. Computational Complexity 11(3-4), 91–108. Preliminary version in ICALP '00

Samuel R. Buss (1998). Lower Bounds on Nullstellensatz Proofs via Designs. In Proof Complexity and Feasible Arithmetics, volume 39 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 59–71. American Mathematical Society. Available at http://www.math.ucsd.edu/~sbuss/ResearchWeb/designs/

Buss, Samuel R., Impagliazzo, Russell, Krajíček, Jan, Pudlák, Pavel, Razborov, Alexander A., Sgall, Jiří: Proof Complexity in Algebraic Systems and Bounded Depth Frege Systems with Modular Counting. Computational Complexity

**6**(3), 256–298 (1997)Samuel R. Buss & Toniann Pitassi (1998). Good Degree Bounds on Nullstellensatz Refutations of the Induction Principle. Journal of Computer and System Sciences 2(57), 162–171. Preliminary version in CCC '96

David A. Carlson & John E. Savage (1980). Graph Pebbling with Many Free Pebbles Can Be Difficult. In Proceedings of the 12th Annual ACM Symposium on Theory of Computing (STOC '80), 326–332

Carlson, David A., Savage, John E.: Extreme Time-Space Tradeoffs for Graphs with Small Space Requirements. Information Processing Letters

**14**(5), 223–227 (1982)Siu Man Chan, Massimo Lauria, Jakob Nordström & Marc Vinyals (2015). Hardness of Approximation in PSPACE and Separation Results for Pebble Games (Extended Abstract). In Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS '15), 466–485

Siu Man Chan & Aaron Potechin (2014). Tight Bounds for Monotone Switching Networks via Fourier Analysis. Theory of Computing 10, 389–419. Preliminary version in STOC '12

Ashok K. Chandra (1973). Efficient Compilation of Linear Recursive Programs. In Proceedings of the 14th Annual Symposium on Switching and Automata Theory (SWAT '73), 16–25

Matthew Clegg, Jeffery Edmonds & Russell Impagliazzo (1996). Using the Groebner Basis Algorithm to Find Proofs of Unsatisfiability. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing (STOC '96), 174–183

Stephen A. Cook (1974). An Observation on Time-Storage Trade Off. Journal of Computer and System Sciences 9(3), 308–316. Preliminary version in STOC '73

Dantchev, Stefan S., Martin, Barnaby, Rhodes, Martin: Tight Rank Lower Bounds for the Sherali-Adams Proof System. Theoretical Computer Science

**410**(21–23), 2054–2063 (2009)Susanna F. de Rezende, Or Meir, Jakob Nordström, Toniann Pitassi, Robert Robere & Marc Vinyals (2020). Lifting with Simple Gadgets and Applications to Circuit and Proof Complexity. In Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science (FOCS '20). To appear

Susanna F. de Rezende, Jakob Nordström, Or Meir & Robert Robere (2019). Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling. In Proceedings of the 34th Annual Computational Complexity Conference (CCC '19), volume 137 of Leibniz International Proceedings in Informatics (LIPIcs), 18:1–18:16

Cynthia Dwork, Moni Naor & Hoeteck Wee (2005). Pebbling and Proofs of Work. In Proceedings of the 25th Annual International Cryptology Conference (CRYPTO '05), volume 3621 of Lecture Notes in Computer Science, 37–54. Springer

Yuval Filmus, Toniann Pitassi, Robert Robere & Stephen A Cook (2013). Average Case Lower Bounds for Monotone Switching Networks. In Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS '13), 598–607

Ofer Gabber & Zvi Galil: Explicit Constructions of Linear- Sized Superconcentrators. Journal of Computer and System Sciences

**22**(3), 407–420 (1981)John R. Gilbert, Thomas Lengauer & Robert Endre Tarjan (1980). The Pebbling Problem is Complete in Polynomial Space. SIAM Journal on Computing 9(3), 513–524. Preliminary version in STOC '79

Dima Grigoriev, Edward A. Hirsch & Dmitrii V. Pasechnik (2002). Exponential Lower Bound for Static Semi-algebraic Proofs. In Proceedings of the 29th International Colloquium on Automata, Languages and Programming (ICALP '02), volume 2380 of Lecture Notes in Computer Science, 257–268. Springer

Mika Göös, Pritish Kamath, Robert Robere & Dmitry Sokolov (2019). Adventures in Monotone Complexity and TFNP. In Proceedings of the 10th Innovations in Theoretical Computer Science Conference (ITCS '19), volume 124 of Leibniz International Proceedings in Informatics (LIPIcs), 38:1–38:19

Isidore Heller & Charles B. Tompkins (1957). An Extension of a Theorem of Dantzig's. In Linear Inequalities and Related Systems. (AM-38), Annals of Mathematics Studies, 247–254. Princeton University Press

John Hopcroft, Wolfgang Paul & Leslie Valiant (1977). On Time Versus Space. Journal of the ACM 24(2), 332–337. Preliminary version in FOCS '75

Impagliazzo, Russell, Pudlák, Pavel, Sgall, Jiří: Lower Bounds for the Polynomial Calculus and the Gröbner Basis Algorithm. Computational Complexity

**8**(2), 127–144 (1999)Arist Kojevnikov & Dmitry Itsykson (2006). Lower Bounds of Static Lovász–Schrijver Calculus Proofs for Tseitin Tautologies. In Proceedings of the 33rd International Colloquium on Automata, Languages and Programming (ICALP '06), volume 4051 of Lecture Notes in Computer Science, 323–334. Springer

Komarath, Balagopal, Sarma, Jayalal, Sawlani, Saurabh: Pebbling meets coloring: Reversible pebble game on trees. Journal of Computer and System Sciences

**91**, 33–41 (2018)Královič, Richard: Time and Space Complexity of Reversible Pebbling. RAIRO - Theoretical Informatics and Applications

**38**(02), 137–161 (2004)Guillaume Lagarde, Jakob Nordström, Dmitry Sokolov & Joseph Swernofsky (2020). Trade-offs Between Size and Degree in Polynomial Calculus. In Proceedings of the 11th Innovations in Theoretical Computer Science Conference (ITCS '20), volume 151 of Leibniz International Proceedings in Informatics (LIPIcs), 72:1–72:16

Lange, Klaus-Jörn, McKenzie, Pierre, Tapp, Alain: Reversible Space Equals Deterministic Space. Journal of Computer and System Sciences

**60**(2), 354–367 (2000)Thomas Lengauer & Robert Endre Tarjan (1982). Asymptotically Tight Bounds on Time-Space Trade-offs in a Pebble Game. Journal of the ACM 29(4), 1087–1130. Preliminary version in STOC '79

Levin, Robert Y., Sherman, Alan T.: A Note on Bennett's Time-Space Tradeoff for Reversible Computation. SIAM Journal on Computing

**19**(4), 673–677 (1990)Li, Ming, Tromp, John, Vitányi, Paul: Reversible Simulation of Irreversible Computation. Physica D: Nonlinear Phenomena

**120**(1–2), 168–176 (1998)Ming Li & Paul Vitányi: Reversibility and Adiabatic Computation: Trading Time and Space for Energy. Proceedings of the Royal Society of London, Series A

**452**(1947), 769–789 (1996)De Loera, Jesús A., Lee, Jon, Margulies, Susan, Onn, Shmuel: Expressing Combinatorial Problems by Systems of Polynomial Equations and Hilbert's Nullstellensatz. Combinatorics, Probability and Computing

**18**(4), 551–582 (2009)Giulia Meuli, Mathias Soeken, Martin Roetteler, Nikolaj Bjørner & Giovanni De Micheli (2019). Reversible Pebbling Game for Quantum Memory Management. In Proceedings of the Design, Automation & Test in Europe Conference & Exhibition (DATE '19), 288–291

Nordström, Jakob: A Simplified Way of Proving Tradeoff Results for Resolution. Information Processing Letters

**109**(18), 1030–1035 (2009)Jakob Nordström (2012). On the Relative Strength of Pebbling and Resolution. ACM Transactions on Computational Logic 13(2), 16:1–16:43. Preliminary version in CCC '10

Jakob Nordström (2013). Pebble Games, Proof Complexity and Time-Space Trade-offs. Logical Methods in Computer Science 9(3), 15:1–15:63

Jakob Nordström (2020). New Wine into Old Wineskins: A Survey of Some Pebbling Classics with Supplemental Results. Manuscript in preparation. To appear in Foundations and Trends in Theoretical Computer Science. Current draft version available at http://www.csc.kth.se/~jakobn/research/

Michael S. Paterson & Carl E. Hewitt (1970). Comparative Schematology. In Record of the Project MAC Conference on Concurrent Systems and Parallel Computation, 119–127

Pippenger, Nicholas: Superconcentrators. SIAM Journal on Computing

**6**(2), 298–304 (1977)Nicholas Pippenger (1980). Pebbling. Technical Report RC8258, IBM Watson Research Center. In Proceedings of the 5th IBM Symposium on Mathematical Foundations of Computer Science, Japan

Toniann Pitassi & Robert Robere (2017). Strongly Exponential Lower Bounds for Monotone Computation. In Proceedings of the 49th Annual ACM Symposium on Theory of Computing (STOC '17), 1246–1255

Toniann Pitassi & Robert Robere (2018). Lifting Nullstellensatz to Monotone Span Programs over Any Field. In Proceedings of the 50th Annual ACM Symposium on Theory of Computing (STOC '18), 1207–1219

Aaron Potechin (2010). Bounds on Monotone Switching Networks for Directed Connectivity. In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS '10), 553–562

Pavel Pudlák & Jiří Sgall (1998). Algebraic Models of Computation and Interpolation for Algebraic Proof Systems. In Proof Complexity and Feasible Arithmetics, volume 39 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 279–296. American Mathematical Society. Available at http://users.math.cas.cz/~pudlak/span.pdf

Robert Robere, Toniann Pitassi, Benjamin Rossman & Stephen A. Cook (2016). Exponential Lower Bounds for Monotone Span Programs. In Proceedings of the 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS '16), 406–415

John E. Savage (1998). Models of Computation: Exploring the Power of Computing. Addison-Wesley. Available at http://www.modelsofcomputation.org

John E. Savage & Sowmitri Swamy (1979). Space-Time Tradeoffs for Oblivious Integer Multiplications. In Proceedings of the 6th International Colloquium on Automata, Languages and Programming (ICALP '79), 498–504

Sethi, Ravi: Complete Register Allocation Problems. SIAM Journal on Computing

**4**(3), 226–248 (1975)John E. Savage & Sowmitri Swamy (1978). Space-Time Trade-offs on the FFT-algorithm. IEEE Transactions on Information Theory 24(5), 563–568

Swamy, Sowmitri, Savage, John E.: Space-Time Tradeoffs for Linear Recursion. Mathematical Systems Theory

**16**(1), 9–27 (1983)Thapen, Neil: A Trade-off Between Length and Width in Resolution. Theory of Computing

**12**(5), 1–14 (2016)Martin Tompa (1978). Time-Space Tradeoffs for Computing Functions, Using Connectivity Properties of Their Circuits. In Proceedings of the 10th Annual ACM Symposium on Theory of Computing (STOC '78), 196–204

Jacobo Torán & Florian Wörz (2020). Reversible Pebble Games and the Relation Between Tree-Like and General Resolution Space. In Proceedings of the 37th International Symposium on Theoretical Aspects of Computer Science (STACS '20), volume 154 of Leibniz International Proceedings in Informatics (LIPIcs), 60:1–60:18

Ryan Williams (2000). Space-Efficient Reversible Simulations. Technical report, Cornell University. Available at http://web.stanford.edu/~rrwill/spacesim9_22.pdf

## Acknowledgements

We are grateful for many interesting discussions about matters pebbling-related (and not-so-pebbling-related) with Arkadev Chattopadhyay, Toniann Pitassi, and Marc Vinyals. We would also like to thank the anonymous reviewers for suggestions that improved the presentation of this work, and in particular for suggesting the perspective of totally unimodular matrices.

This work was mostly carried out while the authors were visiting the Simons Institute for the Theory of Computing in association with the DIMACS/Simons Collaboration on Lower Bounds in Computational Complexity, which is conducted with support from the National Science Foundation. Or Meir was supported by the Israel Science Foundation (grant No. 1445/16). Robert Robere was supported by NSERC, and also conducted part of this work at DIMACS with support from the National Science Foundation under grant number CCF-1445755. Susanna F. de Rezende and Jakob Nordström were supported by the Knut and Alice Wallenberg grant KAW 2016.0066 Approximation and Proof Complexity. Jakob Nordström was also supported by the Swedish Research Council grant 2016-00782 and by the Independent Research Fund Denmark grant 9040-00389B. A preliminary version (de Rezende et al. 2019) of this work appeared in CCC 2019.

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De Rezende, S.F., Meir, O., Nordström, J. *et al.* Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling.
*comput. complex.* **30**, 4 (2021). https://doi.org/10.1007/s00037-020-00201-y

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DOI: https://doi.org/10.1007/s00037-020-00201-y

### Keywords

- Proof complexity
- Nullstellensatz
- Trade-offs
- Pebbling

### Subject classification

- 68Q17