| 1. |
- De Rezende, Susanna F., et al.
(författare)
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Automating algebraic proof systems is NP-hard
- 2021
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Ingår i: STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing. - New York, NY, USA : ACM. - 0737-8017. - 9781450380539 ; , s. 209-222
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Konferensbidrag (refereegranskat)abstract
- We show that algebraic proofs are hard to find: Given an unsatisfiable CNF formula F, it is NP-hard to find a refutation of F in the Nullstellensatz, Polynomial Calculus, or Sherali-Adams proof systems in time polynomial in the size of the shortest such refutation. Our work extends, and gives a simplified proof of, the recent breakthrough of Atserias and Müller (JACM 2020) that established an analogous result for Resolution.
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| 2. |
- De Rezende, Susanna F., et al.
(författare)
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KRW composition theorems via lifting
- 2020
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Ingår i: Proceedings - 2020 IEEE 61st Annual Symposium on Foundations of Computer Science, FOCS 2020. - 0272-5428. - 9781728196220 - 9781728196213 ; 2020-November, s. 43-49
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Konferensbidrag (refereegranskat)abstract
- One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., mathrm{P} nsubseteq text{NC}{1}). Karchmer, Raz, and Wigderson [13] suggested to approach this problem by proving that depth complexity behaves'as expected' with respect to the composition of functions f diamond g. They showed that the validity of this conjecture would imply that mathrm{P} nsubseteq text{NC}{1}. Several works have made progress toward resolving this conjecture by proving special cases. In particular, these works proved the KRW conjecture for every outer function, but only for few inner functions. Thus, it is an important challenge to prove the KRW conjecture for a wider range of inner functions. In this work, we extend significantly the range of inner functions that can be handled. First, we consider the monotone version of the KRW conjecture. We prove it for every monotone inner function whose depth complexity can be lower bounded via a query-to-communication lifting theorem. This allows us to handle several new and well-studied functions such as the s-t-connectivity, clique, and generation functions. In order to carry this progress back to the non-monotone setting, we introduce a new notion of semi-monotone composition, which combines the non-monotone complexity of the outer function with the monotone complexity of the inner function. In this setting, we prove the KRW conjecture for a similar selection of inner functions, but only for a specific choice of the outer function f.
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| 3. |
- de Rezende, Susanna F., 1989-, et al.
(författare)
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Lifting with Simple Gadgets and Applications to Circuit and Proof Complexity
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We significantly strengthen and generalize the theorem lifting Nullstellensatz degree to monotone span program size by Pitassi and Robere (2018) so that it works for any gadget with high enough rank, in particular, for useful gadgets such as equality and greater-than. We apply our generalized theorem to solve two open problems:We present the first result that demonstrates a separation in proof power for cutting planes with unbounded versus polynomially bounded coefficients. Specifically, we exhibit CNF formulas that can be refuted in quadratic length and constant line space in cutting planes with unbounded coefficients, but for which there are no refutations in subexponential length and subpolynomialline space if coefficients are restricted to be of polynomial magnitude.We give the first explicit separation between monotone Boolean formulas and monotone real formulas. Specifically, we give an explicit family of functions that can be computed with monotone real formulas of nearly linear size but require monotone Boolean formulas of exponential size. Previously only a non-explicit separation was known.An important technical ingredient, which may be of independent interest, is that we show that the Nullstellensatz degree of refuting the pebbling formula over a DAG G over any field coincides exactly with the reversible pebbling price of G. In particular, this implies that the standard decision tree complexity and the parity decision tree complexity of the corresponding falsified clause search problem are equal.
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| 4. |
- de Rezende, Susanna F., et al.
(författare)
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Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling
- 2019
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Ingår i: Proceedings of the 34th Annual Computational Complexity Conference (CCC ’19). - : Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. - 9783959771160 ; , s. 18:1-18:16
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Konferensbidrag (refereegranskat)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 an 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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| 5. |
- De Rezende, Susanna F., et al.
(författare)
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Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling
- 2021
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Ingår i: Computational Complexity. - : Springer Science and Business Media LLC. - 1016-3328 .- 1420-8954. ; 30:1
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Tidskriftsartikel (refereegranskat)abstract
- We establish an exactly tight relation between reversiblepebblings 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 formulaover G in size t + 1 and degree s (independently of the field in whichthe Nullstellensatz refutation is made). We use this correspondenceto prove a number of strong size-degree trade-offs for Nullstellensatz,which to the best of our knowledge are the first such results for thisproof system.
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| 6. |
- De Rezende, Susanna, et al.
(författare)
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Lifting with simple gadgets and applications to circuit and proof complexity
- 2020
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Ingår i: Proceedings - 2020 IEEE 61st Annual Symposium on Foundations of Computer Science, FOCS 2020. - 0272-5428. - 9781728196213 - 9781728196220 ; 2020-November, s. 24-30
-
Konferensbidrag (refereegranskat)abstract
- We significantly strengthen and generalize the theorem lifting Nullstellensatz degree to monotone span program size by Pitassi and Robere (2018) so that it works for any gadget with high enough rank, in particular, for useful gadgets such as equality and greater-than. We apply our generalized theorem to solve three open problems: •We present the first result that demonstrates a separation in proof power for cutting planes with unbounded versus polynomially bounded coefficients. Specifically, we exhibit CNF formulas that can be refuted in quadratic length and constant line space in cutting planes with unbounded coefficients, but for which there are no refutations in subexponential length and subpolynomial line space if coefficients are restricted to be of polynomial magnitude. •We give the first explicit separation between monotone Boolean formulas and monotone real formulas. Specifically, we give an explicit family of functions that can be computed with monotone real formulas of nearly linear size but require monotone Boolean formulas of exponential size. Previously only a non-explicit separation was known. •We give the strongest separation to-date between monotone Boolean formulas and monotone Boolean circuits. Namely, we show that the classical GEN problem, which has polynomial-size monotone Boolean circuits, requires monotone Boolean formulas of size 2{Omega(n text{polylog}(n))}. An important technical ingredient, which may be of independent interest, is that we show that the Nullstellensatz degree of refuting the pebbling formula over a DAG G over any field coincides exactly with the reversible pebbling price of G. In particular, this implies that the standard decision tree complexity and the parity decision tree complexity of the corresponding falsified clause search problem are equal. This is an extended abstract. The full version of the paper is available at https://arxiv.org/abs/2001.02144.
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| 7. |
- Rezende, Susanna F.de, et al.
(författare)
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KRW Composition Theorems via Lifting
- 2024
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Ingår i: Computational Complexity. - 1016-3328. ; 33:1
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Tidskriftsartikel (refereegranskat)abstract
- One of the major open problems in complexity theory is proving super-logarithmiclower bounds on the depth of circuits (i.e., P⊈NC1). Karchmer et al. (Comput Complex 5(3/4):191–204, 1995) suggested to approach thisproblem by proving that depth complexity behaves “as expected”with respect to the composition of functions f◊g. They showedthat the validity of this conjecture would imply that P⊈NC1.Several works have made progress toward resolving this conjectureby proving special cases. In particular, these works proved the KRWconjecture for every outer function f, but only for few innerfunctions g. Thus, it is an important challenge to prove the KRWconjecture for a wider range of inner functions.In this work, we extend significantly the range of inner functionsthat can be handled. First, we consider the monotone versionof the KRW conjecture. We prove it for every monotone inner function gwhose depth complexity can be lower-bounded via a query-to-communicationlifting theorem. This allows us to handle several new and well-studiedfunctions such as the s-t-connectivity, clique,and generation functions.In order to carry this progress back to the non-monotone setting,we introduce a new notion of semi-monotone composition, whichcombines the non-monotone complexity of the outer function f withthe monotone complexity of the inner function g. In this setting,we prove the KRW conjecture for a similar selection of inner functions g,but only for a specific choice of the outer function f.
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