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1. Automating algebraic proof systems is NP-Hard

   Göös, M ; Nordström, J ; Pitassi, T ; Robere, R ; de Rezende, Susanna F ; Sokolov, D. ( May 2020 , Electronic colloquium on computational complexity)

   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 Muller (FOCS 2019) that established an analogous result for Resolution. 

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2. On Disperser/Lifting Properties of the Index and Inner-Product Functions

   Beame, Paul ; Koroth, Sajin ( January 2023 , 14th Innovations in Theoretical Computer Science Conference (ITCS 2023))

   Query-to-communication lifting theorems, which connect the query complexity of a Boolean function to the communication complexity of an associated “lifted” function obtained by composing the function with many copies of another function known as a gadget, have been instrumental in resolving many open questions in computational complexity. A number of important complexity questions could be resolved if we could make substantial improvements in the input size required for lifting with the Index function, which is a universal gadget for lifting, from its current near-linear size down to polylogarithmic in the number of inputs N of the original function or, ideally, constant. The near-linear size bound was recently shown by Lovett, Meka, Mertz, Pitassi and Zhang [20] using a recent breakthrough improvement on the Sunflower Lemma to show that a certain graph associated with an Index function of that size is a disperser. They also stated a conjecture about the Index function that is essential for further improvements in the size required for lifting with Index using current techniques. In this paper we prove the following; - The conjecture of Lovett et al. is false when the size of the Index gadget is less than logarithmic in N . - The same limitation applies to the Inner-Product function. More precisely, the Inner-Product function, which is known to satisfy the disperser property at size O(log N ), also does not have this property when its size is less than log N . - Notwithstanding the above, we prove a lifting theorem that applies to Index gadgets of any size at least 4 and yields lower bounds for a restricted class of communication protocols in which one of the players is limited to sending parities of its inputs. - Using a modification of the same idea with improved lifting parameters we derive a strong lifting theorem from decision tree size to parity decision tree size. We use this, in turn, to derive a general lifting theorem in proof complexity from tree-resolution size to tree-like Res(⊕) refutation size, which yields many new exponential lower bounds on such proofs. 

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3. On Disperser/Lifting Properties of the Index and Inner-Product Functions

   Beame, Paul ; Koroth, Sajin ( December 2022 , arXivorg)

   Query-to-communication lifting theorems, which connect the query complexity of a Boolean function to the communication complexity of an associated ‘lifted’ function obtained by composing the function with many copies of another function known as a gadget, have been instrumental in resolving many open questions in computational complexity. A number of important complexity questions could be resolved if we could make substantial improvements in the input size required for lifting with the Index function, which is a universal gadget for lifting, from its current near-linear size down to polylogarithmic in the number of inputs N of the original function or, ideally, constant. The near-linear size bound was recently shown by Lovett, Meka, Mertz, Pitassi and Zhang using a recent breakthrough improvement on the Sunflower Lemma to show that a certain graph associated with an Index function of that size is a disperser. They also stated a conjecture about the Index function that is essential for further improvements in the size required for lifting with Index using current techniques. In this paper we prove the following; • The conjecture of Lovett et al. is false when the size of the Index gadget is less than logarithmic in N. • The same limitation applies to the Inner-Product function. More precisely, the Inner-Product function, which is known to satisfy the disperser property at size O(log N), also does not have this property when its size is less than log N. • Notwithstanding the above, we prove a lifting theorem that applies to Index gadgets of any size at least 4 and yields lower bounds for a restricted class of communication protocols in which one of the players is limited to sending parities of its inputs. • Using a modification of the same idea with improved lifting parameters we derive a strong lifting theorem from decision tree size to parity decision tree size. We use this, in turn, to derive a general lifting theorem in proof complexity from tree-resolution size to tree-like Res(⊕) refutation size, which yields many new exponential lower bounds on such proofs. 

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4. On Disperser/Lifting Properties of the Index and Inner-Product Functions

   Beame, Paul ; Koroth, Sajin ( December 2022 , Electronic colloquium on computational complexity)

   Query-to-communication lifting theorems, which connect the query complexity of a Boolean function to the communication complexity of an associated `lifted' function obtained by composing the function with many copies of another function known as a gadget, have been instrumental in resolving many open questions in computational complexity. Several important complexity questions could be resolved if we could make substantial improvements in the input size required for lifting with the Index function, from its current near-linear size down to polylogarithmic in the number of inputs N of the original function or, ideally, constant. The near-linear size bound was shown by Lovett, Meka, Mertz, Pitassi and Zhang using a recent breakthrough improvement on the Sunflower Lemma to show that a certain graph associated with the Index function of near-linear size is a disperser. They also stated a conjecture about the Index function that is essential for further improvements in the size required for lifting with Index using current techniques. In this paper we prove the following; 1) The conjecture of Lovett et al. is false when the size of the Index gadget is logN−\omega(1). 2) Also, the Inner-Product function, which satisfies the disperser property at size O(logN), does not have this property when its size is log N−\omega(1). 3) Nonetheless, using Index gadgets of size at least 4, we prove a lifting theorem for a restricted class of communication protocols in which one of the players is limited to sending parities of its inputs. 4) Using the ideas from this lifting theorem, we derive a strong lifting theorem from decision tree size to parity decision tree size. We use this to derive a general lifting theorem in proof complexity from tree-resolution size to tree-like Res(\oplus) refutation size, which yields many new exponential lower bounds on such proofs. 

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5. Automating Regular or Ordered Resolution is NP-Hard

   Bell, Zoe ( July 2020 , Electronic colloquium on computational complexity)

   We show that is hard to find regular or even ordered (also known as Davis-Putnam) Resolution proofs, extending the breakthrough result for general Resolution from Atserias and Muller to these restricted forms. Namely, regular and ordered Resolution are automatable if and only if P = NP. Specifically, for a CNF formula F the problem of distinguishing between the existence of a polynomial-size ordered Resolution refutation of F and an at least exponential-size general Resolution proof being required to refute F is NP-complete. 

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