Periodic Reporting for period 2 - EPRICOT (Efficient Proofs and Computation: A Unified Algebraic Approach)
Berichtszeitraum: 2023-01-01 bis 2024-06-30
We address fundamental hardness problems in the theory of computing, relating to the runtime and size of certain computational models. These are notoriously hard open questions at the heart of computer science, the most famous of which is the P vs NP problem. Throughout the years, from the 1960s till now, several approaches have been proposed to tackle the fundamental hardness questions in complexity. Specifically, we are interested to investigate the NP vs coNP question from the perspective of resource bounded provability, namely Proof Complexity. Our approach is to integrate the theory of algebraic complexity and algebraic computation into the theory of proof complexity. This is a new approach, which was sketched initially by Pitassi 1996, and then taken first steps by Grigoriev and Hirsch 2003 and then the PI and Raz in 2008, 2010. It matured to a new approach in Grochow and Pitassi 2018 (JACM) work which introduced the now standard strong algebraic proof system IPS (for Ideal Proof System).
The importance of these questions stems from their fundamental nature: they reveal the basic nature of efficient computation. Which problems can be efficiently computed and what cannot. In parallel, which statements possess short proofs and which cannot, or equivalently: which problems cannot be efficiently solved with nondeterministic algorithms, namely computational machines that are allowed even making arbitrary guesses. It is known that these questions are important to computer science, science and technology, and society at large, due to their relevance in secure computation (cryptography), randomized algorithms and verification.
We ask more precisely: what is the strength of algebraic proof systems such as IPS and its fragments? Can we prove general lower bounds (i.e. the non-existence of short proofs) against such strong proof system? We know that for formulas in Conjunctive Normal Form (CNF), such general lower bounds would imply breakthrough results like separating the algebraic versions of P from NP (formally, the VP vs VNP questions). While also being a strong step towards separating P from NP (and NP from coNP).
We know also that weaker lower bounds against IPS would constitute a technical breakthrough in proof complexity, by establishing lower bounds against propositional proof systems operating with constant depth Boolean circuits equipped with counting gates. This question is open for over three decades.
The importance of these questions stems from their fundamental nature: they reveal the basic nature of efficient computation. Which problems can be efficiently computed and what cannot. In parallel, which statements possess short proofs and which cannot, or equivalently: which problems cannot be efficiently solved with nondeterministic algorithms, namely computational machines that are allowed even making arbitrary guesses. It is known that these questions are important to computer science, science and technology, and society at large, due to their relevance in secure computation (cryptography), randomized algorithms and verification.
We ask more precisely: what is the strength of algebraic proof systems such as IPS and its fragments? Can we prove general lower bounds (i.e. the non-existence of short proofs) against such strong proof system? We know that for formulas in Conjunctive Normal Form (CNF), such general lower bounds would imply breakthrough results like separating the algebraic versions of P from NP (formally, the VP vs VNP questions). While also being a strong step towards separating P from NP (and NP from coNP).
We know also that weaker lower bounds against IPS would constitute a technical breakthrough in proof complexity, by establishing lower bounds against propositional proof systems operating with constant depth Boolean circuits equipped with counting gates. This question is open for over three decades.
Summary of Achievements to date:
RESULTS RELATED TO OBJECTIVE 2
Immediately after the project started a breakthrough in algebraic complexity by Limaye, Srinivasan and Tavenas (STOC 2021) appeared. It obtained the first super polynomial lower bounds against constant-depth algebraic circuits (over large fields). This was a decades-long open problem. We have studied carefully their result and succeeded in using it to obtain the first constant-depth algebraic super-polynomial proof-size lower bound (over large fields).
Our result was obtained by accommodating Limaye et al.'s result to the Functional Lower Bound method of Forbes et al. It was presented at the FOCS 2022 (Govindasamy, Hakoniemi and Tzameret, "Simple Hard Instances for Low Depth Algebraic Proofs", FOCS 2022).
Continuing on the success of this work, together with Limaye we produced further progress on low-depth algebraic proofs. We obtained major improvements in all aspects of algebraic proofs lower bounds compared to Forbes, Shpilka, Tzameret and Wigderson (ToC'21/CCC'16), and our prior result from FOCS 2022. Specifically, in Tuomas Hakoniemi, Nutan Limaye and Iddo Tzameret, Functional Lower Bounds in Algebraic Proofs: Symmetry, Lifting, and Barriers, to appear in STOC 2024, we show how to use symmetry to demonstrate in a general way hard instances for algebraic proof systems, as follows:
1- All purely symmetric (single formula) instances are hard for Nullstellensatz degree as well as size in many algebraic proof systems (operating with read-once ABPs (algebraic branching programs) and multilinear formulas. This includes also the first lower bounds over *finite fields*. This is the first such lower bounds over finite fields, solving an open problem in the field.
2- Greatly improved low-depth algebraic proof-size lower bounds. This improved the result of my group in FOCS 22 to much stronger proof systems.
3- Barriers: we show that the functional lower bound method, which is the most successful method to date for proving lower bounds against IPS proofs cannot lead to a solution of the major open problems on constant-depth propositional proof systems with counting gates.
The above results match the expectations in Objective 2 (Descr. of Action): a characterisation of hard instances in low-depth algebraic proofs.
A different approach towards lower bounds against propositional proofs with counting gates was taken by Dr Michal Garlik (funded by the project). He established a lower bound against propositional proofs operating with ORs of linear equations (over the two-element fields). Together with Efermenko and Itsykson, Garlik achieved the first lower bounds against this proof system under the restriction of "regularity". This work is to appear in STOC 2024 and has already spawned follow-up works.
RESULTS RELATED TO OBJECTIVE 3
We initiated several structural frameworks in which to study algebraic proof-size lower bounds.
My work with Santhanam (Iterated Lower Bound Formulas: A Diagonalization-Based Approach to Proof Complexity; Santhanam and Tzameret. SIAM Journal on Computing [invited; Under 2nd round review]) introduced a novel approach to proof size lower bounds: we devise a self-referential statement to get new conditional lower bounds. This shows that under the assumption that the permanent polynomial does not admit small algebraic circuits, there are no (short) proofs of certain CNF formulas in some strong proof systems.
Together with my student Luming Zhang, we solved an open problem about the ability to stretch certain kinds of pseudorandom generators against nondeterministic machines ("demi-bits"). This work is relevant to propositional proof size lower bounds, and we apply it also to yield new results in average-case complexity. This work appears in the 15th Innovations in Theoretical Computer Science Conference (ITCS), 2024 ("Stretching Demi-Bits and Nondeterministic-Secure Pseudorandomness." Tzameret, Iddo; Zhang, Lu-Ming).
RESULTS RELATED TO OBJECTIVE 2
Immediately after the project started a breakthrough in algebraic complexity by Limaye, Srinivasan and Tavenas (STOC 2021) appeared. It obtained the first super polynomial lower bounds against constant-depth algebraic circuits (over large fields). This was a decades-long open problem. We have studied carefully their result and succeeded in using it to obtain the first constant-depth algebraic super-polynomial proof-size lower bound (over large fields).
Our result was obtained by accommodating Limaye et al.'s result to the Functional Lower Bound method of Forbes et al. It was presented at the FOCS 2022 (Govindasamy, Hakoniemi and Tzameret, "Simple Hard Instances for Low Depth Algebraic Proofs", FOCS 2022).
Continuing on the success of this work, together with Limaye we produced further progress on low-depth algebraic proofs. We obtained major improvements in all aspects of algebraic proofs lower bounds compared to Forbes, Shpilka, Tzameret and Wigderson (ToC'21/CCC'16), and our prior result from FOCS 2022. Specifically, in Tuomas Hakoniemi, Nutan Limaye and Iddo Tzameret, Functional Lower Bounds in Algebraic Proofs: Symmetry, Lifting, and Barriers, to appear in STOC 2024, we show how to use symmetry to demonstrate in a general way hard instances for algebraic proof systems, as follows:
1- All purely symmetric (single formula) instances are hard for Nullstellensatz degree as well as size in many algebraic proof systems (operating with read-once ABPs (algebraic branching programs) and multilinear formulas. This includes also the first lower bounds over *finite fields*. This is the first such lower bounds over finite fields, solving an open problem in the field.
2- Greatly improved low-depth algebraic proof-size lower bounds. This improved the result of my group in FOCS 22 to much stronger proof systems.
3- Barriers: we show that the functional lower bound method, which is the most successful method to date for proving lower bounds against IPS proofs cannot lead to a solution of the major open problems on constant-depth propositional proof systems with counting gates.
The above results match the expectations in Objective 2 (Descr. of Action): a characterisation of hard instances in low-depth algebraic proofs.
A different approach towards lower bounds against propositional proofs with counting gates was taken by Dr Michal Garlik (funded by the project). He established a lower bound against propositional proofs operating with ORs of linear equations (over the two-element fields). Together with Efermenko and Itsykson, Garlik achieved the first lower bounds against this proof system under the restriction of "regularity". This work is to appear in STOC 2024 and has already spawned follow-up works.
RESULTS RELATED TO OBJECTIVE 3
We initiated several structural frameworks in which to study algebraic proof-size lower bounds.
My work with Santhanam (Iterated Lower Bound Formulas: A Diagonalization-Based Approach to Proof Complexity; Santhanam and Tzameret. SIAM Journal on Computing [invited; Under 2nd round review]) introduced a novel approach to proof size lower bounds: we devise a self-referential statement to get new conditional lower bounds. This shows that under the assumption that the permanent polynomial does not admit small algebraic circuits, there are no (short) proofs of certain CNF formulas in some strong proof systems.
Together with my student Luming Zhang, we solved an open problem about the ability to stretch certain kinds of pseudorandom generators against nondeterministic machines ("demi-bits"). This work is relevant to propositional proof size lower bounds, and we apply it also to yield new results in average-case complexity. This work appears in the 15th Innovations in Theoretical Computer Science Conference (ITCS), 2024 ("Stretching Demi-Bits and Nondeterministic-Secure Pseudorandomness." Tzameret, Iddo; Zhang, Lu-Ming).
All results to date clearly are advancing the research field of proof complexity and complexity in general beyond the state-of-the art.
low depth algebraic proof lower bounds were unexpected as explained above, because they build on a recent breakthrough by Limaye et al. from 2021.
low depth algebraic proof lower bounds were unexpected as explained above, because they build on a recent breakthrough by Limaye et al. from 2021.