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Non-Commutative Formulas and Frege Lower Bounds: a New Characterization of Propositional Proofs

Fu Li, Iddo Tzameret, Zhengyu Wang, Marc Herbstritt
2015 Computational Complexity Conference  
This gives a characterization of propositional Frege proofs in terms of (non-commutative) arithmetic formulas that is tighter than (the formula version of IPS) in Grochow and Pitassi [10], in the following  ...  The argument is a characterization of Frege proofs as non-commutative formulas: we show that the Frege system is (quasi-) polynomially equivalent to a non-commutative Ideal Proof System (IPS), following  ...  This gives a fairly compelling and simple new characterization of the proof complexity of propositional Frege proofs.  ... 
doi:10.4230/lipics.ccc.2015.412 dblp:conf/coco/LiTW15 fatcat:hhbf52sujbehvgumh62uolm4y4

On the proof complexity of logics of bounded branching [article]

Emil Jeřábek
2022 arXiv   pre-print
We investigate the proof complexity of extended Frege (EF) systems for basic transitive modal logics (K4, S4, GL, ...) augmented with the bounded branching axioms 𝐁𝐁_k.  ...  Next, we use this characterization to prove superpolynomial (or even exponential, with stronger hypotheses) separations between EF and substitution Frege (SF) systems for all transitive logics contained  ...  Acknowledgements I want to thank Pavel Pudlák and Pavel Hrubeš for clarifying discussion, and the anonymous reviewer for helpful suggestions to improve the presentation.  ... 
arXiv:2004.11282v2 fatcat:xw2sehl4org25fvdhfuolr52ty

Proof complexity in algebraic systems and bounded depth Frege systems with modular counting

S. Buss, R. Impagliazzo, J. Krajíček, P. Pudlák, A. A. Razborov, J. Sgall
1996 Computational Complexity  
As a corollary, using Beame et al. (1994) we obtain a lower bound of the form 2 N Ω(1) for the number of formulas in a constant-depth Frege proof of the modular counting principle Count N q from instances  ...  Further we show that a lower bound for proofs in a bounded depth Frege system in the language with the modular counting connective MOD p follows from a lower bound on the degree of Nullstellensatz proofs  ...  Sgall were partially supported by grants A119107 and A1019602 of the AVČR; and A. A. Razborov was partially supported by RBRF grant 93-011-16015 and by an AMS-FSU grant.  ... 
doi:10.1007/bf01294258 fatcat:pa354ea7sbatdc4vfzudhizm6i

The canonical pairs of bounded depth Frege systems [article]

Pavel Pudlak
2019 arXiv   pre-print
The canonical pair of a proof system P is the pair of disjoint NP sets where one set is the set of all satisfiable CNF formulas and the other is the set of CNF formulas that have P-proofs bounded by some  ...  We give a combinatorial characterization of the canonical pairs of depth d Frege systems.  ...  Bounded depth Frege is a well-studied hierarchy of proof systems above Resolution.  ... 
arXiv:1912.03013v1 fatcat:emhofukqindg7okszftfebtfxu

Algebraic Proof Complexity: Progress, Frontiers and Challenges [article]

Tonnian Pitassi, Iddo Tzameret
2016 arXiv   pre-print
In particular, we focus on tight connections between proof complexity lower bounds (namely, lower bounds on the size of proofs of certain tautologies), algebraic circuit lower bounds, and the Polynomial  ...  open questions, solutions to old problems, and new directions of research.  ...  Acknowledgements We thank Stephen Cook, Kaveh Ghasemloo, Amir Shpilka and Avi Wigderson for very helpful discussions and clarifications, and Michael Forbes for very useful discussions and comments on a  ... 
arXiv:1607.00443v1 fatcat:xdihpreuvfbyrnpkkxxv6znnra

Characterizing Propositional Proofs as Noncommutative Formulas

Fu Li, Iddo Tzameret, Zhengyu Wang
2018 SIAM journal on computing (Print)  
Here, a propositional-calculus (i.e., Frege) proof is any proof starting from a set of axioms and deriving new Boolean formulas using a fixed set of sound derivation rules.  ...  This also gives a characterization of propositional Frege proofs in terms of (non-commutative) arithmetic formulas that is tighter than (the formula version of IPS) in Grochow and Pitassi [FOCS 2014].  ...  Our results thus give a compelling and simple new characterization of the proof complexity of propositional Frege proofs and brings new hope for achieving lower bounds on strong proof systems, by reducing  ... 
doi:10.1137/16m1107632 fatcat:wlvtdrtze5bmpbsc7l5qja7ua4

Characterizing Propositional Proofs as Non-Commutative Formulas [article]

Fu Li, Iddo Tzameret, Zhengyu Wang
2015 arXiv   pre-print
Frege) proof is a proof starting from a set of axioms and deriving new Boolean formulas using a set of fixed sound derivation rules.  ...  In this work we show that Frege lower bounds in fact follow from corresponding size lower bounds on non-commutative formulas computing certain polynomials (and that such lower bounds on non-commutative  ...  Our results thus give a compelling and simple new characterization of the proof complexity of propositional Frege proofs and brings new hope for achieving lower bounds on strong proof systems, by reducing  ... 
arXiv:1412.8746v4 fatcat:aojz652g4jdi3flmefoaxxmrmi

Iterated lower bound formulas: a diagonalization-based approach to proof complexity

Rahul Santhanam, Iddo Tzameret
2021 Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing  
More generally, for any strong enough propositional proof system R we propose a new explicit hard candidate, the iterated R-lower bound formulas, which inductively asserts the non-existence of short R  ...  Our proof of this fact uses the implication from IPS lower bounds to algebraic complexity lower bounds due to Grochow and Pitassi together with a diagonalization argument: the formulas φ n themselves assert  ...  We also thank Jan Krajíček for helpful comments on an earlier draft of this work and anonymous reviewers of this work for very useful comments that improved the exposition.  ... 
doi:10.1145/3406325.3451010 fatcat:n7c3szaru5awtkpbe4jqni7jwy

Ideals, Determinants, and Straightening: Proving and Using Lower Bounds for Polynomial Ideals [article]

Robert Andrews, Michael A. Forbes
2021 arXiv   pre-print
This extends the recent breakthrough low-depth circuit lower bounds of Limaye, Srinivasan, and Tavenas to the setting of proof complexity.  ...  We give several applications of our result, two of which are highlighted below. ∙ We prove super-polynomial lower bounds for Ideal Proof System refutations computed by low-depth circuits.  ...  imply a lower bound on the number of proof lines.  ... 
arXiv:2112.00792v1 fatcat:etstvbels5atfmflir7xyxsiki

How to lie without being (easily) convicted and the lengths of proofs in propositional calculus [chapter]

Pavel Pudlák, Samuel R. Buss
1995 Lecture Notes in Computer Science  
We shall describe two general methods for proving lower bounds on the lengths of proofs in propositional calculus and give examples of such lower bounds.  ...  For the first method, a log n + log log n − O(log log log n) lower bound is given on the lengths of interactive proofs of certain permutation tautologies.  ...  This gives a Ω(n log n) lower bound on the length of tree-like Frege proofs of the tautologies s n,π . Theorem 2.  ... 
doi:10.1007/bfb0022253 fatcat:msdzmo6nqrf2barxsxxsiqob2u

Algebraic Proofs over Noncommutative Formulas [article]

Iddo Tzameret
2010 arXiv   pre-print
We conclude by proposing an approach for establishing lower bounds on PC over ordered formulas proofs, and related systems, based on properties of lower bounds on noncommutative formulas.  ...  We observe that a simple formulation gives rise to systems at least as strong as Frege---yielding a semantic way to define a Cook-Reckhow (i.e., polynomially verifiable) algebraic analog of Frege proofs  ...  I also wish to thank Ran Raz for suggesting this research direction, and Jan Krajíček for inviting me to give a talk at TAMC 2010 on this subject.  ... 
arXiv:1004.2159v2 fatcat:y4bunmfvcfevthvl64zhge3hue

Proof Complexity of Substructural Logics [article]

Raheleh Jalali
2020 arXiv   pre-print
The lower bound also extends to the number of proof-lines (proof-lengths) in any Frege system (extended Frege system) for a logic between 𝖥𝖫 and any infinite branching super-intuitionistic logic.  ...  lower bound on the proof lengths.  ...  I am very grateful to Rosalie Iemhoff and thankful for the hospitality of the Department of Philosophy of Utrecht University where part of this research was done while I was visiting there.  ... 
arXiv:2006.09705v2 fatcat:smp22re7d5g2vmm26fxmyf4dee

Proof complexity of substructural logics

Raheleh Jalali
2021 Annals of Pure and Applied Logic  
The lower bound also extends to the number of proof lines (proof lengths) in any Frege system (extended Frege system) for a logic between FL and any superintuitionistic logic of infinite branching.  ...  lower bound on the proof lengths.  ...  I am very grateful to Rosalie Iemhoff and thankful for the hospitality of the Department of Philosophy of Utrecht University where part of this research was done while I was visiting there.  ... 
doi:10.1016/j.apal.2021.102972 fatcat:toqgduxj3vc7doayfhucosxas4

Circuit Complexity, Proof Complexity, and Polynomial Identity Testing

Joshua A. Grochow, Toniann Pitassi
2014 2014 IEEE 55th Annual Symposium on Foundations of Computer Science  
AC 0 [p]-Frege cannot efficiently prove the depth d PIT axioms, and hence we have a lower bound on AC 0 [p]-Frege.  ...  We introduce a new and very natural algebraic proof system, which has tight connections to (algebraic) circuit complexity.  ...  We thank Pascal Koiran for providing the second half of the proof of Proposition 2.4. We thank Iddo Tzameret for useful discussions that led to Proposition 2.2.  ... 
doi:10.1109/focs.2014.20 dblp:conf/focs/GrochowP14 fatcat:mhmjd72cr5gmfdgtfl4zgsfhsi

Circuit complexity, proof complexity, and polynomial identity testing [article]

Joshua A. Grochow, Toniann Pitassi
2014 arXiv   pre-print
the difficulty of lower bounds on AC^0[p]-Frege, or b) AC^0[p]-Frege cannot efficiently prove the depth d PIT axioms, and hence we have a lower bound on AC^0[p]-Frege.  ...  We introduce a new algebraic proof system, which has tight connections to (algebraic) circuit complexity.  ...  We thank Pascal Koiran for providing the second half of the proof of Proposition 2.4. We thank Iddo Tzameret for useful discussions that led to Proposition 2.2.  ... 
arXiv:1404.3820v1 fatcat:2nerc6uatvdh5pqwtrhelt6jvq
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