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

2015
*
Computational Complexity Conference
*

This gives

doi:10.4230/lipics.ccc.2015.412
dblp:conf/coco/LiTW15
fatcat:hhbf52sujbehvgumh62uolm4y4
*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*. ...##
###
On the proof complexity of logics of bounded branching
[article]

2022
*
arXiv
*
pre-print

We investigate the

arXiv:2004.11282v2
fatcat:xw2sehl4org25fvdhfuolr52ty
*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. ...##
###
Proof complexity in algebraic systems and bounded depth Frege systems with modular counting

1996
*
Computational Complexity
*

As

doi:10.1007/bf01294258
fatcat:pa354ea7sbatdc4vfzudhizm6i
*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. ...##
###
The canonical pairs of bounded depth Frege systems
[article]

2019
*
arXiv
*
pre-print

The canonical pair

arXiv:1912.03013v1
fatcat:emhofukqindg7okszftfebtfxu
*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. ...##
###
Algebraic Proof Complexity: Progress, Frontiers and Challenges
[article]

2016
*
arXiv
*
pre-print

In particular, we focus on tight connections between

arXiv:1607.00443v1
fatcat:xdihpreuvfbyrnpkkxxv6znnra
*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*...##
###
Characterizing Propositional Proofs as Noncommutative Formulas

2018
*
SIAM journal on computing (Print)
*

Here,

doi:10.1137/16m1107632
fatcat:wlvtdrtze5bmpbsc7l5qja7ua4
*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 ...##
###
Characterizing Propositional Proofs as Non-Commutative Formulas
[article]

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 ...

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

2021
*
Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing
*

More generally, for any strong enough

doi:10.1145/3406325.3451010
fatcat:n7c3szaru5awtkpbe4jqni7jwy
*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. ...##
###
Ideals, Determinants, and Straightening: Proving and Using Lower Bounds for Polynomial Ideals
[article]

2021
*
arXiv
*
pre-print

This extends the recent breakthrough low-depth circuit

arXiv:2112.00792v1
fatcat:etstvbels5atfmflir7xyxsiki
*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. ...##
###
How to lie without being (easily) convicted and the lengths of proofs in propositional calculus
[chapter]

1995
*
Lecture Notes in Computer Science
*

We shall describe two general methods for proving

doi:10.1007/bfb0022253
fatcat:msdzmo6nqrf2barxsxxsiqob2u
*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. ...##
###
Algebraic Proofs over Noncommutative Formulas
[article]

2010
*
arXiv
*
pre-print

We conclude by proposing an approach for establishing

arXiv:1004.2159v2
fatcat:y4bunmfvcfevthvl64zhge3hue
*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. ...##
###
Proof Complexity of Substructural Logics
[article]

2020
*
arXiv
*
pre-print

The

arXiv:2006.09705v2
fatcat:smp22re7d5g2vmm26fxmyf4dee
*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. ...##
###
Proof complexity of substructural logics

2021
*
Annals of Pure and Applied Logic
*

The

doi:10.1016/j.apal.2021.102972
fatcat:toqgduxj3vc7doayfhucosxas4
*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. ...##
###
Circuit Complexity, Proof Complexity, and Polynomial Identity Testing

2014
*
2014 IEEE 55th Annual Symposium on Foundations of Computer Science
*

AC 0 [p]-

doi:10.1109/focs.2014.20
dblp:conf/focs/GrochowP14
fatcat:mhmjd72cr5gmfdgtfl4zgsfhsi
*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. ...##
###
Circuit complexity, proof complexity, and polynomial identity testing
[article]

2014
*
arXiv
*
pre-print

the difficulty

arXiv:1404.3820v1
fatcat:2nerc6uatvdh5pqwtrhelt6jvq
*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. ...
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