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Better Lower Bounds for Monotone Threshold Formulas

1997
*
Journal of computer and system sciences (Print)
*

Snir's result implies an 0(kn log(nÂ(k&1)) lower bound

doi:10.1006/jcss.1997.1287
fatcat:pp4eh4quefdbdihtpiphq23u5q
*on**the**size**of*certain restricted depth three*formulas*computing TH k, n (see [17, 23] ). ... Lower bounds*on**the**size**of*such*formulas*were shown by Hansel [8], Krichevskii [13], and Khrapchenko [12] . Hansel and Krichevskii showed a lower bound*of*0(n log n) for computing TH 2, n . ... Subrahmanyam, Ga bor Tardos, and Avi Wigderson for their comments*on**the*earlier versions*of*this paper. I thank*the*referee for her comments and suggestions. ...##
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Complexity of DNF and Isomorphism of Monotone Formulas
[chapter]

2005
*
Lecture Notes in Computer Science
*

We show PP-completeness

doi:10.1007/11549345_36
fatcat:tpy764k2crdqloyypjqzji2yfu
*of*checking whether*the*minimum*size**of*a DNF for a*monotone**formula*is at most k. For k in unary, we show*the**complexity**of**the*problem to drop to coNP. ... We investigate*the**complexity**of*finding prime implicants and minimal equivalent DNFs for Boolean*formulas*, and*of*testing equivalence and isomorphism*of**monotone**formulas*. ... We show that*the*same question is only NP-complete for*monotone**formulas*.*The**complexity**of*calculating*the**size**of*a smallest DNF depends*on**the*representation*of**the*problem. ...##
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Complexity of DNF minimization and isomorphism testing for monotone formulas

2008
*
Information and Computation
*

We show PP-completeness

doi:10.1016/j.ic.2008.03.002
fatcat:puebv25npbeififw6ygzrwquvm
*of*checking if*the*minimum*size**of*a DNF for a*monotone**formula*is at most k, and for k in unary, we show that*the**complexity**of**the*problem drops to coNP. ... We investigate*the**complexity**of*finding prime implicants and minimum equivalent DNFs for Boolean*formulas*, and*of*testing equivalence and isomorphism*of**monotone**formulas*. ... Acknowledgments*The*authors thank*the*anonymous referee for his valuable comments and suggestions. We also gratefully acknowledge*the*editing done by Tom Dodson. ...##
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Candidate Boolean Functions towards Super-Quadratic Formula Size

2015
*
IEICE transactions on information and systems
*

Next, we consider

doi:10.1587/transinf.2014fcp0011
fatcat:vdirtm3jkrhonofaoohtcvnho4
*the*reason*of**the*difficulty*of*resolving*the**formula**complexity**of**the*majority function in contrast with*the*parity function. ... In particular, we discuss*the*structure*of*an optimal protocol partition for*the*Karchmer-Wigderson communication game. key words: Boolean function, computational*complexity*,*formula**complexity*... Acknowledgments*The*author is grateful to anonymous referees for their helpful comments to improve*the*presentation*of**the*paper. ...##
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Negation-limited formulas

2017
*
Theoretical Computer Science
*

We prove that every

doi:10.1016/j.tcs.2016.11.027
fatcat:lvawujcwtbegxkixyxuhy3tzzy
*formula*that contains t negation gates can be shrunk using a random restriction to a*formula**of**size*O(t) with*the*shrinkage exponent*of**monotone**formulas*. ... Using*the*average-case lower bound for polynomial-*size**monotone**formulas**of*Rossman (CCC '15), we obtain an average-case lower bound for polynomial-*size**formulas*with n 1/2−o(1) negations, where n is*the*... Acknowledgments We thank Avishay Tal for helpful discussions and, in particular, for a discussion that led to*the*proof*of*Theorem 3.1. ...##
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Graph Homomorphism Polynomials: Algorithms and Complexity
[article]

2020
*
arXiv
*
pre-print

We discover that, in

arXiv:2011.04778v2
fatcat:23bzgro5snf6bazzsykubzzfou
*the**monotone*setting,*the**formula**complexity*,*the*ABP*complexity*, and*the*circuit*complexity**of*such polynomial families are exactly characterized by*the*treedepth,*the*pathwidth, ... For instance, we attain superpolynomial separations between circuits, ABPs, and*formulas*in*the**monotone*setting, where*the*polynomial families separating*the*classes all correspond to well-studied combinatorial ... (See [Sni80, Theorem 3.2]) For n ≥ 2, there is a polynomial family*of*degree n − 1*on*n 3 − n variables such that*the*family has linear*size**monotone*ABPs but require n log 2 (n+1)*size**monotone**formulas*...##
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On the Complexity of Symbolic Finite-State Automata
[article]

2021
*
arXiv
*
pre-print

number

arXiv:2011.05389v3
fatcat:u4rzea3asnd7rgmnqzukd5db4m
*of*transitions exiting a state, and*the**size**of**the*most*complex*transition predicate. ... We revisit*the**complexity**of*procedures*on*SFAs (such as intersection, emptiness, etc.) and analyze them according to*the*measures we find suitable for symbolic automata:*the*number*of*states,*the*maximal ... There, we showed that*the*conjunction*of*two DNF*formulas**of**sizes*k and l has*size*k + l, which implies that*the*conjunction*of*general*formulas*has linear*size*. ...##
###
Boolean Function Complexity Advances and Frontiers

2014
*
Bulletin of the European Association for Theoretical Computer Science
*

Acknowledgement I am thankful to Sasha Razborov for his comments

dblp:journals/eatcs/Jukna14
fatcat:x3y2bootivdd7mx4xswv7myrhy
*on*this summary. ... , and*the*relation*of**formula**size*with*the*affine dimension*of*graphs. • Lower bounds for*monotone**formulas*were obtained by using rank as well as communication*complexity*arguments. ...*On**the*other hand,*the*length*of*cutting-plane proofs can be lower-bounded using some communication*complexity*arguments or using*the*interpolation theorem together with lower bounds*on**the**size**of*circuits ...##
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Some Properties of Several Proof Systems for Intuitionistic, Johanssons and Monotone Propositional Logics

2018
*
Journal of Asian Scientific Research
*

In this paper we investigate two properties

doi:10.18488/journal.2.2018.82.61.72
fatcat:n2h4inso3zflvhv5fporg3klq4
*of*some propositional systems*of*Intuitionistic, Johansson's and*Monotone*logics: 1)*the*relations between*the*proofs*complexities**of*strongly equal tautologies ... minimal tautology (valid sequents), therefore*the*systems, which are considered in this paper, are no*monotonous*neither by lines nor by*size*. ... In*the*theory*of*proof*complexity*two main characteristics*of**the*proof are: t-*complexity*(length), defined as*the*number*of*proof steps, -*complexity*(*size*), defined as sum*of**sizes*for all*formulas*in ...##
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An Improved Homomorphism Preservation Theorem From Lower Bounds in Circuit Complexity
[article]

2016
*
arXiv
*
pre-print

In this paper, we give a new proof

arXiv:1612.08192v1
fatcat:y23tkqrz6bhwxmslgtmswmew2a
*of*this result via a reduction to lower bounds in circuit*complexity*, specifically*on**the*AC^0*formula**size**of**the*colored subgraph isomorphism problem. ... Quantitatively, this improves*the*result*of*[39], where*the*upper bound*on**the*quantifier-rank*of*Ψ is a non-elementary function*of*k. ... This is a semantic class, akin to*the*class*Monotone**of*all*monotone*languages. ...##
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A Solution of the P versus NP Problem
[article]

2017
*
arXiv
*
pre-print

Berg and Ulfberg and Amano and Maruoka have used CNF-DNF-approximators to prove exponential lower bounds for

arXiv:1708.03486v2
fatcat:qazjea2sxbe63hck4jpfp64uny
*the**monotone*network*complexity**of**the*clique function and*of*Andreev's function. ... We show that these approximators can be used to prove*the*same lower bound for their non-*monotone*network*complexity*. This implies P not equal NP. ...*of*its*monotone**complexity*. ...##
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On the pigeonhole and related principles in deep inference and monotone systems

2014
*
Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) - CSL-LICS '14
*

so

doi:10.1145/2603088.2603164
dblp:conf/csl/Das14
fatcat:dbsbhbqtxrf4hoc4ykpqrmyj3i
*the**complexity**of*normalization. ... We make significant use*of**monotone**formulae*computing boolean threshold functions, an idea previously considered in works*of*Atserias et al. ... This answers questions previously raised in [9] [18] [25] [13]*on**the**complexity**of*KS proofs*of*PHP m n by matching*the*best known bound for*the*more general class*of**monotone*proofs [2] . ...##
###
Page 6074 of Mathematical Reviews Vol. , Issue 2001I
[page]

2001
*
Mathematical Reviews
*

We show that

*the*PC refutations*of*homogenized*formulas*give crucial information about*the**complexity**of**the*original*formulas*. ... We show that mono- tone proofs for this sequent can be easily reduced to*monotone*proofs*of**the**one*-to-*one*and onto PHP, and so CLIQUE also has quasipolynomia!-*size**monotone*proofs. ...##
###
Karchmer-Wigderson Games for Hazard-free Computation
[article]

2021
*
arXiv
*
pre-print

We present a Karchmer-Wigderson game to study

arXiv:2107.05128v1
fatcat:a777vs2krnev7gpgyy4jtr45yy
*the**complexity**of*hazard-free*formulas*. ... Using this game, we prove hazard-free*formula**size*and depth lower bounds that are provably stronger than those possible by*the*standard technique*of*transferring results from*monotone**complexity*in a ... · 3 n )*on**the*hazard-free*formula**complexity*. ...##
###
Remarks on Graph Complexity
[chapter]

1998
*
Lecture Notes in Computer Science
*

2 (l) lower bound

doi:10.1007/978-3-540-49382-2_29
fatcat:etjqktwdkrbfpiwax6mxcxu4re
*on**the**formula**size**of*another explicit function g*on*l variables, where l = (log n). ... Combining our observations with those from PRS], we can show, for instance, that a lower bound*of*n 1+ (1)*on**the*(*monotone*)*formula**size**of*an explicit 2-slice function f*on*n variables would imply a ... (1),*on**the**monotone**formula**size**of*a 2-slice function implies a lower bound*of*(n)*on**the**formula**complexity**of**the*corresponding graph. Similar results hold for circuit*size*. ...
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