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Better Lower Bounds for Monotone Threshold Formulas

Jaikumar Radhakrishnan
1997 Journal of computer and system sciences (Print)  
Snir's result implies an 0(kn log(nÂ(k&1)) lower bound 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.  ... 
doi:10.1006/jcss.1997.1287 fatcat:pp4eh4quefdbdihtpiphq23u5q

Complexity of DNF and Isomorphism of Monotone Formulas [chapter]

Judy Goldsmith, Matthias Hagen, Martin Mundhenk
2005 Lecture Notes in Computer Science  
We show PP-completeness 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.  ... 
doi:10.1007/11549345_36 fatcat:tpy764k2crdqloyypjqzji2yfu

Complexity of DNF minimization and isomorphism testing for monotone formulas

Judy Goldsmith, Matthias Hagen, Martin Mundhenk
2008 Information and Computation  
We show PP-completeness 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.  ... 
doi:10.1016/j.ic.2008.03.002 fatcat:puebv25npbeififw6ygzrwquvm

Candidate Boolean Functions towards Super-Quadratic Formula Size

Kenya UENO
2015 IEICE transactions on information and systems  
Next, we consider 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.  ... 
doi:10.1587/transinf.2014fcp0011 fatcat:vdirtm3jkrhonofaoohtcvnho4

Negation-limited formulas

Siyao Guo, Ilan Komargodski
2017 Theoretical Computer Science  
We prove that every 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.  ... 
doi:10.1016/j.tcs.2016.11.027 fatcat:lvawujcwtbegxkixyxuhy3tzzy

Graph Homomorphism Polynomials: Algorithms and Complexity [article]

Balagopal Komarath, Anurag Pandey, C. S. Rahul
2020 arXiv   pre-print
We discover that, in 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  ... 
arXiv:2011.04778v2 fatcat:23bzgro5snf6bazzsykubzzfou

On the Complexity of Symbolic Finite-State Automata [article]

Dana Fisman and Hadar Frenkel and Sandra Zilles
2021 arXiv   pre-print
number 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.  ... 
arXiv:2011.05389v3 fatcat:u4rzea3asnd7rgmnqzukd5db4m

Boolean Function Complexity Advances and Frontiers

Stasys Jukna
2014 Bulletin of the European Association for Theoretical Computer Science  
Acknowledgement I am thankful to Sasha Razborov for his comments 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  ... 
dblp:journals/eatcs/Jukna14 fatcat:x3y2bootivdd7mx4xswv7myrhy

Some Properties of Several Proof Systems for Intuitionistic, Johanssons and Monotone Propositional Logics

Chubaryan Anahit, Karabakhtsyan Arman, Petrosyan Garik
2018 Journal of Asian Scientific Research  
In this paper we investigate two properties 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  ... 
doi:10.18488/journal.2.2018.82.61.72 fatcat:n2h4inso3zflvhv5fporg3klq4

An Improved Homomorphism Preservation Theorem From Lower Bounds in Circuit Complexity [article]

Benjamin Rossman
2016 arXiv   pre-print
In this paper, we give a new proof 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.  ... 
arXiv:1612.08192v1 fatcat:y23tkqrz6bhwxmslgtmswmew2a

A Solution of the P versus NP Problem [article]

Norbert Blum
2017 arXiv   pre-print
Berg and Ulfberg and Amano and Maruoka have used CNF-DNF-approximators to prove exponential lower bounds for 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.  ... 
arXiv:1708.03486v2 fatcat:qazjea2sxbe63hck4jpfp64uny

On the pigeonhole and related principles in deep inference and monotone systems

Anupam Das
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 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] .  ... 
doi:10.1145/2603088.2603164 dblp:conf/csl/Das14 fatcat:dbsbhbqtxrf4hoc4ykpqrmyj3i

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]

Christian Ikenmeyer, Balagopal Komarath, Nitin Saurabh
2021 arXiv   pre-print
We present a Karchmer-Wigderson game to study 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.  ... 
arXiv:2107.05128v1 fatcat:a777vs2krnev7gpgyy4jtr45yy

Remarks on Graph Complexity [chapter]

Satyanarayana V. Lokam
1998 Lecture Notes in Computer Science  
2 (l) lower bound 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.  ... 
doi:10.1007/978-3-540-49382-2_29 fatcat:etjqktwdkrbfpiwax6mxcxu4re
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