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Some Dichotomy Theorems for Neural Learning Problems

2004
*
Journal of machine learning research
*

The results are established as

dblp:journals/jmlr/Schmitt04
fatcat:bl3cwzg7xbbr7lzgl4qk37edam
*dichotomy**theorems*showing that each*problem*is either NP-complete or solvable in polynomial time. ... In particular, we consider consistency and*maximum*consistency*problems**for*neurons with binary weights, and*maximum*consistency*problems**for*neurons with arbitrary weights. ... Acknowledgments I thank the anonymous reviewers*for*their helpful comments and suggestions. ...##
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An Upper Bound on the Minimum Number of Monomials Required to Separate Dichotomies of {−1, 1}n

2006
*
Neural Computation
*

In

doi:10.1162/neco.2006.18.12.3119
pmid:17052161
fatcat:hfrp3zzuh5gxjoqtoprgrw2bmu
*general*, less than 2 n monomials are sufficient to solve*a*given*dichotomy*. ... The main*theorem*here states that*for*any binary classification*problem*in {−1, 1} n (n > 1), one can always find*a*polynomial function solution with 2 n − 2 n /4 or fewer monomials. ... I was introduced to the*problem*of establishing*a*bound on the*maximum*density by Marifi Guler. I thank Junmei Zhu and Jun Nakanishi*for*their comments on an earlier version of the manuscript. ...##
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Complexity of Homogeneous Co-Boolean Constraint Satisfaction Problems
[article]

2010
*
arXiv
*
pre-print

In this short note we show

arXiv:1011.4744v1
fatcat:5ptx3rduizfplfin4keg34iypm
*a**dichotomy**theorem**for*every finite domain D of CSP built upon graphs of homogeneous co-Boolean functions, i.e., unary functions sharing the Boolean range 0, 1. ... It has been proved that*for*each CSP*problem*over*a*given set of relations there exists*a*corresponding CSP*problem*over graphs of unary functions belonging to the same complexity class. ... Similarly to our*dichotomy**theorem*, which differs from Schaefer's*Dichotomy**Theorem*in [7] , we conjecture that*a**dichotomy**theorem**for*the latter*problem*will also be different from Bulatov's*Dichotomy*...##
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Dichotomy Theorems for Alternation-Bounded Quantified Boolean Formulas
[article]

2004
*
arXiv
*
pre-print

In 1978, Schaefer proved his famous

arXiv:cs/0406006v1
fatcat:scmdburmvvc6dbljacs6ozfufy
*dichotomy**theorem**for**generalized**satisfiability**problems*. ... In this paper, we prove*dichotomy**theorems**for*alternation-bounded quantified*generalized*Boolean formulas, by showing that these*problems*are either Σ_i^p-complete or in P, and we give*a*simple criterion ... Introduction In 1978, Schaefer proved his famous*dichotomy**theorem**for**generalized**satisfiability**problems*. ...##
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Minimum cost and list homomorphisms to semicomplete digraphs

2006
*
Discrete Applied Mathematics
*

The list homomorphism

doi:10.1016/j.dam.2005.11.006
fatcat:w7ym6atkvzf4djesr4hhrd7age
*problem**for*H is*a**generalization*of the homomorphism*problem**for*H, where every vertex x ∈ V (D) is assigned*a*set L x of possible colors (vertices of H). ... Our*dichotomy**for*the list homomorphism*problem*coincides with the one obtained by Bang-Jensen, Hell and MacGillivray in 1988*for*the homomorphism*problem*when H is*a*semicomplete digraph: both*problems*... We are very thankful to Dave Cohen, Martin Green, Pavol Hell and Andrei Krokhin*for*useful discussions on the topic of the paper. ...##
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Inverse Circumscription

2003
*
International Joint Conference on Artificial Intelligence
*

Inverse (or identification)

dblp:conf/ijcai/Chen03
fatcat:asla3oerx5civdmbz3tivwj2ye
*problems*involve decid ing whether or not an explicitly given set of data points have an implicit description,*for*instance, in the form of*a*constraint network. ... This pa per formalizes and studies the inverse circumscription*problem*, which (roughly speaking) is to de cide, given*a*set of models, if there exists*a*formula whose circumscription describes the input ... The author wishes to thank Bart Selman*for*useful discussions and suggestions, and Joe Halpern*for*his advice on the preparation of the final version of this paper. ...##
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Steady-state and periodic exponential turnpike property for optimal control problems in Hilbert spaces
[article]

2016
*
arXiv
*
pre-print

*A*similar statement holds true as well when replacing an optimal steady-state by an optimal periodic trajectory. ... In this work, we study the steady-state (or periodic) exponential turnpike property of optimal control

*problems*in Hilbert spaces. ...

*maximum*principle

*for*optimal control

*problems*is valid. ...

##
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A Free Boundary Problem for a Leslie-Gower Predator-Prey Model in Higher Dimensions and Heterogeneous Environment

2020
*
American Journal of Applied Mathematics
*

We establish that

doi:10.11648/j.ajam.20200805.17
fatcat:grijqs6wxnc6tiwk6y27zd3egi
*a*spreading-vanishing*dichotomy*is held*for*this model. We use the comparison principle. we will give the existence, uniqueness and some estimates of the solution to the*problem*. ... This paper is mainly concerned with some free boundary*problems**for**a*modified Leslie-Gower predator-prey model in higher dimensional and heterogeneous environment. ... We have proved that*for*both*problems*,*a*spreading-vanishing*dichotomy*holds (*Theorems*3.1, 3.3 and 3.4), and when spreading occurs the spreading fronts expand at*a*nearly constant speed*for*large time ...##
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Steady-State and Periodic Exponential Turnpike Property for Optimal Control Problems in Hilbert Spaces

2018
*
SIAM Journal of Control and Optimization
*

*A*similar statement holds true as well when replacing an optimal steady-state by an optimal periodic trajectory. ... In this work, we study the steady-state (or periodic) exponential turnpike property of optimal control

*problems*in Hilbert spaces. ...

*maximum*principle

*for*optimal control

*problems*is valid. ...

##
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On First-Species Counterpoint Theory

2021
*
MusMat|: Brazilian Journal of Music and Mathematics
*

We

doi:10.46926/musmat.2021v5n2.1-40
fatcat:idtvifgcgvc7zko2spetgeo6re
*generalize*first-species counterpoint theory to arbitrary rings and obtain some new counting and maximization results that enrich the theory of admitted successors, pointing to*a*structural approach ... The original motivations of the theory, as well as all technical passages, are carefully reviewed so as to provide*a*complete exposition. ... Thus,*for**a**general*strong*dichotomy*, we locally characterize contrapuntal dissonances and consonances with 1. ...##
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The Complexity of Computing Optimal Assignments of Generalized Propositional Formulae
[article]

1998
*
arXiv
*
pre-print

We consider the

arXiv:cs/9809116v1
fatcat:p4raoiwtc5gndl5flxx74ficfe
*problems*of finding the lexicographically minimal (or maximal)*satisfying*assignment of propositional formulae*for*different restricted formula classes. ... It turns out that*for*each class from our framework, the above*problem*is either polynomial time solvable or complete*for*OptP. ... We are extremely grateful to Nadia Creignou, Caen,*for**a*lot of helpful hints. ...##
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The Complexity of Homomorphisms of Signed Graphs and Signed Constraint Satisfaction
[chapter]

2014
*
Lecture Notes in Computer Science
*

We also introduce the more

doi:10.1007/978-3-642-54423-1_46
fatcat:cudazm72vre5pjgoq3c24revui
*general*concept of signed constraint satisfaction*problems*and show that*a**dichotomy**for*such*problems*is equivalent to the statement of the Feder-Vardi*Dichotomy*Conjecture. ... We prove*a**dichotomy*result*for*the class of all (C k , Σ)-Coloring*problems*(where C k is*a*cycle of length k ≥ 3): (C k , Σ)-Coloring is NP-complete, unless both k and the size of Σ are even. ... While it is*a*difficult*problem*to prove*a**dichotomy**for*S-CSP (equivalently,*for*CSP or*for*digraph homomorphism*problems*), it will be of interest to prove*a**dichotomy**for*signed graph homomorphism*problems*...##
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Constraint satisfaction problem and universal algebra

2014
*
ACM SIGLOG News
*

This column gives

doi:10.1145/2677161.2677165
fatcat:k5dstcfax5fujky2h5ekhogk2q
*a*brief survey of current research on the complexity of the constraint satisfaction*problem*(CSP) over fixed constraint languages. ... -The*dichotomy**theorem*of Schaefer*for*CSPs over*a*two-element domain was*generalized*to*a*three-element domain [Bulatov 2011] . ... ., one that*satisfies*the*maximum*possible number of constraints. ...##
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A combinatorial constraint satisfaction problem dichotomy classification conjecture

2010
*
European journal of combinatorics (Print)
*

The extension in this paper gives

doi:10.1016/j.ejc.2009.02.007
fatcat:s3s5i7nbmvhxtnfrefgt2y7u6y
*a*polynomial-time reduction of CSP(H)*for*any relational system H to CSP(P )*for*any relational system P that meets*a*certain technical partition condition, that of being ... This follows immediately from the*theorem*by the fact, shown in [12] , that CSP(K 3 ) is N P -complete (even*for*undirected instances). ...*A*substantial*generalization*of this*dichotomy*result was obtained in [1]*for*the H-colouring*problems*over digraphs with no sources and no sinks. ...##
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Fine-Grained Dichotomies for the Tutte Plane and Boolean #CSP

2018
*
Algorithmica
*

Another

doi:10.1007/s00453-018-0472-z
fatcat:i2ys5fwwkngxbmy557eybmtbti
*dichotomy**theorem*we strengthen is the one of Creignou and Hermann [6]*for*counting the number of*satisfying*assignments to*a*constraint satisfaction*problem*instance over the Boolean domain. ... We complete the*dichotomy**theorem**for*the Tutte polynomial under #ETH by proving that the number of all acyclic subgraphs of*a*given n-vertex graph cannot be determined in time exp o(n) unless #ETH fails ...*dichotomies*" work group at the Simons Institute in the spring of 2016. ...
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