TY - GEN

T1 - On the combinatorial representation of information

AU - Ratsaby, Joel

PY - 2006

Y1 - 2006

N2 - Kolmogorov introduced a combinatorial measure of the information I(x : y) about the unknown value of a variable y conveyed by an input variable x taking a given value x. The paper extends this definition of information to a more general setting where 'x = x' may provide a vaguer description of the possible value of y. As an application, the space P({0, 1}n) of classes of binary functions f : [n] → {0, 1}, [n] = {1,..., n], is considered where y represents an unknown function t ∈ {0, 1}[n] and as input, two extreme cases are considered: x = xMd and x = xM′d which indicate that t is an element of a set G ⊆ {0, 1}n that satisfies a property Md or M′d respectively. Property Md (or M′d) means that there exists an E ⊆ [n], |E| = d, such that |trE(G)| = 1 (or 2d) where trE(G) denotes the trace of G on E. Estimates of the information value I(xMd : t) and I(xM′d: t) are obtained. When d is fixed, it is shown that I(xMd : t) ≈ d and I(x M′d : t) ≈ 1 as n → ∞.

AB - Kolmogorov introduced a combinatorial measure of the information I(x : y) about the unknown value of a variable y conveyed by an input variable x taking a given value x. The paper extends this definition of information to a more general setting where 'x = x' may provide a vaguer description of the possible value of y. As an application, the space P({0, 1}n) of classes of binary functions f : [n] → {0, 1}, [n] = {1,..., n], is considered where y represents an unknown function t ∈ {0, 1}[n] and as input, two extreme cases are considered: x = xMd and x = xM′d which indicate that t is an element of a set G ⊆ {0, 1}n that satisfies a property Md or M′d respectively. Property Md (or M′d) means that there exists an E ⊆ [n], |E| = d, such that |trE(G)| = 1 (or 2d) where trE(G) denotes the trace of G on E. Estimates of the information value I(xMd : t) and I(xM′d: t) are obtained. When d is fixed, it is shown that I(xMd : t) ≈ d and I(x M′d : t) ≈ 1 as n → ∞.

KW - Combinatorial complexity

KW - Information theory

KW - VC-dimension

UR - http://www.scopus.com/inward/record.url?scp=33749546981&partnerID=8YFLogxK

U2 - 10.1007/11809678_50

DO - 10.1007/11809678_50

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AN - SCOPUS:33749546981

SN - 3540369252

SN - 9783540369257

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 479

EP - 488

BT - Computing and Combinatorics - 12th Annual International Conference, COCOON 2006, Proceedings

T2 - 12th Annual International Conference on Computing and Combinatorics, COCOON 2006

Y2 - 15 August 2006 through 18 August 2006

ER -