Kolmogorov Capacity with Overlap
The notion of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula>-mutual information between non-stochastic uncertain variables is introduced as...
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MDPI AG
2025-04-01
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| author | Anshuka Rangi Massimo Franceschetti |
| author_facet | Anshuka Rangi Massimo Franceschetti |
| author_sort | Anshuka Rangi |
| collection | DOAJ |
| description | The notion of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula>-mutual information between non-stochastic uncertain variables is introduced as a generalization of Nair’s non-stochastic information functional. Several properties of this new quantity are illustrated and used in a communication setting to show that the largest <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula>-mutual information between received and transmitted codewords over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>-noise channels equals the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ϵ</mi><mo>,</mo><mi>δ</mi><mo>)</mo></mrow></semantics></math></inline-formula>-capacity. This notion of capacity generalizes the Kolmogorov <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>-capacity to packing sets of overlap at most <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula> and is a variation of a previous definition proposed by one of the authors. Results are then extended to more general noise models, including non-stochastic, memoryless, and stationary channels. The presented theory admits the possibility of decoding errors, as in classical information theory, while retaining the worst-case, non-stochastic character of Kolmogorov’s approach. |
| format | Article |
| id | doaj-art-f33dd4e556e34b18be3c1fcbfdbe353d |
| institution | Kabale University |
| issn | 1099-4300 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj-art-f33dd4e556e34b18be3c1fcbfdbe353d2025-08-20T03:47:52ZengMDPI AGEntropy1099-43002025-04-0127547210.3390/e27050472Kolmogorov Capacity with OverlapAnshuka Rangi0Massimo Franceschetti1Department of Electrical and Computer Engineering, University of California at San Diego, 9500 Gilman Drive, Mail Code 0407, La Jolla, CA 92093-0407, USADepartment of Electrical and Computer Engineering, University of California at San Diego, 9500 Gilman Drive, Mail Code 0407, La Jolla, CA 92093-0407, USAThe notion of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula>-mutual information between non-stochastic uncertain variables is introduced as a generalization of Nair’s non-stochastic information functional. Several properties of this new quantity are illustrated and used in a communication setting to show that the largest <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula>-mutual information between received and transmitted codewords over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>-noise channels equals the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ϵ</mi><mo>,</mo><mi>δ</mi><mo>)</mo></mrow></semantics></math></inline-formula>-capacity. This notion of capacity generalizes the Kolmogorov <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>-capacity to packing sets of overlap at most <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula> and is a variation of a previous definition proposed by one of the authors. Results are then extended to more general noise models, including non-stochastic, memoryless, and stationary channels. The presented theory admits the possibility of decoding errors, as in classical information theory, while retaining the worst-case, non-stochastic character of Kolmogorov’s approach.https://www.mdpi.com/1099-4300/27/5/472<i>ϵ</i>-capacity(<i>ϵ</i>,<i>δ</i>)-capacitymutual informationnon-stochastic uncertainty |
| spellingShingle | Anshuka Rangi Massimo Franceschetti Kolmogorov Capacity with Overlap Entropy <i>ϵ</i>-capacity (<i>ϵ</i>,<i>δ</i>)-capacity mutual information non-stochastic uncertainty |
| title | Kolmogorov Capacity with Overlap |
| title_full | Kolmogorov Capacity with Overlap |
| title_fullStr | Kolmogorov Capacity with Overlap |
| title_full_unstemmed | Kolmogorov Capacity with Overlap |
| title_short | Kolmogorov Capacity with Overlap |
| title_sort | kolmogorov capacity with overlap |
| topic | <i>ϵ</i>-capacity (<i>ϵ</i>,<i>δ</i>)-capacity mutual information non-stochastic uncertainty |
| url | https://www.mdpi.com/1099-4300/27/5/472 |
| work_keys_str_mv | AT anshukarangi kolmogorovcapacitywithoverlap AT massimofranceschetti kolmogorovcapacitywithoverlap |