{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,29]],"date-time":"2025-10-29T19:39:10Z","timestamp":1761766750097,"version":"build-2065373602"},"reference-count":65,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2020,7,4]],"date-time":"2020-07-04T00:00:00Z","timestamp":1593820800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Information"],"abstract":"<jats:p>Defining computation as information processing (information dynamics) with information as a relational property of data structures (the difference in one system that makes a difference in another system) makes it very suitable to use operator formulation, with similarities to category theory. The concept of the operator is exceedingly important in many knowledge areas as a tool of theoretical studies and practical applications. Here we introduce the operator theory of computing, opening new opportunities for the exploration of computing devices, processes, and their networks.<\/jats:p>","DOI":"10.3390\/info11070349","type":"journal-article","created":{"date-parts":[[2020,7,6]],"date-time":"2020-07-06T03:19:27Z","timestamp":1594005567000},"page":"349","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Prolegomena to an Operator Theory of Computation"],"prefix":"10.3390","volume":"11","author":[{"given":"Mark","family":"Burgin","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, CA 90095, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9881-400X","authenticated-orcid":false,"given":"Gordana","family":"Dodig-Crnkovic","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Engineering, Chalmers University of Technology, 41296 Gothenburg, Sweden"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"480","DOI":"10.2307\/1969165","article-title":"Relations between homology and homotopy groups of spaces","volume":"46","author":"Eilenberg","year":"1945","journal-title":"Ann. Math."},{"key":"ref_2","unstructured":"Bourbaki, N. (1960). Elements de Mathematique. Theorie des Ensembles, Hermann."},{"key":"ref_3","unstructured":"Robinson, A. (1963). Introduction to Model Theory and Metamathematics of Algebra, North-Holland Publishing."},{"key":"ref_4","unstructured":"Goldblatt, R. (1979). Topoi: The Categorical Analysis of Logic, North Holland Publishing."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"697","DOI":"10.3390\/info2040697","article-title":"Epistemic Information in Stratified M-Spaces","volume":"2","author":"Burgin","year":"2011","journal-title":"Information"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1140\/epjst\/e2016-60362-9","article-title":"Nature as a Network of Morphological Infocomputational Processes for Cognitive Agents","volume":"226","year":"2017","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_7","unstructured":"Von Neumann, J. (1955). Mathematical Foundations of Quantum Mechanics, Princeton University Press."},{"key":"ref_8","unstructured":"Exner, P., and Havl\u00ed\u010dek, M. (2008). Hilbert Space Operators in Quantum Physics, Springer."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Brown, A., and Pearcy, C. (1977). Introduction to Operator Theory I: Elements of Functional Analysis, Springer-Verlag.","DOI":"10.1007\/978-1-4612-9926-4"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Ball, J.A., Bolotnikov, V., Helton, J.W., and Rodman, L. (2010). Topics in Operator Theory (Operator Theory: Advances and Applications), Birkh\u00e4userVerlag.","DOI":"10.1007\/978-3-0346-0161-0"},{"key":"ref_11","unstructured":"Burgin, M. (2017). Semitopological Vector Spaces: Hypernorms, Hyperseminorms and Operators, Apple Academic Press."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"357","DOI":"10.3390\/info5020357","article-title":"Weighted E-Spaces and Epistemic Information Operators","volume":"5","author":"Burgin","year":"2014","journal-title":"Information"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Harris, Z. (1991). A Theory of Language and Information: A Mathematical Approach, Oxford University Press.","DOI":"10.1093\/oso\/9780198242246.001.0001"},{"key":"ref_14","first-page":"33","article-title":"Information as a Natural and Social Operator","volume":"18","author":"Brenner","year":"2011","journal-title":"Inform. Theor. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Burgin, M., and Brenner, J. (2017). Operators in Nature, Science, Technology, and Society: Mathematical, Logical, and Philosophical Issues. Philosophies, 2.","DOI":"10.3390\/philosophies2030021"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Pingali, K., Nguyen, D., Kulkarni, M., Burtscher, M., Hassaan, M.A., Kaleem, R., Lee, T.H., Lenharth, A., Manevich, R., and Mendez-Lojo, M. (2011). The Tao of Parallelism in Algorithms. PLDI \u201911: Proceedings of the 32nd ACM SIGPLAN Conference on Programming Language Design and Implementation, Association for Computing Machinery.","DOI":"10.1145\/1993498.1993501"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Rozenberg, G., Back, T., and Kok, J. (2012). Handbook of Natural Computing, Springer Verlag.","DOI":"10.1007\/978-3-540-92910-9"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Adamatzky, A. (2018). Unconventional Computing. A Volume in Encyclopedia of Complexity and Systems Science, Springer Nature. [2nd ed.].","DOI":"10.1007\/978-1-4939-6883-1"},{"key":"ref_19","first-page":"1","article-title":"Structural Machines as a Mathematical Model of Biological and Chemical Computers","volume":"7","author":"Burgin","year":"2017","journal-title":"Theor. Appl. Math. Comput. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1080\/03081079.2017.1300585","article-title":"Structural machines and slime mold computation","volume":"45","author":"Burgin","year":"2017","journal-title":"Int. J. Gen. Syst."},{"key":"ref_21","unstructured":"Levine, I.N. (1991). Quantum Chemistry, Prentice Hall. [4th ed.]."},{"key":"ref_22","unstructured":"McQuarrie, D.A. (1983). Quantum Chemistry, University Science Books."},{"key":"ref_23","unstructured":"(2020, May 08). What Does Operator Mean (in Computer Programming)?. Available online: https:\/\/www.techopedia.com\/definition\/3485\/operator-programming."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Burgin, M. (2018). Triadic Structures in Interpersonal Communication. Information, 9.","DOI":"10.3390\/info9110283"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Burgin, M., and Dodig-Crnkovi\u0107, G. (2019). A Multiscale Taxonomy of Information in the World. Theoretical Information Studies. Information in the World, World Scientific.","DOI":"10.1142\/11191"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"M\u00fcller, V. (2018). Cognition as Embodied Morphological Computation. Philosophy and Theory of Artificial Intelligence 2017, Springer. PT-AI 2017. Studies in Applied Philosophy, Epistemology and Rational Ethics.","DOI":"10.1007\/978-3-319-96448-5"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"413","DOI":"10.3389\/fncel.2019.00413","article-title":"Axonal Computations","volume":"13","author":"Alcami","year":"2019","journal-title":"Front. Cell. Neurosci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"474","DOI":"10.1038\/nrn2864","article-title":"Neuronal arithmetic","volume":"11","author":"Silver","year":"2010","journal-title":"Nat. Rev. Neurosci."},{"key":"ref_29","first-page":"185","article-title":"Three Levels of the Symbolosphere","volume":"160","author":"Burgin","year":"2006","journal-title":"Semiotica"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1016\/0004-3702(91)90053-M","article-title":"Intelligence without representation","volume":"47","author":"Brooks","year":"1991","journal-title":"Artif. Intell."},{"key":"ref_31","first-page":"175","article-title":"On the Concept of Algorithm","volume":"8","author":"Kolmogorov","year":"1953","journal-title":"Rus. Math. Surv."},{"key":"ref_32","unstructured":"Codd, E.F. (1968). Cellular Automata, Academic Press."},{"key":"ref_33","first-page":"147","article-title":"Semantic Information","volume":"4","author":"Carnap","year":"1958","journal-title":"Br. J. Philos. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"G\u00e4rdenfors, P. (2000). Conceptual Spaces: The Geometry of Thought, MIT Press.","DOI":"10.7551\/mitpress\/2076.001.0001"},{"key":"ref_35","first-page":"9","article-title":"Conceptual Spaces as a Framework for Knowledge Representation","volume":"2","year":"2004","journal-title":"Mind Matter"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Krois, J.M., Rosengren, M., Steidele, A., and Westerkamp, D. (2007). Cognitive semantics and image schemas with embodied forces. Embodiment in Cognition and Culture, Benjamins.","DOI":"10.1075\/aicr.71"},{"key":"ref_37","unstructured":"Shreider, Y.A. (1967). On Semantic Aspects of Information Theory. Inform. Cybern., 15\u201347. (In Russian)."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Burgin, M., and D\u00edaz-Nafr\u00eda, J.M. (2019, January 7\u20139). Introduction to the Mathematical Theory of Knowledge Conceptualization: Conceptual Systems and Structures. Proceedings of the Second International Conference on Applied Informatics (ICAI 2019), Communications in Computer and Information Science Book Series (CCIS, Volume 1051), Madrid, Spain.","DOI":"10.1007\/978-3-030-32475-9_34"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Dodig-Crnkovic, G., and Giovagnoli, R. (2013). Computing Nature\u2015A Network of Networks of Concurrent Information Processes. Computing Nature, Springer.","DOI":"10.1007\/978-3-642-37225-4"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"460","DOI":"10.3390\/info2030460","article-title":"Dynamics of Information as Natural Computation","volume":"2","year":"2011","journal-title":"Information"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Burgin, M., and Dodig-Crnkovic, G. (2013). From the Closed Classical Algorithmic Universe to an Open World of Algorithmic Constellations. Computing Nature, Springer. arXiv:1211.4547.","DOI":"10.1007\/978-3-642-37225-4_16"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Pfeifer, R., and Bongard, J. (2006). How the Body Shapes the Way We Think: A New View of Intelligence, MIT Press.","DOI":"10.7551\/mitpress\/3585.001.0001"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1023\/B:MIND.0000005133.87521.5c","article-title":"On Communication and Computation","volume":"14","year":"2004","journal-title":"Minds Mach."},{"key":"ref_44","unstructured":"Dodig-Crnkovic, G. (2006). Investigations into Information Semantics and Ethics of Computing, M\u00e4lardalen University Press."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Burgin, M., and Dodig-Crnkovic, G. (2011). Information and Computation\u2015Omnipresent and Pervasive. Information and Computation, World Scientific.","DOI":"10.1142\/7637"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/S0304-3975(98)00096-6","article-title":"Achilles and the tortoise climbing up the hyper-arithmetical hierarchy","volume":"210","author":"Bournez","year":"1999","journal-title":"Theor. Comput. Sci."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/S0167-6423(97)00006-3","article-title":"Computing with Continuous Change","volume":"30","author":"Gupta","year":"1999","journal-title":"Sci. Comput. Program."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1002\/sapm1941201337","article-title":"Mathematical Theory of the Differential Analyzer","volume":"20","author":"Shannon","year":"1941","journal-title":"J. Math. Phys. MIT"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/0304-3975(95)00248-0","article-title":"Recursion Theory on the Reals and Continuous-time Computation: Real numbers and computers","volume":"162","author":"Moore","year":"1996","journal-title":"Theor. Comput. Sci."},{"key":"ref_50","unstructured":"Kelemen, J. (2006, January 24\u201325). On a Possible Future of Computationalism. Proceedings of the 7th International Symposium of Hungarian Researchers on Computational Intelligence, HUCI\u201906, Budapest, Hungary."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Bera, R.K. (2020). The Amazing World of Quantum Computing, Springer.","DOI":"10.1007\/978-981-15-2471-4"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"2155","DOI":"10.1098\/rspa.2009.0063","article-title":"The physical character of information","volume":"465","author":"Karnani","year":"2009","journal-title":"Proc. R. Soc. A"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1703","DOI":"10.3390\/e14091703","article-title":"MENS, an Info-Computational Model for (Neuro-)cognitive Systems Capable of Creativity","volume":"14","author":"Ehresmann","year":"2012","journal-title":"Entropy"},{"key":"ref_54","unstructured":"Burgin, M. (2005). Super-Recursive Algorithms, Springer."},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Burgin, M., and Dodig-Crnkovic, G. (2015, January 7\u201311). A Taxonomy of Computation and Information Architecture. Proceedings of the 2015 European Conference on Software Architecture Workshops, Dubrovnik\/Cavtat, Croatia.","DOI":"10.1145\/2797433.2797440"},{"key":"ref_56","first-page":"654","article-title":"Universal Limit Turing Machines","volume":"325","author":"Burgin","year":"1992","journal-title":"Not. Russ. Acad. Sci."},{"key":"ref_57","first-page":"793","article-title":"Multiple computations and Kolmogorov complexity for such processes","volume":"27","author":"Burgin","year":"1983","journal-title":"Not. Acad. Sci. USSR"},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Zenil, H. (2012). A Computable Universe. Understanding Computation & Exploring Nature as Computation, World Scientific Publishing Company\/Imperial College Press.","DOI":"10.1142\/8306"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"204","DOI":"10.3390\/info3020204","article-title":"Physical Computation as Dynamics of Form that Glues Everything Together","volume":"3","year":"2012","journal-title":"Information"},{"key":"ref_60","unstructured":"Burgin, M. (2010). Measuring Power of Algorithms, Computer Programs, and Information Automata, Nova Science Publishers."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"1465","DOI":"10.1142\/S012905411240059X","article-title":"Decidability and Universality in the Axiomatic Theory of Computability and Algorithms","volume":"23","author":"Burgin","year":"2012","journal-title":"Int. J. Found. Comput. Sci."},{"key":"ref_62","unstructured":"Dodig-Crnkovic, G., and Burgin, M. (2012, January 2\u20136). Axiomatic Tools versus Constructive approach to Unconventional Algorithms. Proceedings of the Symposium on Natural Computing\/Unconventional Computing and its Philosophical Significance, AISB\/IACAP World Congress, Birmingham, UK."},{"key":"ref_63","doi-asserted-by":"crossref","unstructured":"Burgin, M. (2020). Information Processing by Structural Machines. Theoretical Information Studies: Information in the World, World Scientific.","DOI":"10.1142\/9789813277496_0015"},{"key":"ref_64","doi-asserted-by":"crossref","unstructured":"Blackadar, B. (2005). Operator Algebras, Springer.","DOI":"10.1007\/3-540-28517-2"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"366","DOI":"10.1006\/inco.1994.1037","article-title":"A Completeness Theorem for Kleene Algebras and the Algebra of Regular Events","volume":"110","author":"Kozen","year":"1994","journal-title":"Inform. Comput."}],"container-title":["Information"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2078-2489\/11\/7\/349\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:47:21Z","timestamp":1760176041000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2078-2489\/11\/7\/349"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,4]]},"references-count":65,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2020,7]]}},"alternative-id":["info11070349"],"URL":"https:\/\/doi.org\/10.3390\/info11070349","relation":{},"ISSN":["2078-2489"],"issn-type":[{"type":"electronic","value":"2078-2489"}],"subject":[],"published":{"date-parts":[[2020,7,4]]}}}