{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T16:44:13Z","timestamp":1778690653171,"version":"3.51.4"},"reference-count":51,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,2,13]],"date-time":"2020-02-13T00:00:00Z","timestamp":1581552000000},"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>Algorithms and abstract automata (abstract machines) are used to describe, model, explore and improve computers, cell phones, computer networks, such as the Internet, and processes in them. Traditional models of information processing systems\u2014abstract automata\u2014are aimed at performing transformations of data. These transformations are performed by their hardware (abstract devices) and controlled by their software (programs)\u2014both of which stay unchanged during the whole computational process. However, in physical computers, their software is also changing by special tools such as interpreters, compilers, optimizers and translators. In addition, people change the hardware of their computers by extending the external memory. Moreover, the hardware of computer networks is incessantly altering\u2014new computers and other devices are added while other computers and other devices are disconnected. To better represent these peculiarities of computers and computer networks, we introduce and study a more complete model of computations, which is called a triadic automaton or machine. In contrast to traditional models of computations, triadic automata (machine) perform computational processes transforming not only data but also hardware and programs, which control data transformation. In addition, we further develop taxonomy of classes of automata and machines as well as of individual automata and machines according to information they produce.<\/jats:p>","DOI":"10.3390\/info11020102","type":"journal-article","created":{"date-parts":[[2020,2,14]],"date-time":"2020-02-14T10:09:57Z","timestamp":1581674997000},"page":"102","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Triadic Automata and Machines as Information Transformers"],"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"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,13]]},"reference":[{"key":"ref_1","unstructured":"Burgin, M. (2005). Superrecursive Algorithm, Springer."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Adamatzky, A. (2018). Inductive Turing Machines. Unconventional Computing\u2014A volume in the Encyclopedia of Complexity and Systems Science, Springer.","DOI":"10.1007\/978-1-4939-6883-1"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Dymond, P.W., and Cook, S.A. (1980, January 13\u201315). Hardware complexity and parallel computation. Proceedings of the 21st Annual Symposium on Foundations of Computer Science, Syracuse, NY, USA.","DOI":"10.1109\/SFCS.1980.22"},{"key":"ref_4","first-page":"99","article-title":"Towards a complexity theory of parallel computation","volume":"XXVII","author":"Cook","year":"1981","journal-title":"L\u2019Enseignement Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/0304-3975(86)90139-8","article-title":"On nondeterminism in parallel computation","volume":"47","author":"Dymond","year":"1986","journal-title":"Theor. Comput. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1449","DOI":"10.1080\/00207160310001606034","article-title":"Nonlinear Phenomena in Spaces of Algorithms","volume":"80","author":"Burgin","year":"2003","journal-title":"Int. J. Comput. Math."},{"key":"ref_7","unstructured":"Kleene, S. (1958, January 14\u201321). Constructive and Non-constructive Operations. Proceedings of the International Congress of Mathematicians, Edinburg, UK."},{"key":"ref_8","unstructured":"Burgin, M. (1992). Reflexive Calculi and Logic of Expert Systems. Creative Processes Modeling by Means of Knowledge Bases, Institute of Mathematics. (In Russian)."},{"key":"ref_9","first-page":"94","article-title":"Reflexive Turing Machines and Calculi","volume":"148","author":"Burgin","year":"1993","journal-title":"Vychislitelnyye Syst. (Log. Methods Comput. Sci.)"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Schroeder, M.J. (2013). Dualism of Selective and Structural Manifestations of Information in Modelling of Information Dynamics. Computing Nature, Springer. SAPERE 7.","DOI":"10.1007\/978-3-642-37225-4_7"},{"key":"ref_11","unstructured":"Schroeder, M.J. (2013, January 2\u20135). From Proactive to Interactive Theory of Computation. Proceedings of the 6th AISB Symposium on Computing and Philosophy: The Scandal of Computation\u2014What is Computation?, Exeter, UK."},{"key":"ref_12","unstructured":"Burgin, M. (2019, January 2\u20137). Processing information by symmetric inductive machines. Proceedings of the IS4SI Summit Berkeley 2019\u2014Where is the I in AI and the Meaning in Information, Berkeley, CA, USA."},{"key":"ref_13","unstructured":"Smith, B.C. (1982). Procedural Reflection in Programming Languages. [Ph.D. Thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology]."},{"key":"ref_14","unstructured":"Demers, F.-N., and Malenfant, J. (1995, January 20\u201325). Reflection in logic, functional and object-oriented programming: A Short Comparative Study. Proceedings of the IJCAI\u201995 Workshop on Reflection and Metalevel Architectures and their Applications in AI, Montreal, QC, Canada."},{"key":"ref_15","unstructured":"Chlipala, A. (2010, January 22\u201328). Ur: statically-typed metaprogramming with type-level record computation. Proceedings of the ACM SIGPLAN 2010 Conference on Programming Language Design and Implementation (PLDI\u201910), Phoenix, AZ, USA."},{"key":"ref_16","unstructured":"Burgin, M. (2012). Structural Reality, Nova Science Publishers."},{"key":"ref_17","unstructured":"Mandler, J.M. (1984). Stories, Scripts, and Scenes: Aspects of Schema Theory, Lawrence Erlbaum Associates."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Arbib, M.A. (2005). Modules, Brains and Schemas, Formal Methods, Springer. LNCS 3393.","DOI":"10.1007\/978-3-540-31847-7_9"},{"key":"ref_19","unstructured":"Burgin, M. (2006, January 2\u20136). Mathematical Schema Theory for Modeling in Business and Industry. Proceedings of the 2006 Spring Simulation Multi Conference (SpringSim\u201806), Huntsville, AL, USA."},{"key":"ref_20","unstructured":"Burgin, M., and Debnath, N. (2010, January 22\u201324). Reusability as Design of Second-Level Algorithms. Proceedings of the ISCA 25th International Conference \u201cComputers and their Applications\u201d (CATA-2010), ISCA, Honolulu, HI, USA."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"694","DOI":"10.1007\/s00224-011-9369-9","article-title":"Second-level Algorithms, Superrecursivity, and Recovery Problem in Distributed Systems","volume":"50","author":"Burgin","year":"2012","journal-title":"Theory Comput. Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"390","DOI":"10.1162\/jocn.1993.5.4.390","article-title":"Development of elementary numerical abilities: A neuronal model","volume":"5","author":"Dehaene","year":"1993","journal-title":"J. Cogn. Neurosci."},{"key":"ref_23","unstructured":"Dehaene, S. (1997). The Number Sense: How the Mind Creates Mathematics, Oxford University Press."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Schroeder, M.J. (2017). Computing with Nature. Proceedings, 1.","DOI":"10.3390\/IS4SI-2017-04022"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Dodig-Crnkovic, G., and Burgin, M. (2011). Information and Computation, World Scientific.","DOI":"10.1142\/7637"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Zuse, K. (1969). Rechnender Raum, Friedrich Vieweg&Sohn.","DOI":"10.1007\/978-3-663-02723-2"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1016\/0167-2789(90)90186-S","article-title":"Digital Mechanics","volume":"45","author":"Fredkin","year":"1990","journal-title":"Phys. D"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1007\/s11023-011-9235-1","article-title":"Significance of Models of Computation from Turing Model to Natural Computation","volume":"21","year":"2011","journal-title":"Minds Mach."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Burgin, M., and Dodig-Crnkovic, G. (2011). Information and Computation\u2014Omnipresent and Pervasive. Information and Computation, World Scientific.","DOI":"10.1142\/7637"},{"key":"ref_30","unstructured":"Shiva, S.G. (2005). Advanced Computer Architectures, CRC Press."},{"key":"ref_31","unstructured":"Haykin, S. (1994). Neural Networks: A Comprehensive Foundation, Macmillan."},{"key":"ref_32","first-page":"175","article-title":"On the concept of algorithm","volume":"8","author":"Kolmogorov","year":"1953","journal-title":"Uspehi Mat. Nauk."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"490","DOI":"10.1137\/0209036","article-title":"Storage Modification Machines","volume":"9","year":"1980","journal-title":"SIAM J. Comput."},{"key":"ref_34","unstructured":"Kleinberg, J., and Tardos, E. (2006). Algorithm Design, Pearson-Addison-Wesley."},{"key":"ref_35","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_36","first-page":"1","article-title":"Structural Machines as a Mathematical Model of Biological and Chemical Computers","volume":"7","author":"Burgin","year":"2017","journal-title":"Theory Appl. Math. Comput. Sci."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Burgin, M., Mikkilineni, R., and Mittal, S. (2017). Knowledge processing as structure transformation. Proceedings, 1.","DOI":"10.3390\/IS4SI-2017-03988"},{"key":"ref_38","unstructured":"Burgin, M. (2010). Measuring Power of Algorithms, Computer Programs, and Information Automata, Nova Science Publishers."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1145\/321160.321170","article-title":"Computability of Recursive Functions","volume":"10","author":"Shepherdson","year":"1963","journal-title":"J. ACM"},{"key":"ref_40","unstructured":"von Neumann, J. (1949). Theory of Self-Reproducing Automata, University of Illinois Press. University of Illinois Lectures on the Theory and Organization of Complicated Automata, Edited and completed by Arthur W. Burks."},{"key":"ref_41","unstructured":"Minsky, M. (1967). Computation: Finite and Infinite Machines, Prentice-Hall."},{"key":"ref_42","unstructured":"Hopcroft, J.E., Motwani, R., and Ullman, J.D. (2007). Introduction to Automata Theory, Languages, and Computation, Addison Wesley."},{"key":"ref_43","first-page":"6","article-title":"Periodic Turing Machines","volume":"5","author":"Burgin","year":"2014","journal-title":"J. Comput. Technol. Appl. (JoCTA)"},{"key":"ref_44","first-page":"1","article-title":"Inductive Cellular Automata","volume":"1","author":"Burgin","year":"2015","journal-title":"Int. J. Data Struct. Algorithms"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Siegelman, H.T. (1999). Neural Networks and Analog Computation: Beyond the Turing Limit, Birkhauser.","DOI":"10.1007\/978-1-4612-0707-8"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/S0273-0979-1989-15750-9","article-title":"On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines","volume":"21","author":"Blum","year":"1989","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Blum, L., Cucker, F., Shub, M., and Smale, S. (1998). Complexity and Real Computation, Springer.","DOI":"10.1007\/978-1-4612-0701-6"},{"key":"ref_48","first-page":"1289","article-title":"Inductive Turing Machines","volume":"270","author":"Burgin","year":"1983","journal-title":"Not. Acad. Sci. USSR"},{"key":"ref_49","unstructured":"Burgin, M. (2003, January 13\u201315). Cluster Computers and Grid Automata. Proceedings of the ISCA 17th International Conference \u201cComputers and their Applications\u201d, International Society for Computers and their Applications, Honolulu, HI, USA."},{"key":"ref_50","unstructured":"Burgin, M. (2007, January 25\u201328). Interactive Hypercomputation. Proceedings of the 2007 International Conference on Foundations of Computer Science (FCS\u201907), Las Vegas, NV, USA."},{"key":"ref_51","unstructured":"Burgin, M. (1999, January 12\u201315). Super-recursive Algorithms as a Tool for High Performance Computing. Proceedings of the High Performance Computing Symposium, San Diego, CA, USA."}],"container-title":["Information"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2078-2489\/11\/2\/102\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T08:57:36Z","timestamp":1760173056000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2078-2489\/11\/2\/102"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,13]]},"references-count":51,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,2]]}},"alternative-id":["info11020102"],"URL":"https:\/\/doi.org\/10.3390\/info11020102","relation":{},"ISSN":["2078-2489"],"issn-type":[{"value":"2078-2489","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,13]]}}}