{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T17:36:16Z","timestamp":1775669776332,"version":"3.50.1"},"reference-count":43,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,5]],"date-time":"2022-08-05T00:00:00Z","timestamp":1659657600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002726","name":"Beijing Normal University","doi-asserted-by":"publisher","award":["28705-310432106"],"award-info":[{"award-number":["28705-310432106"]}],"id":[{"id":"10.13039\/501100002726","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The notion of information and complexity are important concepts in many scientific fields such as molecular biology, evolutionary theory and exobiology. Many measures of these quantities are either difficult to compute, rely on the statistical notion of information, or can only be applied to strings. Based on assembly theory, we propose the notion of a ladderpath, which describes how an object can be decomposed into hierarchical structures using repetitive elements. From the ladderpath, two measures naturally emerge: the ladderpath-index and the order-index, which represent two axes of complexity. We show how the ladderpath approach can be applied to both strings and spatial patterns and argue that all systems that undergo evolution can be described as ladderpaths. Further, we discuss possible applications to human language and the origin of life. The ladderpath approach provides an alternative characterization of the information that is contained in a single object (or a system) and could aid in our understanding of evolving systems and the origin of life in particular.<\/jats:p>","DOI":"10.3390\/e24081082","type":"journal-article","created":{"date-parts":[[2022,8,7]],"date-time":"2022-08-07T22:51:46Z","timestamp":1659912706000},"page":"1082","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Ladderpath Approach: How Tinkering and Reuse Increase Complexity and Information"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2640-6490","authenticated-orcid":false,"given":"Yu","family":"Liu","sequence":"first","affiliation":[{"name":"International Academic Center of Complex Systems, Beijing Normal University, Zhuhai 519087, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zengru","family":"Di","sequence":"additional","affiliation":[{"name":"International Academic Center of Complex Systems, Beijing Normal University, Zhuhai 519087, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8503-0177","authenticated-orcid":false,"given":"Philip","family":"Gerlee","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Chalmers University of Technology, 405 30 Gothenburg, Sweden"},{"name":"Department of Mathematical Sciences, University of Gothenburg, 405 30 Gothenburg, Sweden"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,5]]},"reference":[{"key":"ref_1","unstructured":"Smith, J.M. (2022, August 02). 20th Century Biology as A Science of Information (95\/102). YouTube Video. Available online: https:\/\/youtube.com\/watch?v=78ikxE5-POY."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"20180041","DOI":"10.1098\/rsfs.2018.0041","article-title":"Semantic information, autonomous agency and non-equilibrium statistical physics","volume":"8","author":"Kolchinsky","year":"2018","journal-title":"Interface Focus"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1103\/PhysRevLett.63.105","article-title":"Inferring statistical complexity","volume":"63","author":"Crutchfield","year":"1989","journal-title":"Phys. Rev. Lett."},{"key":"ref_4","unstructured":"Fellerman, H., D\u00f6rr, M., Hanczy, M., Ladegaard Laursen, L., Mauer, S., Merkle, D., Monnard, P.A., St\u00f8y, K., and Rasmussen, S. (2010). Weak Emergence and Complexity. Artificial Life XII, Proceedings of the Twelfth International Conference on the Synthesis and Simulation of Living Systems, MIT Press."},{"key":"ref_5","first-page":"3","article-title":"Three approaches to the quantitative definition of information","volume":"1","author":"Kolmogorov","year":"1965","journal-title":"Probl. Peredachi Inf. [Probl. Inf. Transm.]"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1145\/321356.321363","article-title":"On the length of programs for computing finite binary sequences","volume":"13","author":"Chaitin","year":"1966","journal-title":"J. ACM"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1145\/321495.321506","article-title":"On the length of programs for computing finite binary sequences: Statistical considerations","volume":"16","author":"Chaitin","year":"1969","journal-title":"J. ACM"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Chaitin, G.J. (1992). Information-Theoretic Incompleteness, World Scientific.","DOI":"10.1142\/1861"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1070\/RM1970v025n06ABEH001269","article-title":"The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms","volume":"25","author":"Zvonkin","year":"1970","journal-title":"Russ. Math. Surv."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Vit\u00e1nyi, P.M.B. (2020). How incomputable is Kolmogorov complexity?. Entropy, 22.","DOI":"10.3390\/e22040408"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1109\/TIT.1976.1055501","article-title":"On the Complexity of Finite Sequences","volume":"22","author":"Lempel","year":"1976","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1109\/TIT.1977.1055714","article-title":"A universal algorithm for sequential data compression","volume":"23","author":"Ziv","year":"1977","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"530","DOI":"10.1109\/TIT.1978.1055934","article-title":"Compression of individual sequences via variable-rate coding","volume":"24","author":"Ziv","year":"1978","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1016\/S0167-2789(99)00179-7","article-title":"Physical complexity of symbolic sequences","volume":"137","author":"Adami","year":"2000","journal-title":"Phys. Nonlinear Phenom."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"907","DOI":"10.1007\/BF00668821","article-title":"Towards a quantitative theory of self-generated complexity","volume":"25","author":"Grassberger","year":"1986","journal-title":"Int. J. Theory Phys."},{"key":"ref_17","first-page":"51","article-title":"The information complexity of learning tasks, their structure and their distance","volume":"10","author":"Achille","year":"2021","journal-title":"Inf. Inference J. IMA"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Makowski, M., Piotrowski, E.W., Fr\u0105ckiewicz, P., and Szopa, M. (2021). Transactional Interpretation for the Principle of Minimum Fisher Information. Entropy, 23.","DOI":"10.3390\/e23111464"},{"key":"ref_19","unstructured":"Li, M., and Vitanyi, P. (2008). An Introduction to Kolmogorov Complexity and Its Applications, Springer. [3rd ed.]."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Finkel, A., and Jantzen, M. (1992, January 13\u201315). Compression and entropy. Proceedings of the STACS 92: 9th Annual Symposium on Theoretical Aspects of Computer Science, Cachan, France.","DOI":"10.1007\/3-540-55210-3"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1270","DOI":"10.1109\/18.243444","article-title":"A measure of relative entropy between individual sequences with application to universal classification","volume":"39","author":"Ziv","year":"1993","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1161","DOI":"10.1126\/science.860134","article-title":"Evolution and tinkering","volume":"196","author":"Jacob","year":"1977","journal-title":"Science"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3192","DOI":"10.1073\/pnas.0409515102","article-title":"Inferring network mechanisms: The Drosophila melanogaster protein interaction network","volume":"102","author":"Middendorf","year":"2005","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"921","DOI":"10.1038\/nrg2267","article-title":"The road to modularity","volume":"8","author":"Wagner","year":"2007","journal-title":"Nat. Rev. Genet."},{"key":"ref_25","unstructured":"Valverde, S., and Sole, R. (2007). Hierarchical Small Worlds in Software Architecture. arXiv."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"20190325","DOI":"10.3389\/fphys.2017.00497","article-title":"Breakdown of Modularity in Complex Networks","volume":"8","author":"Valverde","year":"2017","journal-title":"Front. Physiol."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"20190325","DOI":"10.1098\/rstb.2019.0325","article-title":"Evolving complexity: How tinkering shapes cells, software and ecological networks","volume":"375","author":"Valverde","year":"2020","journal-title":"Philos. Trans. R. Soc. Biol. Sci."},{"key":"ref_28","unstructured":"Knuth, D. (1997). Evaluation of Powers. Art of Computer Programming, Volume 2: Seminumerical Algorithms, Addison-Wesley Professional. [3rd ed.]."},{"key":"ref_29","first-page":"20160342","article-title":"A probabilistic framework for identifying biosignatures using Pathway Complexity","volume":"375","author":"Marshall","year":"2017","journal-title":"Philos. Trans. R. Soc. Math. Phys. Eng. Sci."},{"key":"ref_30","unstructured":"Murray, A., Marshall, S., and Cronin, L. (2018). Defining Pathway Assembly and Exploring its Applications. arXiv."},{"key":"ref_31","unstructured":"Marshall, S.M., Moore, D., Murray, A.R.G., Walker, S.I., and Cronin, L. (2019). Quantifying the pathways to life using assembly spaces. arXiv."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"3033","DOI":"10.1038\/s41467-021-23258-x","article-title":"Identifying molecules as biosignatures with assembly theory and mass spectrometry","volume":"12","author":"Marshall","year":"2021","journal-title":"Nat. Commun."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"eabj2465","DOI":"10.1126\/sciadv.abj2465","article-title":"Exploring and mapping chemical space with molecular assembly trees","volume":"7","author":"Liu","year":"2021","journal-title":"Sci. Adv."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"638","DOI":"10.1137\/0210047","article-title":"Computing sequences with addition chains","volume":"10","author":"Downey","year":"1981","journal-title":"SIAM J. Comput."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1016\/j.jtbi.2003.11.020","article-title":"Detecting autocatalytic, self-sustaining sets in chemical reaction systems","volume":"227","author":"Hordijk","year":"2004","journal-title":"J. Theor. Biol."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"18854","DOI":"10.1074\/jbc.RA118.003795","article-title":"Mathematical modeling reveals spontaneous emergence of self-replication in chemical reaction systems","volume":"293","author":"Liu","year":"2018","journal-title":"J. Biol. Chem."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1186\/s13062-020-00269-0","article-title":"On the definition of a self-sustaining chemical reaction system and its role in heredity","volume":"15","author":"Liu","year":"2020","journal-title":"Biol. Direct"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"9","DOI":"10.2174\/1874196700801010009","article-title":"Finite universe of discourse: The systems biology of Walter Elsasser (1904\u20131991)","volume":"1","author":"Gatherer","year":"2008","journal-title":"Open Biol. J."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1002\/cbic.200800554","article-title":"Biological implications of the ribosome\u2019s stunning stereochemistry","volume":"10","author":"Zimmerman","year":"2009","journal-title":"ChemBioChem"},{"key":"ref_40","unstructured":"Schr\u00f6dinger, E. (1944). What Is Life? The Physical Aspect of the Living Cell, Cambridge University Press."},{"key":"ref_41","first-page":"8769060","article-title":"Philosophical analysis of the meaning and nature of entropy and negative entropy theories","volume":"2020","author":"Wu","year":"2020","journal-title":"Complexity"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"034113","DOI":"10.1063\/1.5111333","article-title":"The generalized Boltzmann distribution is the only distribution in which the Gibbs-Shannon entropy equals the thermodynamic entropy","volume":"151","author":"Gao","year":"2019","journal-title":"J. Chem. Phys."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Schroeder, D.V. (2021). An Introduction to Thermal Physics, Oxford University Press.","DOI":"10.1093\/oso\/9780192895547.001.0001"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/8\/1082\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:04:39Z","timestamp":1760141079000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/8\/1082"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,5]]},"references-count":43,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["e24081082"],"URL":"https:\/\/doi.org\/10.3390\/e24081082","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints202207.0255.v1","asserted-by":"object"}]},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,8,5]]}}}