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This definition can be extended to define a notion of randomness for ''infinite'' sequences from a finite alphabet. These [[algorithmically random sequence]]s can be defined in three equivalent ways. One way uses an effective analogue of [[measure theory]]; another uses effective [[Martingale (probability theory)|martingales]]. The third way defines an infinite sequence to be random if the prefix-free Kolmogorov complexity of its initial segments grows quickly enough — there must be a constant ''c'' such that the complexity of an initial segment of length ''n'' is always at least ''n''−''c''. This definition, unlike the definition of randomness for a finite string, is not affected by which universal machine is used to define prefix-free Kolmogorov complexity.<ref>{{cite journal |doi=10.1016/s0019-9958(66)80018-9 |title=The definition of random sequences |journal=Information and Control |volume=9 |issue=6 |pages=602–619 |year=1966 |last=Martin-Löf |first=Per |doi-access=free }}</ref>
== Applications ==
Kolmogorov complexity has found increasing use in the analysis of real-world systems, particularly for uncovering structure and [[causality]] in data-rich but poorly understood domains.<ref>{{cite journal |last1=Zenil |first1=Hector |last2=Kiani |first2=Narsis A. |last3=Zea |first3=Allan A. |last4=Tegnér |first4=Jesper |title=Causal deconvolution by algorithmic generative models |journal=Nature Machine Intelligence |volume=1 |issue=1 |year=2019 |pages=58-66 |doi=10.1038/s42256-018-0005-0 }}</ref> One notable example is the conceptual framework introduced by Zenil et al. (2019) for extracting generative rules from complex [[Dynamical system|dynamical systems]]. Based on [[Algorithmic information theory]] (AIT) and an associated algorithmic information calculus (AIC),<ref>{{cite journal |last1=Zenil |first1=Hector |last2=Kiani |first2=N. A. |last3=Marabita |first3=F. |last4=Deng |first4=Y. |last5=Elias |first5=S. |last6=Schmidt |first6=A. |last7=Ball |first7=G. |last8=Tegnér |first8=J. |title=An Algorithmic Information Calculus for Causal Discovery and Reprogramming Systems |journal=Science |year=2019 |doi=10.1016/j.sci.2019.30270-6 }}</ref> the method—called Algorithmic Information Dynamics (AID)<ref>{{cite journal | last=Zenil | first=Hector | title=Algorithmic Information Dynamics | journal=Scholarpedia | date=25 July 2020 | volume=15 | issue=7 | doi=10.4249/scholarpedia.53143 | doi-access=free | bibcode=2020SchpJ..1553143Z | hdl=10754/666314 | hdl-access=free }}</ref>—was used to reconstruct phase spaces and identify causal mechanisms of discrete systems (including cellular automata). Their approach applied perturbation analysis to quantify the algorithmic complexity of system components, enabling reconstruction of the system’s generative rules without requiring explicit kinetic equations. This method provided insights into the causal structure of systems and their reprogrammability toward desired states.<ref> {{cite book | last1=Zenil | first1=Hector | last2=Kiani | first2=Narsis A. | last3=Tegner | first3=Jesper | title=Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems | publisher=Cambridge University Press | year=2023 | doi=10.1017/9781108596619 | isbn=978-1-108-59661-9 | url=https://doi.org/10.1017/9781108596619}}</ref>
==Relation to entropy==
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