Пер Эрик Мартин Лёф: Логик, философ және математик-статистик ғалымның өмірі мен еңбектері

Swedish logician, philosopher, and mathematical statistician

Per Erik Rutger Martin Löf (/l//ɒ//f/; ; born 8 May 1942) is a Swedish logician, philosopher, and mathematical statistician. He is internationally renowned for his work on the foundations of probability, statistics, mathematical logic, and computer science. Since the late 1970s, Martin Löf's publications have been mainly in logic. In philosophical logic, Martin Löf has wrestled with the philosophy of logical consequence and judgment, partly inspired by the work of Brentano, Frege, and Husserl. In mathematical logic, Martin Löf has been active in developing intuitionistic type theory as a constructive foundation of mathematics; Martin Löf's work on type theory has influenced computer science. Until his retirement in 2009, Per Martin Löf held a joint chair for Mathematics and Philosophy at Stockholm University. His brother Anders Martin Löf is now emeritus professor of mathematical statistics at Stockholm University; the two brothers have collaborated in research in probability and statistics. The research of Anders and Per Martin Löf has influenced statistical theory, especially concerning exponential families, the expectation maximization method for missing data, and model selection. Per Martin Löf received his PhD in 1970 from Stockholm University, under Andrey Kolmogorov. Martin Löf is an enthusiastic bird watcher; his first scientific publication was on the mortality rates of ringed birds.

In 1964 and 1965, Martin Löf studied in Moscow under the supervision of Andrei N. Kolmogorov. He wrote a 1966 article The definition of random sequences that gave the first suitable definition of a random sequence. Earlier researchers such as Richard von Mises had attempted to formalize the notion of a test for randomness in order to define a random sequence as one that passed all tests for randomness; however, the precise notion of a randomness test was left vague. Martin Löf's key insight was to use the theory of computation to define formally the notion of a test for randomness. This contrasts with the idea of randomness in probability; in that theory, no particular element of a sample space can be said to be random. Martin Löf randomness has since been shown to admit many equivalent characterizations — in terms of compression, randomness tests, and gambling — that bear little outward resemblance to the original definition, but each of which satisfies our intuitive notion of properties that random sequences ought to have: random sequences should be incompressible, they should pass statistical tests for randomness, and it should be impossible to make money betting on them. The existence of these multiple definitions of Martin Löf randomness, and the stability of these definitions under different models of computation, give evidence that Martin Löf randomness is a fundamental property of mathematics and not an accident of Martin Löf's particular model. The thesis that the definition of Martin Löf randomness "correctly" captures the intuitive notion of randomness has been called the "Martin Löf–Chaitin Thesis"; it is somewhat similar to the Church–Turing thesis. Following Martin Löf's work, algorithmic information theory defines a random string as one that cannot be produced from any computer program that is shorter than the string (Chaitin–Kolmogorov randomness); i. e. a string whose Kolmogorov complexity is at least the length of the string. This is a different meaning from the usage of the term in statistics. Whereas statistical randomness refers to the process that produces the string (e. g. flipping a coin to produce each bit will randomly produce a string), algorithmic randomness refers to the string itself. Algorithmic information theory separates random from nonrandom strings in a way that is relatively invariant to the model of computation being used. An algorithmically random sequence is an infinite sequence of characters, all of whose prefixes (except possibly a finite number of exceptions) are strings that are "close to" algorithmically random (their length is within a constant of their Kolmogorov complexity).

Per Martin Löf has done important research in mathematical statistics, which (in the Swedish tradition) includes probability theory and statistics.

Per Martin Löf began bird watching in his youth and remains an enthusiastic bird watcher. As a teenager, he published an article on estimating the mortality rates of birds, using data from bird ringing, in a Swedish zoological journal: This paper was soon cited in leading international journals, and this paper continues to be cited. In the biology and statistics of birds, there are several problems of missing data. Martin Löf's first paper discussed the problem of estimating the mortality rates of the Dunlin species, using capture recapture methods. The problem of determining the biological sex of a bird, which is extremely difficult for humans, is one of the first examples in Martin Löf's lectures on statistical models.

Martin Löf wrote a licenciate thesis on probability on algebraic structures, particularly semigroups, while a student of Ulf Grenander at Stockholm University.

Martin Löf developed innovative approaches to statistical theory. In his paper "On Tables of Random Numbers", Kolmogorov observed that the frequency probability notion of the limiting properties of infinite sequences failed to provide a foundation for statistics, which considers only finite samples. Much of Martin Löf's work in statistics was to provide a finite sample foundation for statistics.

In the 1970s, Per Martin Löf made important contributions to statistical theory and inspired further research, especially by Scandinavian statisticians including Rolf Sundberg, Thomas Höglund, and Steffan Lauritzen. In this work, Martin Löf's previous research on probability measures on semigroups led to a notion of "repetitive structure" and a novel treatment of sufficient statistics, in which one parameter exponential families were characterized. He provided a category theoretic approach to nested statistical models, using finite sample principles. Before (and after) Martin Löf, such nested models have often been tested using chi square hypothesis tests, whose justifications are only asymptotic (and so irrelevant to real problems, which always have finite samples). Many of these results reached the international scientific community through the 1976 paper on the expectation maximization (EM) method by Arthur P. Dempster, Nan Laird, and Donald Rubin, which was published in a leading international journal, sponsored by the Royal Statistical Society.

In philosophical logic, Per Martin Löf has published papers on the theory of logical consequence, on judgments, etc. He has been interested in Central European philosophical traditions, especially of the German language writings of Franz Brentano, Gottlob Frege, and of Edmund Husserl.

Martin Löf has worked in mathematical logic for many decades. From 1968 to '69 he worked as an assistant professor at the University of Chicago where he met William Alvin Howard with whom he discussed issues related to the Curry–Howard correspondence. Martin Löf's first draft article on type theory dates back to 1971. This impredicative theory generalized Girard's System F. However, this system turned out to be inconsistent due to Girard's paradox which was discovered by Girard when studying System U, an inconsistent extension of System F. This experience led Per Martin Löf to develop the philosophical foundations of type theory, his meaning explanation, a form of proof theoretic semantics, which justifies predicative type theory as presented in his 1984 Bibliopolis book, and extended in a number of increasingly philosophical texts, such as his influential On the Meanings of the Logical Constants and the Justifications of the Logical Laws. The 1984 type theory was extensional while the type theory presented in the book by Nordström et al. in 1990, which was heavily influenced by his later ideas, intensional, and more amenable to being implemented on a computer. Martin Löf's intuitionistic type theory developed the notion of dependent types and directly influenced the development of the calculus of constructions and the logical framework LF. A number of popular computer based proof systems are based on type theory, for example NuPRL, LEGO, Coq, ALF, Agda, Twelf, Epigram, and Idris.

Martin Löf is a member of the Royal Swedish Academy of Sciences (elected 1990) and of the Academia Europaea (elected 1989).

George A. Barnard, "Gone Birdwatching", New Scientist, 4 December 1999, magazine issue 2215.

Per Martin Löf. "The Definition of Random Sequences." Information and Control, 9(6): 602–619, 1966. Li, Ming and Vitányi, Paul, An Introduction to Kolmogorov Complexity and Its Applications, Springer, 1997. Introduction chapter full text.

Grenander, Ulf. Probability on Algebraic Structures. (Dover reprint)
Martin Löf, P. The continuity theorem on a locally compact group. Teor. Verojatnost. i Primenen. 10 1965 367–371. Martin Löf, Per. Probability theory on discrete semigroups. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 4 1965 78—102
Nitis Mukhopadhyay. "A Conversation with Ulf Grenander". Statist. Sci. Volume 21, Number 3 (2006), 404–426.

Anders Martin Löf. 1963. "Utvärdering av livslängder i subnanosekundsområdet" ("Evaluation of lifetimes in time lengths below one nanosecond"). ("Sundberg formula", according to Sundberg 1971)
Per Martin Löf. 1966. Statistics from the point of view of statistical mechanics. Lecture notes, Mathematical Institute, Aarhus University. ("Sundberg formula" credited to Anders Martin Löf, according to Sundberg 1971)
Per Martin Löf. 1970. Statistika Modeller (Statistical Models): Anteckningar fran seminarier läsåret 1969–1970 (Notes from seminars in the academic year 1969–1970), with the assistance of Rolf Sundberg. Stockholm University. Martin Löf, P. "Exact tests, confidence regions and estimates", with a discussion by A. W. F. Edwards, G. A. Barnard, D. A. Sprott, O. Barndorff Nielsen, D. Basu and G. Rasch. Proceedings of Conference on Foundational Questions in Statistical Inference (Aarhus, 1973), pp. 121–138. Memoirs, No. 1, Dept. Theoret. Statist., Inst. Math., Univ. Aarhus, Aarhus, 1974. Martin Löf, P. Repetitive structures and the relation between canonical and microcanonical distributions in statistics and statistical mechanics. With a discussion by D. R. Cox and G. Rasch and a reply by the author. Proceedings of Conference on Foundational Questions in Statistical Inference (Aarhus, 1973), pp. 271–294. Memoirs, No. 1, Dept. Theoret. Statist., Inst. Math., Univ. Aarhus, Aarhus, 1974. Martin Löf, P. The notion of redundancy and its use as a quantitative measure of the deviation between a statistical hypothesis and a set of observational data. With a discussion by F. Abildgård, A. P. Dempster, D. Basu, D. R. Cox, A. W. F. Edwards, D. A. Sprott, G. A. Barnard, O. Barndorff Nielsen, J. D. Kalbfleisch and G. Rasch and a reply by the author. Proceedings of Conference on Foundational Questions in Statistical Inference (Aarhus, 1973), pp. 1–42. Memoirs, No. 1, Dept. Theoret. Statist., Inst. Math., Univ. Aarhus, Aarhus, 1974. Martin Löf, Per The notion of redundancy and its use as a quantitative measure of the discrepancy between a statistical hypothesis and a set of observational data. Scand. J. Statist. 1 (1974), no. 1, 3—18. Sverdrup, Erling. "Tests without power." Scand. J. Statist. 2 (1975), no. 3, 158–160. Martin Löf, Per Reply to Erling Sverdrup's polemical article: Tests without power (Scand. J. Statist. 2 (1975), no. 3, 158–160). Scand. J. Statist. 2 (1975), no. 3, 161–165. Sverdrup, Erling. A rejoinder to: Tests without power (Scand. J. Statist. 2 (1975), 161—165) by P. Martin Löf. Scand. J. Statist. 4 (1977), no. 3, 136—138. Martin Löf, P. Exact tests, confidence regions and estimates. Foundations of probability and statistics. II. Synthese 36 (1977), no. 2, 195—206. Rolf Sundberg. 1971. Maximum likelihood theory and applications for distributions generated when observing a function of an exponential family variable. Dissertation, Institute for Mathematical Statistics, Stockholm University. Sundberg, Rolf. Maximum likelihood theory for incomplete data from an exponential family. Scand. J. Statist. 1 (1974), no. 2, 49—58. Sundberg, Rolf An iterative method for solution of the likelihood equations for incomplete data from exponential families. Comm. Statist.—Simulation Comput. B5 (1976), no. 1, 55—64. Sundberg, Rolf Some results about decomposable (or Markov type) models for multidimensional contingency tables: distribution of marginals and partitioning of tests. Scand. J. Statist. 2 (1975), no. 2, 71—79. Höglund, Thomas. The exact estimate — a method of statistical estimation. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 29 (1974), 257—271. Lauritzen, Steffen L. Extremal families and systems of sufficient statistics. Lecture Notes in Statistics, 49. Springer Verlag, New York, 1988. xvi+268 pp.

Per Martin Löf. A theory of types. Preprint, Stockholm University, 1971. Per Martin Löf. An intuitionistic theory of types. In G. Sambin and J. Smith, editors, Twenty Five Years of Constructive Type Theory. Oxford University Press, 1998. Reprinted version of an unpublished report from 1972. Per Martin Löf. An intuitionistic theory of types: Predicative part. In H. E. Rose and J. C. Shepherdson, editors, Logic Colloquium ‘73, pages 73–118. North Holland, 1975. Per Martin Löf. Constructive mathematics and computer programming. In Logic, Methodology and Philosophy of Science VI, 1979. Eds. Cohen, et al. North Holland, Amsterdam. pp. 153–175, 1982. Per Martin Löf. Intuitionistic type theory. (Notes by Giovanni Sambin of a series of lectures given in Padua, June 1980). Napoli, Bibliopolis, 1984. Per Martin Löf. Philosophical implications of type theory, Unpublished notes, 1987? Per Martin Löf. Substitution calculus, 1992. Notes from a lecture given in Göteborg. Bengt Nordström, Kent Petersson, and Jan M. Smith. Programming in Martin Löf's Type Theory. Oxford University Press, 1990. (The book is out of print, but a free version has been made available.) Per Martin Löf. On the Meanings of the Logical Constants and the Justifications of the Logical Laws. Nordic Journal of Philosophical Logic, 1(1): 11–60, 1996. Per Martin Löf. Logic and Ethics. In T. Piecha and P. Schroeder Heister, editors, Proof Theoretic Semantics: Assessment and Future Perspectives. Proceedings of the Third Tübingen Conference on Proof Theoretic Semantics, 27–30 March 2019, pages 227 235. URI: http://dx. doi. org/10.15496/publikation 35319. University of Tübingen 2019.