-: Per Martin-Löf :-
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| Per Martin-Löf |
Per Erik Rutger Martin-Löf (born 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.
Per Martin-Löf holds 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 regarding exponential families, the expectation-maximization method for missing data, and model selection.
Per Martin-Löf is an enthusiastic bird-watcher, whose first scientific publication was on the mortality rates of ringed birds.
In 1964–65 Martin-Löf was studying in Moscow under the supervision of Andrei N. Kolmogorov. During this time, Martin-Löf wrote his 1966 article On the definition of random sequences, which 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 formally define 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 satisfy 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 difficult 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.
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).
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.
Per Martin-Löf holds 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 regarding exponential families, the expectation-maximization method for missing data, and model selection.
Per Martin-Löf is an enthusiastic bird-watcher, whose first scientific publication was on the mortality rates of ringed birds.
In 1964–65 Martin-Löf was studying in Moscow under the supervision of Andrei N. Kolmogorov. During this time, Martin-Löf wrote his 1966 article On the definition of random sequences, which 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 formally define 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 satisfy 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 difficult 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.
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).
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