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.

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.

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 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.

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

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 system presented in this article is a minor variation of Gentzen's or Prawitz's formulation, but with a closer adherence to Martin-Löf's description of logical judgments and connectives ( Martin-Löf, 1996 ).

Intuitionistic type theory was introduced by Per Martin-Löf, a Swedish mathematician and philosopher, in 1972.

In particular, a very detailed treatment of the EM method for exponential families was published by Rolf Sundberg in his thesis and several papers following his collaboration with Per Martin-Löf and Anders Martin-Löf.

Per Martin-Löf is an enthusiastic bird-watcher, whose first scientific publication was on the mortality rates of ringed birds.

In the 1960s Per Martin-Löf noticed that the sets coding such frequency-based stochastic properties are a special kind of measure zero sets, and that a more general and smooth definition can be obtained by considering all effectively measure zero sets.

This provides the foundation for the intuitionistic type theory developed by Per Martin-Löf, and is often extended to a three way correspondence, the third leg of which are the cartesian closed categories.

Reprinted in The Collected Works of Harald Cramér edited by Anders Martin-Löf, 2 volumes Springer 1994.

In Martin-Löf type theory and higher-order Heyting arithmetic, the appropriate statement of the axiom of choice is ( depending on approach ) included as an axiom or provable as a theorem.

In the 1980s, Per Martin-Löf developed intuitionistic type theory ( also called Constructive type theory ), which associated functional programs with constructive proofs of arbitrarily complex mathematical propositions expressed as dependent types.

In 1966 Per Martin-Löf introduced a new notion which is now generally considered the most satisfactory notion of algorithmic randomness.

Intuitionistic type theory, or constructive type theory, or Martin-Löf type theory or just Type Theory is a logical system and a set theory based on the principles of mathematical constructivism.

* Per Martin-Löf.

Per Erik Rutger Martin-Löf ( born 1942 ) is a Swedish logician, philosopher, and mathematical statistician.

Per Martin-Löf holds a joint chair for Mathematics and Philosophy at Stockholm University.

In 1964 – 65 Martin-Löf studied in Moscow under the supervision of Andrei N. Kolmogorov.

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.

In recent weeks, as a result of a sweeping defense policy reappraisal by the Kennedy Administration, basic United States strategy has been modified -- and large new sums allocated -- to meet the accidental-war danger and to reduce it as quickly as possible.

There is one exception to the above statement as has been pointed out, and that is that fluids can relax by flowing into fields of lower rates of shear, so the statement should be modified by stating that the mechanics are similar.

Instead of having a pulp cavity, each tooth has a cluster of thin, upright, parallel tubes of vasodentin ( a modified form of dentine ), with individual pulp canals, held together by cementum.

This has led to their use as insecticides, and more recently to genetically modified crops using Bt genes.

The biotechnology sector has allowed the U. S. farming industry to rapidly increase its supply of corn and soybeans — the main inputs into biofuels — by developing genetically modified seeds which are resistant to pests and drought.

This microprogram is sometimes rewritable so that it can be modified to change the way the CPU decodes instructions even after it has been manufactured.

It has even been suggested that the original bony plates of all the vertebrates are gone and that the present scales are just modified teeth, even if both teeth and the body armor have a common origin a long time ago.

The Chilean government strongly supports foreign investment in the sector and has modified its mining industry laws and regulations to create a favorable investing environment for foreigners.

Where there are historical immigrant Chinese populations, the style of food has evolved and been adapted to local tastes and ingredients, and modified by the local cuisine, to greater or lesser extents.

The cane toad inhabits open grassland and woodland, and has displayed a " distinct preference " for areas that have been modified by humans, such as gardens and drainage ditches.

It entered service in 1990 and has a modified crane arm with bucket instead off the pincers.

This prose version has since been modified and reproduced widely by other authors.

Church subsequently modified his methods to include use of Herbrand – Gödel recursion and then proved ( 1936 ) that the Entscheidungsproblem is unsolvable: There is no generalized " effective calculation " ( method, algorithm ) that can determine whether or not a formula in either the recursive-or λ-calculus is " valid " ( more precisely: no method to show that a well formed formula has a " normal form ").

The order has been renewed and modified slightly in subsequent years, but it has allowed for more stable marketing.

She pilots a generic heavy spacecraft called Red Tail which is pale blue despite the name and has been heavily modified with armament and tracking sensors.

Captain America has also used a custom specialized motorcycle, modified by the S. H. I. E. L. D.

The circuit design has been modified and altered several times:

This seven-judge procedure has been modified as of the 2012 London Olympics: rather than eliminating one high and one low award and then reducing the total by as in previous international events, the two highest awards and the two lowest are disregarded, leaving three to be summed and multiplied by the difficulty rating.

The score is computed similarly to the scores from other diving events, but has been modified starting with the 2012 London Olympics for the use of the larger judging panels.

The song, or a modified version of it, has been used in the Roger Jones musical From Pharaoh to Freedom.

Writing in Cosmos, journalist Wilson da Silva reacted to Greenpeace's destruction of a genetically modified wheat crop in Ginninderra as another sign that the organization has " lost its way " and had degenerated into a " sad, dogmatic, reactionary phalanx of anti-science zealots who care not for evidence, but for publicity ".

A genetically modified organism ( GMO ) or engineered organism ( GEO ) is an organism whose genetic material has been altered using genetic engineering techniques.

Genetically modified crops ( GM crops, or biotech crops ) are plants, the DNA of which has been modified using genetic engineering techniques.