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We prove the effective version of Birkhoff's ergodic theorem for Martin-Löf random points and effectively open sets, improving the results previously obtained in this direction (in particular those of V. Vyugin, Nandakumar and Hoyrup, Rojas). The proof consists of two steps. First, we prove a generalization of Kučera's theorem, which is a particular case of(More)
We show that the uniform validity is equivalent to the non-uniform validity for Blass' semantics of [A game semantics for linear logic. Annals of Pure and Applied Logic 56 (1992) 183–220]. We present a shorter proof (than that of [G. Japaridze. The intuitionistic fragment of computability logic at the propositional level. Annals of Pure and Applied Logic(More)
The Kolmogorov complexity function K can be relativized using any oracle A, and most properties of K remain true for relativized versions K A. In section 1 we provide an explanation for this observation by giving a game-theoretic interpretation and showing that all " natural " properties are either true for all K A or false for all K A if we restrict(More)
A theorem of Kučera states that given a Martin-Löf random infinite binary sequence ω and an effectively open set A of measure less than 1, some tail of ω is not in A. We show that this result can be seen as an effective version of Birkhoff's ergodic theorem (in a special case). We prove several results in the same spirit and generalize them via an effective(More)
We show that the uniform validity is equivalent to the non-uniform validity for both Blass' semantics of [1] and Japaridze's semantics of [5] (thus proving a conjecture from [5]). We present a shorter proof (than that of [10]) of the completeness of the positive fragment of intuitionistic logic for both these semantics. Finally, we show that validity for(More)
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