Randomness for Free

We consider two-player zero-sum games on graphs. These games can be classified on the basis of the information of the players and on the mode of interaction between them. On the basis of information the classification is as follows: (a) partial-observation (both players have partial view of the game); (b) one-sided complete-observation (one player has complete observation); and (c) complete-observation (both players have complete view of the game). On the basis of mode of interaction we have the following classification: (a) concurrent (both players interact simultaneously); and (b) turn-based (both players interact in turn). The two sources of randomness in these games are randomness in transition function and randomness in strategies. In general, randomized strategies are more powerful than deterministic strategies, and randomness in transitions gives more general classes of games. In this work we present a complete characterization for the classes of games where randomness is not helpful in: (a) the transition function probabilistic transition can be simulated by deterministic transition); and (b) strategies (pure strategies are as powerful as randomized strategies). As consequence of our characterization we obtain new undecidability results for these games

Sub-computable Boundedness Randomness

This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for PSPACE functions. These new notions are robust in that there are equivalent formulations in terms of (1) Martin-L\"of tests, (2) Kolmogorov complexity, and (3) martingales. We show these notions can be equivalently defined with prefix-free Kolmogorov complexity. We prove that one direction of van Lambalgen's theorem holds for relative computability, but the other direction fails. We discuss statistical properties of these notions of randomness

Mistaking Randomness for Free Will

Belief in free will is widespread. The present research considered one reason why people may believe that actions are freely chosen rather than determined: they attribute randomness in behavior to free will. Experiment 1 found that participants who were prompted to perform a random sequence of actions experienced their behavior as more freely chosen than those who were prompted to perform a deterministic sequence. Likewise, Experiment 2 found that, all else equal, the behavior of animated agents was perceived to be more freely chosen if it consisted of a random sequence of actions than if it consisted of a deterministic sequence; this was true even when the degree of randomness in agents’ behavior was largely a product of their environments. Together, these findings suggest that randomness in behavior—one’s own or another’s—can be mistaken for free will.Psycholog

Kolmogorov Complexity and Solovay Functions

Solovay proved that there exists a computable upper bound f of the prefix-free Kolmogorov complexity function K such that f (x) = K(x) for infinitely many x. In this paper, we consider the class of computable functions f such that K(x) <= f (x)+O(1) for all x and f (x) <= K(x) + O(1) for infinitely many x, which we call Solovay functions. We show that Solovay functions present interesting connections with randomness notions such as Martin-L\"of randomness and K-triviality