Random tree recursions: Which fixed points correspond to tangible sets of trees?

Random closed sets viewed as random recursions

It is known that the box dimension of any Martin-Löf random closed set of {0, 1}N is log2( 43 ). Barmpalias et al. [Journal of Logic and Computation, Vol. 17, No. 6 (2007)] gave one method of producing such random closed sets and then computed the box dimension, and posed several questions regarding other methods of construction. We outline a method using random recursive constructions for comp...

Mobs, Trees, and Fixed Points

Evaluating Polynomials at Fixed Sets of Points

We investigate the evaluation of an (n-1)st degree polynomial at a sequence of n points. It is shown that such an evaluation reduces directly to a simple convolution if and only if the sequence of points is of the form b, ba, ba2, ., ba for complex numbers a and b (the so-called "chirp transform"). By more complex reductions we develop an O(n log n) evaluation algorithm for sequences of points ...

On Sets, Types, Fixed Points, and Checkerboards

Most current research on automated theorem proving is concerned with proving theorems of first-order logic. We discuss two examples which illustrate the advantages of using higher-order logic in certain contexts. For some purposes type theory is a much more convenient vehicle for formalizing mathematics than axiomatic set theory. Even theorems of first-order logic can sometimes be proven more e...

Common random fixed points of compatible random operators

The study of random fixed point theory was initiated by the Prague school of probabilists in the 1950s [12, 13, 26]. Random fixed point theorems are stochastic generalization of classical fixed point theorems. The survey article by Bharucha-Reid [10] attracted the attention of several mathematicians and gave wings to this theory. Itoh [16] extended Spacek’s and Hans’s theorem to multivalued con...