Fixed points and admissible sets

Counting Lattice Points in Admissible Adelic Sets

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...

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 ...

Diagonal arguments and fixed points

‎A universal schema for diagonalization was popularized by N.S‎. ‎Yanofsky (2003)‎, ‎based on a pioneering work of F.W‎. ‎Lawvere (1969)‎, ‎in which the existence of a (diagonolized-out and contradictory) object implies the existence of a fixed-point for a certain function‎. ‎It was shown that many self-referential paradoxes and diagonally proved theorems can fit in that schema‎. ‎Here‎, ‎we fi...

Fixed points and maximal independent sets in AND-OR networks

We study the maximum number of 2xed points of boolean networks with local update function AND–OR. We prove that this number for networks with connected digraph is 2(n−1)=2 for n odd and 2(n−2)=2 + 1 for n even if the digraph has not loops; and 2n−1 + 1 otherwise, where n is the number of nodes of the digraph. We also exhibit some networks reaching these bounds. To obtain these results we constr...