Input-Driven Tissue P Automata

We introduce several variants of input-driven tissue P automata where the rules to be applied only depend on the input symbol. Both strings and multisets are considered as input objects; the strings are either read from an input tape or defined by the sequence of symbols taken in, and the multisets are given in an input cell at the beginning of a computation, enclosed in a vesicle. Additional symbols generated during a computation are stored in this vesicle, too. An input is accepted when the vesicle reaches a final cell and it is empty. The computational power of some variants of input-driven tissue P automata is illustrated by examples and compared with the power of the input-driven variants of other automata as register machines and counter automata

Using Automata Theory to Solve Problems in Additive Number Theory

Additive number theory is the study of the additive properties of integers. Perhaps the best-known theorem is Lagrange’s result that every natural number is the sum of four squares. We study numbers whose base-k representations have certain interesting proper- ties. In particular, we look at palindromes, which are numbers whose base-k representations read the same forward and backward, and binary squares, which are numbers whose binary representation is some block repeated twice (like (36)_2 = 100100). We show that all natural numbers are the sum of four binary palindromes. We also show that all natural numbers are the sum of three base-3 palindromes, and are also the sum of three base-4 palindromes. We also show that every sufficiently large natural number is the sum of four binary squares. We establish these results using virtually no number theory at all. Instead, we construct automated proofs using automata. The general proof technique is to build an appropriate machine, and then run decision algorithms to establish our theorems