Using a result of Gurevich and Lewis on the word problem for finite semigroups, we give short proofs that the following theories are hereditarily undecidable: (1) finite graphs of vertex-degree at most 3; (2) finite nonvoid sets with two distinguished permutations; (3) finite-dimensional vector spaces over a finite field with two distinguished endomorphisms.

Let K be a finite graph with vertex set V = {x0, x1, …, xσ–1} and automorphism group G. It is assumed that G acts transitively on V. We can imagine that the vertices of K represent σ cities and a traveler visits the cities in a series of random flights. The traveler starts at a given city and in each flight, independently of the past journey, chooses a city at random as the destination. Denote by vn (n = 1, 2, …) the location of the traveler at the end of the nth flight, and by v0 the initial location. It is assumed that the transition probabilities P{vn = xj | vn–1 = xi}, xi ϵ V, xj ϵ V, do not depend on n and are invariant under the action of G on V. The main result of this paper consists in determining p(n), the probability that the traveler returns to the initial position at the end of the nth flight.