We present a two-parameter family $(G_{m,k})_{m, k \in \mathbb{N}_{\geq 2}}$, of finite, non-abelian random groups and propose that, for each fixed k, as m → ∞ the commuting graph of Gm,k is almost surely connected and of diameter k. We present heuristic arguments in favour of this conjecture, following the lines of classical arguments for the Erdős–Rényi random graph. As well as being of independent interest, our groups would, if our conjecture is true, provide a large family of counterexamples to the conjecture of Iranmanesh and Jafarzadeh that the commuting graph of a finite group, if connected, must have a bounded diameter. Simulations of our model yielded explicit examples of groups whose commuting graphs have all diameters from 2 up to 10.

We address the problem to know whether the relation induced by a one-rulelength-preserving rewrite system is rational. We partially answer to a conjecture of ÉricLilin who conjectured in 1991 that a one-rule length-preserving rewrite system is arational transduction if and only if the left-hand side u and theright-hand side v of the rule of the system are not quasi-conjugate orare equal, that means if u and v are distinct, there donot exist words x, y and z such thatu = xyz and v = zyx.We prove the only if part of this conjecture and identify two non trivialcases where the if part is satisfied.

We present a new incremental algorithm for minimising deterministic finite automata. Itruns in quadratic time for any practical application and may be halted at any point,returning a partially minimised automaton. Hence, the algorithm may be applied to a givenautomaton at the same time as it is processing a string for acceptance. We also includesome experimental comparative results.

In a 1976 paper published in Science, Knuth presented an algorithm to sample (non-uniform) self-avoiding walks crossing a square of side k. From this sample, he constructed an estimator for the number of such walks. The quality of this estimator is directly related to the (relative) variance of a certain random variable Xk. From his experiments, Knuth suspected that this variance was extremely large (so that the estimator would not be very efficient). But how large? For the analogous Rosenbluth algorithm, which samples unconfined self-avoiding walks of length n, the variance of the corresponding estimator is believed to be exponential in n.

A few years ago, Bassetti and Diaconis showed that, for a sampler à la Knuth that generates walks crossing a k × k square and consisting of North and East steps, the relative variance is only $O(\sqrt k)$. In this note we take one step further and show that, for walks consisting of North, South and East steps, the relative variance jumps to $2^{k(k+1)}/(k+1)^{2k}$. This is exponential in the average length of the walks, which is of order k2. We also obtain partial results for general self-avoiding walks crossing a square, suggesting that the relative variance could be exponential in k2 (which is again the average length of these walks).

Knuth's algorithm is a basic example of a widely used technique called sequential importance sampling. The present paper, following the paper by Bassetti and Diaconis, is one of very few examples where the variance of the estimator can be found.

A rational polyhedron$P\subseteq {\mathbb{R^n}}$ is a finite union of simplexes in ${\mathbb{R^n}}$ with rational vertices. P is said to be $\mathbb Z$-homeomorphic to the rational polyhedron $Q\subseteq {\mathbb{R^{\it m}}}$ if there is a piecewise linear homeomorphism η of P onto Q such that each linear piece of η and η−1 has integer coefficients. When n=m, $\mathbb Z$-homeomorphism amounts to continuous $\mathcal{G}_n$-equidissectability, where $\mathcal{G}_n=GL(n,\mathbb Z) \ltimes \mathbb Z^{n}$ is the affine group over the integers, i.e., the group of all affinities on $\mathbb{R^{n}}$ that leave the lattice $\mathbb Z^{n}$ invariant. $\mathcal{G}_n$ yields a geometry on the set of rational polyhedra. For each d=0,1,2,. . ., we define a rational measure λd on the set of rational polyhedra, and show that any two $\mathbb Z$-homeomorphic rational polyhedra $$P\subseteq {\mathbb{R^n}}$$ and $Q\subseteq {\mathbb{R^{\it m}}}$ satisfy $\lambda_d(P)=\lambda_d(Q)$. $\lambda_n(P)$ coincides with the n-dimensional Lebesgue measure of P. If 0 ≤ dim P=d < n then λd(P)>0. For rational d-simplexes T lying in the same d-dimensional affine subspace of ${\mathbb{R^{\it n}}, $\lambda_d(T)$$ is proportional to the d-dimensional Hausdorff measure of T. We characterize λd among all unimodular invariant valuations.

Based on the formal framework of reaction systems by Ehrenfeucht and Rozenberg[Fund. Inform. 75 (2007) 263–280], reaction automata (RAs)have been introduced by Okubo et al. [Theoret. Comput. Sci.429 (2012) 247–257], as language acceptors with multiset rewritingmechanism. In this paper, we continue the investigation of RAs with a focus on the twomanners of rule application: maximally parallel and sequential. Considering restrictionson the workspace and the λ-input mode, we introduce the correspondingvariants of RAs and investigate their computation powers. In order to explore Turingmachines (TMs) that correspond to RAs, we also introduce a new variant of TMs withrestricted workspace, called s(n)-restricted TMs. Themain results include the following: (i) for a language L and a functions(n), L is accepted by ans(n)-bounded RA with λ-input mode insequential manner if and only if L is accepted by alog s(n)-bounded one-way TM; (ii) if a languageL is accepted by a linear-bounded RA in sequential manner, thenL is also accepted by a P automaton [Csuhaj−Varju and Vaszil, vol. 2597of Lect. Notes Comput. Sci. Springer (2003) 219–233.] in sequentialmanner; (iii) the class of languages accepted by linear-bounded RAs in maximally parallelmanner is incomparable to the class of languages accepted by RAs in sequential manner.

Several types of systems of parallel communicating restarting automata are introduced andstudied. The main result shows that, for all types of restarting automata X, centralizedsystems of restarting automata of type X have the same computational power asnon-centralized systems of restarting automata of the same type and the same number ofcomponents. This result is proved by a direct simulation. In addition, it is shown thatfor one-way restarting automata without auxiliary symbols, systems of degree at least twoare more powerful than the component automata themselves. Finally, a lower and an upperbound are given for the expressive power of systems of parallel communicating restartingautomata.

The recently introduced model of transducing by observing is compared with traditionalmodels for computing transductions on the one hand and the recently introduced restartingtransducers on the other hand. Most noteworthy, transducing observer systems withlength-reducing rules are almost equivalent to RRWW-transducers. With painter rules weobtain a larger class of relations that additionally includes nearly all rationalrelations.