section titles are poorly
chosen (Section 3 is called "One-way multi-head automata", despite the fact that Definition 3 is about two-way multi-head automata).

verb tenses change between sentences (p. 1: "Balodis et al. have studied .. They show ... ")

Page 8-9, Definition 5: there must be some missing assumption that the
machine is linearly bounded (it does not write to the right of the
input word); otherwise the positions of the heads are not bounded by
log n, which is used in many places.

Page 10, "space of size O(kn)": the definition of ultrametric
automaton gives that it may accept words of arbitrary length, so here
you cannot simulate the automaton on arbitrary input. But in the
following proofs you use only words of bounded size, so you can
reformulate the simulation result to make it both true and useful in
the construction.

Proof of Theorem 2: Please mention that what is happening here is diagonalization.

Sec. 4.2 and on: in my opinion, all the following material should be completely rewritten: too many models, too sketchy, badly written

Nodaļā 4.2 aiz definition 7

$\mathcal{T}$ starts by reading the input word and deterministically (making transitions with amplitude $1$) writes $n$ in binary on the second tape where $n$ is the description of the automaton $\mathcal{A}$.

//tik tālu viss labi. bet tad seko:

Additionally a space of size $O(k \cdot n)$ is reserved (assuming that the length of the input word on which the automaton must be simulated does not exceed $2^n$) for $k$ counters denoting the positions of the heads of the automaton $\mathcal{A}$.

Pirmkārt, vai tas ir tas pats n? Otrkārt, vai to, kas iekavās vispār vajag? (Varbūt bija saistība ar kkādām nākamajām simulācijām, kur bija reģistru mašīna.. neatceros)

Nodaļa ir izmesta, bet appendixam pievienota nav

Page 12: the words u (varbūt domāts v?) and h are not defined.

1. jābūt v_4, nevis kkādam v_m vai v_{m'} ? Tad kāda nozīme daudzpunktēm?

the Authors should much more stress the possible advantages of their approach and give the proper context to understand their results. E.g. in Sec. 4.3 it is proved that languages recognizable by multihead ultrametric automata have similar structure to languages recognizable by probabilistic, nondeterministic and deterministic automata. Were we to expect something different? Any answer to this question is welcome.

pg.3, Sec. 2.2: the notion of ultametrics is referred to metrics in general and not to norms (which is a particular metric). The three lines before Def.1 should be moved after the notion of metric.

sentences are missing words (e.g., on p. 6: "The automaton starts its
in n different starting states .." ),

pg.1, third paragraph of the Introduction: rewrite in more details

The Authors exceed maximal paper lenght by 3 pages. In order to shorten the manuscript discussion of p-adic numbers in subsection 2.1 can be ommitted altogether, therefore revision is suggested.

First sentence gives the Turing machine credit to a 2012 paper and not to Turing 1936? or maybe it is an English issue.

pg.7, Fig.1: even in this case, the reader must figure out the meaning of edge labels
Jāsaraksta bildes apraksts