This paper is the first step in thesolution of the problem of finite completion of comma-free codes.We show that every finite comma-free code is included in afinite comma-free code of particular kind, which we called, for lack of a better term, canonical comma-free code. Certainly, finite maximal comma-free codesare always canonical. The final step of the solution which consistsin proving further that every canonical comma-free code is completedto a finite maximal comma-free code, is intended to be published in a forthcomingpaper.

This paper is a sequel to anearlier paper of the present author, in which it was proved thatevery finite comma-free code is embedded into a so-called (finite)canonical comma-free code. In this paper, it is proved that every(finite) canonical comma-free code is embedded into a finite maximal comma-freecode, which thus achieves the conclusion that every finite comma-freecode has finite completions.


This paper investigates one possible model of reversible computations, animportant paradigm in the context of quantum computing. Introduced byBennett, a reversible pebble game is anabstraction of reversible computation that allows to examine the space andtime complexity of various classes of problems. We present a techniquefor proving lower and upper bounds on time and space complexity for severaltypes of graphs. Using this technique we show that the time needed toachieve optimal space for chain topology is Ω(nlgn) for infinitelymany n and we discusstime-space trade-offs for chain. Further we show a tight optimalspace bound for the binary tree of height h of the form h + Θ(lg*h)and discuss space complexity for the butterfly. These results give anevidence that reversible computations need more resources than standardcomputations. We also show an upper bound on time and space complexity ofthe reversible pebble game based on the time and space complexity of thestandard pebble game, regardless of the topology of the graph.

We study the complexity of the infinite word uβ associated with theRényi expansion of 1 in an irrational base β > 1.When β is the golden ratio, this is the well known Fibonacci word,which is Sturmian, and of complexity C(n) = n + 1.For β such thatdβ(1) = t1t2...tm is finite we provide a simple description ofthe structure of special factors of the word uβ . When tm =1we show thatC(n) = (m - 1)n + 1. In the cases when t1 = t2 = ... tm-1 ort1 > max{t2,...,tm-1 } we show that the first differenceof the complexity function C(n + 1) - C(n ) takes value in{m - 1,m} for every n, and consequently we determine thecomplexity of uβ . We show thatuβ is an Arnoux-Rauzy sequence if and only ifdβ(1) = tt...t1. On the example ofβ = 1 + 2cos(2π/7), solution of X3 = 2X2 + X - 1, we illustratethat the structure of special factors is more complicated fordβ (1) infinite eventually periodic.The complexity for this word is equal to 2n+1.