The properties characterizing Sturmian words are considered forwords on multiliteral alphabets. Wesummarize various generalizations of Sturmian words tomultiliteral alphabets and enlarge the list of knownrelationships among these generalizations.We provide a new equivalent definition of rich wordsand make use of it in the study of generalizations of Sturmian words based on palindromes.We also collect many examples of infinite words to illustrate differences in thegeneralized definitions of Sturmian words.

The paper deals with some classes of two-dimensional recognizablelanguages of “high complexity”,in a sense specified in the paper and motivated by some necessary conditions holding for recognizable andunambiguous languages. For such classes we can solve someopen questions related to unambiguity, finite ambiguityand complementation. Then we reformulate a necessary condition for recognizability stated by Matz, introducing a new complexity function.We solve an open question proposed by Matz, showing that all the known necessary conditions for recognizability of a language and its complement arenot sufficient. The proof relies on a family of languages defined by functions.

Quantum Coherent Spaces were introduced by Girard as a quantum framework where to interpret the exponential-free fragment of Linear Logic. Aim of this paper is to extend Girard's interpretation to a subsystem of linear logic with bounded exponentials. We provide deduction rules for the bounded exponentials and, correspondingly, we introduce the novel notion of bounded exponentials of Quantum Coherent Spaces. We show that the latter provide a categorical model of our system. In order to do that, we first study properties of the category of Quantum Coherent Spaces.

We study the relation between the standard two-way automata andmore powerful devices, namely, two-way finite automata equippedwith some $\ell$ additional “pebbles” that are movable alongthe input tape, but their use is restricted (nested) ina stack-like fashion. Similarly as in the case of the classicaltwo-way machines, it is not known whether there existsa polynomial trade-off, in the number of states, between thenondeterministic and deterministic two-way automata with $\ell$ nested pebbles. However, we show that these two machine modelsare not independent: if there exists a polynomial trade-off forthe classical two-way automata, then, for each $\ell$ ≥ 0,there must also exist a polynomial trade-off for the two-wayautomata with $\ell$ nested pebbles. Thus, we have an upwardcollapse (or a downward separation) from the classical two-wayautomata to more powerful pebble automata, still staying withinthe class of regular languages. The same upward collapse holdsfor complementation of nondeterministic two-way machines. These results are obtained by showing that each pebble machinecan be, by using suitable inputs, simulated by a classicaltwo-way automaton (and vice versa), with only a linear number ofstates, despite the existing exponential blow-up between theclassical and pebble two-way machines.

We consider directed figures defined as labelled polyominoes with designated start andend points, with two types of catenation operations. We are especially interested in codicityverification for sets of figures, and we show that depending on the catenation type the questionwhether a given set of directed figures is a code is decidable or not. In the former case we give aconstructive proof which leads to a straightforward algorithm.

We present a CAT (constant amortized time) algorithm for generating those partitions of n that are in the ice pile model $\mbox{IPM}_k$ (n),a generalization of the sand pile model $\mbox{SPM}$ (n).More precisely, for any fixed integer k, we show thatthe negative lexicographic ordering naturally identifies a tree structure on the lattice $\mbox{IPM}_k$ (n):this lets us design an algorithm which generates all the ice piles of $\mbox{IPM}_k$ (n)in amortized timeO(1)and in spaceO( $\sqrt n$ ).

We introduce a polynomial invariant of graphs on surfaces, PG, generalizing the classical Tutte polynomial. Topological duality on surfaces gives rise to a natural duality result for PG, analogous to the duality for the Tutte polynomial of planar graphs. This property is important from the perspective of statistical mechanics, where the Tutte polynomial is known as the partition function of the Potts model. For ribbon graphs, PG specializes to the well-known Bollobás–Riordan polynomial, and in fact the two polynomials carry equivalent information in this context. Duality is also established for a multivariate version of the polynomial PG. We then consider a 2-variable version of the Jones polynomial for links in thickened surfaces, taking into account homological information on the surface. An analogue of Thistlethwaite's theorem is established for these generalized Jones and Tutte polynomials for virtual links.

Let Ωn be the nn-element set consisting of all functions that have {1, 2, 3, . . ., n} as both domain and codomain. Let T(f) be the order of f, i.e., the period of the sequence f, f(2), f(3), f(4) . . . of compositional iterates. A closely related number, B(f) = the product of the lengths of the cycles of f, has previously been used as an approximation for T. This paper proves that the average values of these two quantities are quite different. The expected value of T iswhere k0 is a complicated but explicitly defined constant that is approximately 3.36. The expected value of B is much larger: