Marcin Piątkowski

The Number of Runs in Sturmian Words

Denote by the class of standard Sturmianwords. It is a class of highly compressible words extensively studied in combinatorics of words, including the well known Fibonacci words. The suffix automata for these words have a very particular structure. This implies a simple characterization (described in the paper by the Structural Lemma) of the periods of runs (maximal repetitions) in Sturmian words. Using this characterization we derive an explicit formula for the number ρ(w) of runs in words , with respect to their recurrences(directive sequences). We show that

Diverse Palindromic Factorization Is NP-complete

\frac{\rho(w)}{|w|}\le \frac{4}{5} \textrm{\ for each\ }\ w\in {\cal S},

Reversible computation vs. reversibility in Petri nets

and there is an infinite sequence of strictly growing words

Reversing Transitions in Bounded Petri Nets

w_k\in {\cal S}

Tighter Bounds for the Sum of Irreducible LCP Values

such that

Reversible Computation vs. Reversibility in Petri Nets

\lim_{k\rightarrow \infty}\ \frac{\rho(w_k)}{|w_k|}\ =\ \frac{4}{5}

Conditions for Petri Net Solvable Binary Words

. The complete understanding of the function ρfor a large class of complicated words is a step towards better understanding of the structure of runs in words. We also show how to compute the number of runs in a standard Sturmian word in linear time with respect to the size of its compressed representation (recurrences describing the word). This is an example of a very fast computation on texts given implicitly in terms of a special grammar-based compressed representation (usually of logarithmic size with respect to the explicit text).

On Generation of Context-Abstract Plans

We prove that it is NP-complete to decide whether a given string can be factored into palindromes that are each unique in the factorization.

Square-free words over partially commutative alphabets

Abstract Petri nets are a general formal model of concurrent systems which supports both action-based and state-based modelling and reasoning. One of important behavioural properties investigated in the context of Petri nets has been reversibility, understood as the possibility of returning to the initial marking from any reachable net marking. Thus reversibility in Petri nets is a global property. Reversible computation, on the other hand, is typically a local mechanism by which a system can undo some of the executed actions. This paper is concerned with the modelling of reversible computation within Petri nets. A key idea behind the proposed construction is to add ‘reverses’ of selected transitions, and the paper discusses its different implementations. Adding reverses can severely impact on the behaviour of a Petri net. Therefore it is important, in particular, to be able to determine whether the modified net has a similar set of states as the initial one. We first prove that the problem of establishing whether the initial and modified nets have the same reachable markings is undecidable, even in the restricted case considered in this paper. We then show that the problem of checking whether the reachability sets of the two nets cover the same markings is decidable.

The Maximal Number of Runs in Standard Sturmian Words

Reversible computation deals with mechanisms for undoing the effects of actions executed by a dynamic system. This paper is concerned with reversibility in the context of Petri nets which are a general formal model of concurrent systems. A key construction we investigate amounts to adding ‘reverse’ versions of selected net transitions. Such a static modification can severely impact on the behaviour of the system, e.g., the problem of establishing whether the modified net has the same states as the original one is undecidable. We therefore concentrate on nets with finite state spaces and show, in particular, that every transition in such nets can be reversed using a suitable finite set of new transitions.