Checking Whether an Automaton Is Monotonic Is NP-complete

An automaton is monotonic if its states can be arranged in a linear order that is preserved by the action of every letter. We prove that the problem of deciding whether a given automaton is monotonic is NP-complete. The same result is obtained for oriented automata, whose states can be arranged in a cyclic order. Moreover, both problems remain hard under the restriction to binary input alphabets.Comment: 13 pages, 4 figures. CIAA 2015. The final publication is available at http://link.springer.com/chapter/10.1007/978-3-319-22360-5_2

Large Aperiodic Semigroups

The syntactic complexity of a regular language is the size of its syntactic semigroup. This semigroup is isomorphic to the transition semigroup of the minimal deterministic finite automaton accepting the language, that is, to the semigroup generated by transformations induced by non-empty words on the set of states of the automaton. In this paper we search for the largest syntactic semigroup of a star-free language having $n$ left quotients; equivalently, we look for the largest transition semigroup of an aperiodic finite automaton with $n$ states. We introduce two new aperiodic transition semigroups. The first is generated by transformations that change only one state; we call such transformations and resulting semigroups unitary. In particular, we study complete unitary semigroups which have a special structure, and we show that each maximal unitary semigroup is complete. For $n \ge 4$ there exists a complete unitary semigroup that is larger than any aperiodic semigroup known to date. We then present even larger aperiodic semigroups, generated by transformations that map a non-empty subset of states to a single state; we call such transformations and semigroups semiconstant. In particular, we examine semiconstant tree semigroups which have a structure based on full binary trees. The semiconstant tree semigroups are at present the best candidates for largest aperiodic semigroups. We also prove that $2^n-1$ is an upper bound on the state complexity of reversal of star-free languages, and resolve an open problem about a special case of state complexity of concatenation of star-free languages.Comment: 22 pages, 1 figure, 2 table

Reset thresholds of automata with two cycle lengths

We present several series of synchronizing automata with multiple parameters, generalizing previously known results. Let p and q be two arbitrary co-prime positive integers, q > p. We describe reset thresholds of the colorings of primitive digraphs with exactly one cycle of length p and one cycle of length q. Also, we study reset thresholds of the colorings of primitive digraphs with exactly one cycle of length q and two cycles of length p.Comment: 11 pages, 5 figures, submitted to CIAA 201

On the Number of Synchronizing Colorings of Digraphs

We deal with $k$-out-regular directed multigraphs with loops (called simply \emph{digraphs}). The edges of such a digraph can be colored by elements of some fixed $k$-element set in such a way that outgoing edges of every vertex have different colors. Such a coloring corresponds naturally to an automaton. The road coloring theorem states that every primitive digraph has a synchronizing coloring. In the present paper we study how many synchronizing colorings can exist for a digraph with $n$ vertices. We performed an extensive experimental investigation of digraphs with small number of vertices. This was done by using our dedicated algorithm exhaustively enumerating all small digraphs. We also present a series of digraphs whose fraction of synchronizing colorings is equal to $1-1/k^d$, for every $d \ge 1$ and the number of vertices large enough. On the basis of our results we state several conjectures and open problems. In particular, we conjecture that $1-1/k$ is the smallest possible fraction of synchronizing colorings, except for a single exceptional example on 6 vertices for $k=2$.Comment: CIAA 2015. The final publication is available at http://link.springer.com/chapter/10.1007/978-3-319-22360-5_1

COD Fractions in Sewage Flowing into Polish Sewage Treatment Plants

The paper presents the results of studies concerning the designation of COD fraction in raw wastewater. The research was conducted in three mechanical-biological sewage treatment plants. The results were compared with data assumed in the ASM models. During the investigation, the following fractions of COD were determined: dissolved non-biodegradable SI, dissolved easily biodegradable SS, in organic suspension slowly degradable XS, and in organic suspension non-biodegradable XI. The methodology fordetermining the COD fraction was based on the ATV- A 131guidelines. The real concentration of fractions in raw wastewater and the percentage of each fraction in total CODare different from data reported in the literature

The fate of nitrogen in the urban area – The case of Zielona Góra, Poland

The anthropogenic change of the nitrogen (N) cycle is strongly triggered by urban demand (such as food and meat consumption, energy demand and transport). As a consequence of high population density, impacts on human health through water and air pollution also concentrate on a city environment. Thus, an urban perspective on a predominantly rural pollution becomes relevant. Urban N budgets may be considered less intrinsically connected, so that separation of an agri-food chain and an industry-combustion chain is warranted. Results have been obtained for Zielona Góra, Poland, a city of 140,000 inhabitants characterized by domestic and transport sources and forest-dominated surroundings. In addition to food imports in Zielona Gora amounting to about 30 %, in the suburban area a significant share of N amounting to 41 % is related to fertilizer imports. The remaining imports are in fuel, electronics, textiles, plastics and paper. Most of the agri-food N (45 %) is denitrified in wastewater treatment. N associated with combustion (mainly NOx emissions from vehicles) represents a much smaller share than N entering via the agri-food system, amounting to 22 % of the total N imports. This overall picture is maintained also when specifically addressing the city center, with the exception of mineral fertilizer that plays a much smaller role, with just 7 % of N imports to the city

Urban nitrogen budgets: Evaluating and comparing the path of nitrogen through cities for improved management

Reactive nitrogen (Nr) released to the environment is a cause of multiple environmental threats. While Nr flows are often only analyzed in an agricultural context, consumption and emission takes place in the urban environment, and opportunities for Nr recycling and effective policy implementation for mitigation often appear in cities. Since little information is available on the bigger picture of Nr flows through the urban environment, these opportunities often remain unexploited. Here we developed a framework to model Nr pathways through urban and surrounding areas, which we applied to four test areas (Beijing and Shijiazhuang (China), Vienna (Austria), and Zielona Góra (Poland)). Using indicators such as recycling rates and Nr surplus, we estimated environmental risks and recycling potentials based on Nr flows and their entry and exit points. Our findings show marked differences between the core and surrounding areas of each city, with the former being a site of Nr consumption with largest flows associated with households, and the latter a site of (agricultural) production with largest flows associated with industry (fertilizers) and urban plants. As a result, Nr transgresses the core areas in a rather linear manner with only 0-5 % being re-used, with inputs from Nr contained in food and fuels and outputs most commonly as non-reactive N2 emissions to the atmosphere from wastewater treatment and combustion processes. While the peri-urban areas show a higher Nr recycling rate (6-14 %), Nr accumulation and emissions from cultivated land pose significant environmental challenges, indicating the need for mitigation measures. We found potential to increase nitrogen use efficiency through improved Nr management on cultivated areas and to increase Nr recycling using urine and sewage sludge as synthetic fertilizer substitutes. Hence our framework for urban nitrogen budgets not only allows for consistent budgeting but helps identify common patterns, potentially harmful flows and Nr recycling potential