The First Order Definability of Graphs with Separators via the Ehrenfeucht Game

We say that a first order formula $\Phi$ defines a graph $G$ if $\Phi$ is true on $G$ and false on every graph $G'$ non-isomorphic with $G$. Let $D(G)$ be the minimal quantifier rank of a such formula. We prove that, if $G$ is a tree of bounded degree or a Hamiltonian (equivalently, 2-connected) outerplanar graph, then $D(G)=O(\log n)$, where $n$ denotes the order of $G$. This bound is optimal up to a constant factor. If $h$ is a constant, for connected graphs with no minor $K_h$ and degree $O(\sqrt n/\log n)$, we prove the bound $D(G)=O(\sqrt n)$. This result applies to planar graphs and, more generally, to graphs of bounded genus.Comment: 17 page

Structure Theorem and Strict Alternation Hierarchy for FO² on Words

It is well-known that every first-order property on words is expressible using at most three variables. The subclass of properties expressible with only two variables is also quite interesting and well-studied. We prove precise structure theorems that characterize the exact expressive power of first-order logic with two variables on words. Our results apply to FO$^2[<]$ and FO$^2[<,suc]$, the latter of which includes the binary successor relation in addition to the linear ordering on string positions. For both languages, our structure theorems show exactly what is expressible using a given quantifier depth, $n$, and using $m$ blocks of alternating quantifiers, for any $mleq n$. Using these characterizations, we prove, among other results, that there is a strict hierarchy of alternating quantifiers for both languages. The question whether there was such a hierarchy had been completely open since it was asked in [Etessami, Vardi, and Wilke 1997]

UTJECAJ SEDIMENTNOG OKRUŽJA NA TEHNIČKA SVOJSTVA DONJOKREDNIH VAPNENACA KAMENOLOMA LAKOVIĆI U ISTRI (HRVATSKA)

The Lower Cretaceous limestones from the Lakovići quarry belong to the basal part of the Aptian limestones. These limestones are the beginning of the second transgressive-regrcssive megasequence in Istria which followed a general Upper Aptian emersion event. Within the approximately 50 m thick limestone sequence that is quarried, four facies units were defined according to their petrographic and sedimen-tological features. The following facies units were devided: A- Micritic limestones, which were deposited as shallowing-upward cycles and which begin with breccias containing clay matrix or terminate with dark-gray clays sporadically accompanied by deposition and redeposi-lion of black terrestrial and swamp clays, as well as sediments from bogs and pools that were developed in isolated bays and/or laggons; B -Grain supported limestone deposited as fine-grained to coarse-grained carbonate sands in a predominantly high-energy shallows, bars and sandy beaches; C - Micritic limestones deposited in restricted low-energy shallow subtidal environments; D - Drain supported limestones deposited as fine grained carbonate sands in high-energy shallows and bars. The results of petrological, sedimentological and technological investigations show that the limestones from each individual facies unit have different technical properties, notably porosity, bulk density and water absorption, i.e. the wide range of technical quality of the limestones quarried is a direct consequence of their facies characteristics. The outlined facies units enable separation of rock mass in the quarry not only by their petrological characteristics but also according to the technical quality of the rock.Donjokredni vapnenci kamenoloma Lakovići pripadaju bazalnom dijelu aplskih vapnenaca kojima, nakon opće gornjoaptske emerzije, počinje druga transgresivno-regresivna megasekvencija u Istri. Unutar 50-ak metara debljine vapnenaca koji se eksploatiraju u kamenolomu Lakovići izdvojene su na osnovu petrografsko-sedimentoloških karakteristika četiri facijesne jedinice. To su: A- Mikritom bogati vapnenci taloženi kao ciklusi oplićavanja naviše koji počinju brečama s glinovitim matriksom ili završavaju tamnosivim glinama; B - Zrnasti vapnenci taloženi kao sitnozrnati do krupnozrnati vapnenački pijesci u plićacima s pretežito visokom energijom vode i na pješčanim plažama; C -Mikritom bogati vapnenci taloženi u zaštićenim potplimnim plićacima s niskom energijom vode i D-Zrnasti vapnenci taloženi kao sitnozrnati vapnenački pijesci u plićacima i plažama s pretežito visokom energijom vode. Rezultati petroloških, sedimentoloških i tehničko-tehnoloških istraživanja pokazuju da vapnenci svake pojedine facijesne jedinice imaju različita tehnička svojstva, posebice ona vezana uz poniznost, prostornu masu i upijanje vode, tj. da je velika varijabilnost tehničke kvalitete kamena u kamenolomu izravna posljedica njihovih facijesnih karakteristika. Izdvojenim facijesnim jedinicama određena je i selektivnost stijenske mase u kamenolomu u odnosu na, kako petrološka svojstva, tako i tehničko-tehnološku kvalitetu kamena

Average-energy games

Two-player quantitative zero-sum games provide a natural framework to synthesize controllers with performance guarantees for reactive systems within an uncontrollable environment. Classical settings include mean-payoff games, where the objective is to optimize the long-run average gain per action, and energy games, where the system has to avoid running out of energy. We study average-energy games, where the goal is to optimize the long-run average of the accumulated energy. We show that this objective arises naturally in several applications, and that it yields interesting connections with previous concepts in the literature. We prove that deciding the winner in such games is in NP inter coNP and at least as hard as solving mean-payoff games, and we establish that memoryless strategies suffice to win. We also consider the case where the system has to minimize the average-energy while maintaining the accumulated energy within predefined bounds at all times: this corresponds to operating with a finite-capacity storage for energy. We give results for one-player and two-player games, and establish complexity bounds and memory requirements.Comment: In Proceedings GandALF 2015, arXiv:1509.0685