Lower Bounds for Monotone Counting Circuits

A {+,x}-circuit counts a given multivariate polynomial f, if its values on 0-1 inputs are the same as those of f; on other inputs the circuit may output arbitrary values. Such a circuit counts the number of monomials of f evaluated to 1 by a given 0-1 input vector (with multiplicities given by their coefficients). A circuit decides $f$ if it has the same 0-1 roots as f. We first show that some multilinear polynomials can be exponentially easier to count than to compute them, and can be exponentially easier to decide than to count them. Then we give general lower bounds on the size of counting circuits.Comment: 20 page

PRATARMĖ

Klaipėdos universiteto (įkurto 1991 metais) leidžiamas žurnalas TILTAI/BRIDGES/ BRÜCKEN yra skirtas  socialiniams mokslams. Jame spausdinamuose straipsniuose nagrinėjamos aktualios ekonomikos, vadybos, demografijos, socialinės geografijos, geopolitikos, politikos, kultūros, švietimo, religijos, krašto tvarkymo ir kitos socialinės problemos. Ypač laukiama straipsnių apie paribių ir tarpvalstybinių regionų ekonomikos bei kultūros plėtrą. Mokslas nepripažįsta sienų, todėl mokslinis bendradarbiavimas – vienas svarbiausių pasaulio bendruomenės pažangos elementų. Dėl to dalyvauti leidžiant žurnalą, publikuoti straipsnius kviečiami mokslininkai iš įvairių šalių. TILTAI yra mokslo žurnalas, kuriame publikuojami moksliniai straipsniai Lietuvos mokslo tarybos nutarimu yra pripažįstami ginant daktaro ir habilituoto daktaro disertacijas, gaunant pedagoginius mokslo vardus

Notes on Boolean Read-k and Multilinear Circuits

A monotone Boolean (OR,AND) circuit computing a monotone Boolean function f is a read-k circuit if the polynomial produced (purely syntactically) by the arithmetic (+,x) version of the circuit has the property that for every prime implicant of f, the polynomial contains at least one monomial with the same set of variables, each appearing with degree at most k. Every monotone circuit is a read-k circuit for some k. We show that already read-1 (OR,AND) circuits are not weaker than monotone arithmetic constant-free (+,x) circuits computing multilinear polynomials, are not weaker than non-monotone multilinear (OR,AND,NOT) circuits computing monotone Boolean functions, and have the same power as tropical (min,+) circuits solving combinatorial minimization problems. Finally, we show that read-2 (OR,AND) circuits can be exponentially smaller than read-1 (OR,AND) circuits.Comment: A throughout revised version. To appear in Discrete Applied Mathematic

PRATARMĖ

Klaipėdos universiteto (įkurto 1991 metais) leidžiamas žurnalas TILTAI/BRIDGES/ BRÜCKEN yra skirtas  socialiniams mokslams. Jame spausdinamuose straipsniuose nagrinėjamos aktualios ekonomikos, vadybos, demografijos, socialinės geografijos, geopolitikos, politikos, kultūros, švietimo, religijos, krašto tvarkymo ir kitos socialinės problemos. Ypač laukiama straipsnių apie paribių ir tarpvalstybinių regionų ekonomikos bei kultūros plėtrą. Mokslas nepripažįsta sienų, todėl mokslinis bendradarbiavimas – vienas svarbiausių pasaulio bendruomenės pažangos elementų. Dėl to dalyvauti leidžiant žurnalą, publikuoti straipsnius kviečiami mokslininkai iš įvairių šalių. TILTAI yra mokslo žurnalas, kuriame publikuojami moksliniai straipsniai Lietuvos mokslo tarybos nutarimu yra pripažįstami ginant daktaro ir habilituoto daktaro disertacijas, gaunant pedagoginius mokslo vardus

Graphs and Circuits: Some Further Remarks

We consider the power of single level circuits in the context of graph complexity. We first prove that the single level conjecture fails for fanin-$2$ circuits over the basis ${oplus,land,1}$. This shows that the (surpisingly tight) phenomenon, established by Mirwald and Schnorr (1992) for quadratic functions, has no analogon for graphs. We then show that the single level conjecture fails for unbounded fanin circuits over ${lor,land,1}$. This partially answers the question of Pudl\u27ak, R"odl and Savick\u27y (1986). We also prove that $Sigma_2 eq Pi_2$ in a restricted version of the hierarhy of communication complexity classes introduced by Babai, Frankl and Simon (1986). Further, we show that even depth-$2$ circuits are surprisingly powerful: every bipartite $n imes n$ graph of maximum degree $Delta$ can be represented by a monotone CNF with $O(Deltalog n)$ clauses. We also discuss a relation between graphs and $ACC$-circuits

Peculiarities of Implantation in the Posterior Mandible

Cilj istraživanja bio je provjeriti kliničko-anatomske osobitosti bezubih segmenata stražnjeg dijela donje čeljusti (edentulous mandibular dental segments, eMDS), kao i procijeniti mogućnost implantacije zuba za svaki kliničko-anatomski tip eMDS. U istraživanje je uključeno 198 ispitanika, od toga 88 muškaraca i 110 žena. Procjena eMDS obavljena je s pomoću ortopantomografije, računalne tomografije i kliničke evaluacije uz uporabu posebnog mjernog šestara za greben, namijenjenog mjerenju širine alveolarnih nastavaka. Postavljena su ukupno 353 jednofazna zubna implantata OSTEOFIX, oblika vijka (screw-shaped single-stage Osteofix Dental Implant System). Implantacija je bila uspješna u 346 (98,02 %) slučajeva. Na temelju rezultata mjerenja eMDS, kao i na temelju literaturnih podataka, predlažemo da se dentalni segmenti stražnjeg dijela donje čeljusti podijele u sljedeća tri kliničko-anatomska tipa: I. tip - bez atrofije eMDS, a ako je i ima, ispravna je implantacija moguća; II. tip - moguća je ispravna implantacija, ali uz istodobno pojačavanje alveolarnog grebena; III. tip - moguća je samo odgođena implantacija nakon pojačavanja alveolarnog grebena.The aim of the study was to investigate the clinical-anatomical peculiarities of edentulous mandibular dental segments (eMDS), situated in the posterior mandible, as well as to estimate the possibility of dental implantation for every clinical-anatomical eMDS type. One hundred and ninety-eight patients were examined, 88 male and 110 female. Orthopantomography, computer tomography, clinical examination using special ridge - mapping calipers for measurement of alveolar process width were employed to estimate eMDS. Three hundred anf fifty-three screw-shaped single-stage OSTEOFIX Dental Implant System implants were inserted. In 346 (98.02%) cases implantation was successful. Concerning the results of the eMDS measurements, the literature data suggested the division of posterior mandible dental segments into the following three clinical - anatomical types: type I - atrophy of eMDS is absent or if present, proper implantation is available; type II - simultaneous implantation is available only with augmentation of the alveolar ridge; type III - only delayed implantation is available after augmentation of the alveolar ridge

Laws of Little in an open queueing network

The object of this research in the queueing theory is theorems about the functional strong laws of large numbers (FSLLN) under the conditions of heavy traffic in an open queueing network (OQN). The FSLLN is known as a fluid limit or fluid approximation. In this paper, FSLLN are proved for the values of important probabilistic characteristics of the OQN investigated as well as the virtual waiting time of a customer and the queue length of customers. As applications of the proved theorems laws of Little in OQN are presented