recognizable picture languages and domino tiling abstract in 2 giammarresi and restivo define the notion of local picture languages by giving a set of authorized 2 2 tiles over where is a boundary symbol which surrounds the pictures then they define the class of recognizable picture languages as the set of languages which can be obtained by projection of a local one this class is of interest since it admits several quite different characterizations 3 here we define the hv local picture languages where 2 2 tiles are replaced by horizontal and vertical dominoes so the horizontal and the vertical scanning can be done separately however we prove that every recognizable picture language can be obtained as a projection of a hv local language

a characterization of recognizable picture languages by tilings by finite sets as extension of the kleene star to pictures we introduce the operation of tiling we give a characterization of recognizable picture languages by intersection of tilings by finite sets of pictures

using entropy based methods to study general constrained parameter optimization problems abstract we propose the use of physics techniques for entropy determination on constrained parameter optimization problems the main feature of such techniques the construction of an unbiased walk on energy space suggests their use on the quest for optimal solutions of an optimization problem moreover the entropy and its associated density of states give us information concerning the feasibility of solutions

scaling and entropy in p median facility location along a line the p median problem is a common model for optimal facility location the task is to place p facilities e g warehouses or schools in a heterogeneously populated space such that the average distance from a person s home to the nearest facility is minimized here we study the special case where the population lives along a line e g a road or a river if facilities are optimally placed the length of the line segment served by a facility is inversely proportional to the square root of the population density this scaling law is derived analytically and confirmed for concrete numerical examples of three us interstate highways and the mississippi river if facility locations are permitted to deviate from the optimum the number of possible solutions increases dramatically using monte carlo simulations we compute how scaling is affected by an increase in the average distance to the nearest facility we find that the scaling exponents change and are most sensitive near the optimum facility distribution

subshifts as models for mso logic we study the monadic second order mso hierarchy over colorings of the discrete plane and draw links between classes of formula and classes of subshifts we give a characterization of existential mso in terms of projections of tilings and of universal sentences in terms of combinations of pattern counting subshifts conversely we characterize logic fragments corresponding to various classes of subshifts subshifts of finite type sofic subshifts all subshifts finally we show by a separation result how the situation here is different from the case of tiling pictures studied earlier by giammarresi et al

characterizations of recognizable picture series the theory of two dimensional languages as a generalization of formal string languages was motivated by problems arising from image processing and pattern recognition and also concerns models of parallel computing here we investigate power series on pictures these are functions that map pictures to elements of a semiring and provide an extension of two dimensional languages to a quantitative setting we assign weights to different devices ranging from picture automata to tiling systems we will prove that for commutative semirings the behaviours of weighted picture automata are precisely alphabetic projections of series defined in terms of rational operations and also coincide with the families of series characterized by weighted tiling or weighted domino systems

determining the density of states for classical statistical models a random walk algorithm to produce a flat histogram we describe an efficient monte carlo algorithm using a random walk in energy space to obtain a very accurate estimate of the density of states for classical statistical models the density of states is modified at each step when the energy level is visited to produce a flat histogram by carefully controlling the modification factor we allow the density of states to converge to the true value very quickly even for large systems from the density of states at the end of the random walk we can estimate thermodynamic quantities such as internal energy and specific heat capacity by calculating canonical averages at any temperature using this method we not only can avoid repeating simulations at multiple temperatures but we can also estimate the free energy and entropy quantities that are not directly accessible by conventional monte carlo simulations this algorithm is especially useful for complex systems with a rough landscape since all possible energy levels are visited with the same probability as with the multicanonical monte carlo technique our method overcomes the tunneling barrier between coexisting phases at first order phase transitions in this paper we apply our algorithm to both first and second order phase transitions to demonstrate its efficiency and accuracy we obtained direct simulational estimates for the density of states for two dimensional ten state potts models on lattices up to 200 x 200 and ising models on lattices up to 256 x 256 our simulational results are compared to both exact solutions and existing numerical data obtained using other methods applying this approach to a three dimensional j spin glass model we estimate the internal energy and entropy at zero temperature and using a two dimensional random walk in energy and order parameter space we obtain the rough canonical distribution and energy landscape in order parameter space preliminary data suggest that the glass transition temperature is about 1 2 and that better estimates can be obtained with more extensive application of the method this simulational method is not restricted to energy space and can be used to calculate the density of states for any parameter by a random walk in the corresponding space

efficient multiple range random walk algorithm to calculate the density of states we present a new monte carlo algorithm that produces results of high accuracy with reduced simulational effort independent random walks are performed concurrently or serially in different restricted ranges of energy and the resultant density of states is modified continuously to produce locally flat histograms this method permits us to directly access the free energy and entropy is independent of temperature and is efficient for the study of both 1st order and 2nd order phase transitions it should also be useful for the study of complex systems with a rough energy landscape

pure galois theory in categories abstract galois theory j pure appl algebra 19 1980 21 42 2 m barr abstract galois theory ii j pure appl algebra 25 1982 227 247 3 m barr and r diaconescu on locally simply connected toposes and their fundamen tal groups cahiers topologie gt om diffprentielle catigoriques 22 i98 i 301 3 14 4 s u chase d k harrison and a rosenberg galois theory and cohomology of commutative rings mem amer mafh sot 52 1965 15 33 5 s u chase and m e sweedler hopf algebras and galois theory in lecture notes in math vol 97 springer verlag new york berlin 1969 6 i diers categories of boolean sheaves of simple algebras in lecture notes in math vol 1187 springer verlag new york berlin 1986 7 a grothendieck revetements itales et groupe fondamental sga 1 in lecture notes in math vol 269 springer verlag new york berlin 1972 8 j f kennison galois theory and theaters of action in a topos j pure appl algebra 18 1980 149 164 9

internal categories and groupoids in congruence modular varieties abstract it is well known that there exist several simple descriptions of the internal categories and groupoids in various classical algebraic categories such as groups rings lie algebras etc one of those descriptions is extended to arbitrary congruence modular varieties we describe all internal categories satisfying a certain commutator condition which always holds for internal groupoids and for all internal categories in mal tsev congruence permutable varieties the results of this paper show a deep connection between internal category theory and commutator theory in universal algebra the so called kiss difference operation also plays a central role

uniform and nonuniform recognizability deterministic and nondeterministic finite state recognizability over finite structures are introduced in an algebraic setting avoiding detailed computational conventions as needed in the definition of finite state acceptors for deterministic recognizability the classical approach is adopted using a uniform homomorphism from the input domain consisting of terms into a finite algebra for the nondeterministic case we refer to relational input structures and to an acceptance via relational homomorphisms which are applied nonuniformly since they depend on the input structures we show how this approach encompasses known models of nondeterministic automata over finite words trees pictures and graphs and present some elementary metaresults connecting uniform recognizability nonuniform recognizability and monadic second order logic

monadic second order logic over rectangular pictures and recognizability by tiling systems abstract it is shown that a set of pictures rectangular arrays of symbols is recognized by a finite tiling system iff it is definable in existential monadic second order logic as a consequence finite tiling systems constitute a notion of recognizability over two dimensional inputs which at the same time generalizes finite state recognizability over strings and also matches a natural logic the proof is based on the ehrenfeucht fraisse technique for first order logic and an implementation of threshold counting within tiling systems

dot depth monadic quantifier alternation and first order closure over grids and pictures this paper presents results from two different areas the first area is monadic second order logic mso over finite structures in particular over the so called grids these are structures whose elements can be arranged as a matrix and which have two binary relations corresponding to vertical and horizontal successors for this logic we study the expressive power of the alternation of existential and universal monadic second order quantifiers i e set quantifiers in matz et al information and computation lics 97 1999 to appear it had been shown that these alternations cannot be limited to a fixed number without loss of expressiveness in this paper we strengthen this result in several ways firstly we show that there are mso formulas that have a very restricted form of k 1 set quantifiers but are not equivalent to a formula with k quantifiers secondly we show that if we fix the number of such alternations the expressive power of formulas that start with a block of universal quantifiers differs from the power of those that start with an existential one this was previously known only for coloured grids thirdly we investigate how an additional prefix of first order i e element quantifiers influences the expressive power of mso formulas the second area that this paper is concerned with is two dimensional formal language theory we study how the alternation of first and monadic second order quantifications on the one hand relates to the dot depth measure of two dimensional i e picture languages on the other hand that measure is the two dimensional version of the classical notion of dot depth for one dimensional starfree word languages we show that the hierarchy induced by this dot depth cuts through the hierarchy given by monadic second order quantifications in particular beyond each level of the monadic second order alternation hierarchy there is a starfree picture language

new monte carlo algorithm entropic sampling we present a new monte carlo sampling algorithm with which one can obtain any desired distribution of the sampling in one monte carlo simulation the free energy and the entropy of a system can thus be obtained from a simple exercise of this algorithm the main idea is to sample directly the entropy of a system at infinite temperature importance sampling is shown to be a particular case of the new algorithm the algorithm is tested against the exact partition function of the ital l 4 simple cubic ising model a comparison with the multicanonical ensemble for the ital l 12 ital q 10 potts model shows that the new algorithm is more general and more efficient

perfectly quilted rectangular snake tilings we introduce a particular form of snake tilings to define picture languages and relate the obtained family to the recognizable picture languages as defined by wang tiles the correspondence for substitution tilings is even closer and hence is applicable to the hilbert curve

precrossed modules and galois theory the adjunction between crossed modules and precrossed modules over a fixed group can be seen as a special case of a more general adjunction between internal groupoids and internal reflexive graphs in a mal tsev variety by using the categorical galois theory we characterize the central extensions with respect to this latter adjunction in terms of the universal algebraic commutator in particular we get a description of the central extensions of precrossed modules and of precrossed rings this characterization provides a natural way to define a categorical notion of peiffer commutator c 2005 elsevier inc all rights reserved

higher central extensions and cohomology abstract we establish a galois theoretic interpretation of cohomology in semi abelian categories cohomology with trivial coefficients classifies central extensions also in arbitrarily high degrees this allows us to obtain a duality in a certain sense between internal homology and external cohomology in semi abelian categories these results depend on a geometric viewpoint of the concept of a higher central extension as well as the algebraic one in terms of commutators

a categorical approach to commutator theory abstract we define a notion of commutator for equivalence relations in a mal cev category and show that it verifies a universal property

higher commutator conditions for extensions in mal tsev categories abstract we define a galois structure on the category of pairs of equivalence relations in an exact mal tsev category and characterize central and double central extensions in terms of higher commutator conditions these results generalize both the ones related to the abelianization functor in exact mal tsev categories and the ones corresponding to the reflection from the category of internal reflexive graphs to the subcategory of internal groupoids some examples and applications are given in the categories of groups precrossed modules precrossed lie algebras and compact groups

on the complexity of some common geometric location problems given n demand points in the plane the p center problem is to find p supply points anywhere in the plane so as to minimize the maximum distance from a demand point to its respective nearest supply point the p median problem is to minimize the sum of distances from demand points to their respective nearest supply points we prove that the p center and the p median problems relative to both the euclidean and the rectilinear metrics are np hard in fact we prove that it is np hard even to approximate the p center problems sufficiently closely the reductions are from 3 satisfiability

complexity of two dimensional patterns in dynamical systems such as cellular automata and iterated maps it is often useful to look at a language or set of symbol sequences produced by the system there are well established classification schemes such as the chomsky hierarchy with which we can measure the complexity of these sets of sequences and thus the complexity of the systems which produce them in this paper we look at the first few levels of a hierarchy of complexity for two or more dimensional patterns we show that several definitions of regular language or local rule that are equivalent in d 1 lead to distinct classes in d 2 we explore the closure properties and computational complexity of these classes including undecidability and l nl and np completeness results we apply these classes to cellular automata in particular to their sets of fixed and periodic points finite time images and limit sets we show that it is undecidable whether a ca in d 2 has a periodic point of a given period and that certain local lattice languages are not finite time images or limit sets of any ca we also show that the entropy of a d dimensional ca s finite time image cannot decrease faster than t d unless it maps every initial condition to a single homogeneous state

the monadic quantifier alternation hierarchy over grids and graphs the monadic second order quantifier alternation hierarchy over the class of finite graphs is shown to be strict the proof is based on automata theoretic ideas and starts from a restricted class of graph like structures namely finite two dimensional grids considering grids where the width is a function of the height we prove that the difference between the levels k 1 and k of the monadic hierarchy is witnessed by a set of grids where this function is k 1 fold exponential we then transfer the hierarchy result to the class of directed or undirected graphs using an encoding technique called strong reduction it is notable that one can obtain sets of graphs which occur arbitrarily high in the monadic hierarchy but are already definable in the first order closure of existential monadic second order logic we also verify that these graph properties even belong to the complexity class nlog which indicates a profound difference between the monadic hierarchy and the polynomial hierarchy

3 3 lemma and protomodularity abstract the classical 3 3 lemma and snake lemma valid in any abelian category still hold in any quasi pointed the map 0 1 is a mono regular and protomodular category some applications are given in this abstract context concerning the denormalization of kernel maps and the normalization of internals groupoids i e associated crossed modules

resolutions higher extensions and the relative mal tsev axiom abstract we study how the concept of higher dimensional extension which comes from categorical galois theory relates to simplicial resolutions for instance an augmented simplicial object is a resolution if and only if its truncation in every dimension gives a higher extension in which sense resolutions are infinite dimensional extensions or higher extensions are finite dimensional resolutions we also relate certain stability conditions of extensions to the kan property for simplicial objects this gives a new proof of the fact that a regular category is mal tsev if and only if every simplicial object is kan using a relative setting of extensions