resolution of a system of the max product fuzzy relation equations using lou factorization in this paper the lu factorization is extended to the fuzzy square matrix with respect to the max product composition operator called l u factorization equivalently we will find two fuzzy lower and upper triangular matrices l and u for a fuzzy square matrix a such that a l u where is the max product composition an algorithm is presented to find the matrices l and u furthermore some necessary and sufficient conditions are proposed for the existence and uniqueness of the l u factorization for a given fuzzy square matrix a an algorithm is also proposed to find the solution set of a square system of fuzzy relation equations fre using the l u factorization the algorithm finds the solution set without finding its minimal solutions and maximum solution it is shown that the two algorithms have a polynomial time complexity as o n 3 since the determination of the minimal solutions is an np hard problem the algorithm can be very important from the practical point of view

a fuzzy approach to select the location of the distribution center abstract the location selection of distribution center dc is one of the most important decision issues for logistics managers owing to vague concept frequently represented in decision data a new multiple criteria decision making method is proposed to solve the distribution center location selection problem under fuzzy environment in the proposed method the ratings of each alternative and the weight of each criterion are described by linguistic variables which can be expressed in triangular fuzzy numbers the final evaluation value of each dc location is also expressed in a triangular fuzzy number by calculating the difference of final evaluation value between each pair of dc locations a fuzzy preference relation matrix is constructed to represent the intensity of the preferences of one plant location over another and then a stepwise ranking procedure is proposed to determine the ranking order of all candidate locations finally a numerical example is solved to illustrate the procedure of the proposed method at the end of this paper

solutions of fuzzy relation equations based on continuous t norms this study is concerned with fuzzy relation equations with continuous t norms in the form atr b where a and b are the fuzzy subsets of x and y respectively email protected xxy is a fuzzy relation and t is a continuous t norm the problem is how to determine a from r and b first an if and only if condition of being solvable is presented novel algorithms are then presented for determining minimal solutions when x and y are finite the proposed algorithms generate all minimal solutions for the equations making them efficient solving procedures

some results on the resolution of fuzzy relation equations abstract in this paper two special cases of the solution of the inverse problem are investigated the necessary and sufficient conditions for the existence of a solution such that the upper bound is the unit vector and for a unique solution of the inverse problem equal to the unit vector are presented an algorithmic approach which derives the output vector and the state matrix in each one of these cases is also proposed

on graph entropy measures based on the number of independent sets and matchings abstract in this paper we consider the graph entropy measures based on the number of independent sets and matchings the reason to study these measures relates to the fact that the independent set and matching problem is computationally demanding however these features can be calculated for smaller networks in case one can establish efficient estimations those measures may be also used for larger graphs so we establish some upper and lower bounds as well as some information inequalities for these information theoretic quantities in order to give further evidence we also generate numerical results to study these measures such as list the extremal graphs for these entropies those results reveal the two entropies possess some new features

the smallest grammar problem this paper addresses the smallest grammar problem what is the smallest context free grammar that generates exactly one given string spl sigma this is a natural question about a fundamental object connected to many fields such as data compression kolmogorov complexity pattern identification and addition chains due to the problem s inherent complexity our objective is to find an approximation algorithm which finds a small grammar for the input string we focus attention on the approximation ratio of the algorithm and implicitly the worst case behavior to establish provable performance guarantees and to address shortcomings in the classical measure of redundancy in the literature our first results are concern the hardness of approximating the smallest grammar problem most notably we show that every efficient algorithm for the smallest grammar problem has approximation ratio at least 8569 8568 unless p np we then bound approximation ratios for several of the best known grammar based compression algorithms including lz78 b isection sequential longest match greedy and re pair among these the best upper bound we show is o n sup 1 2 we finish by presenting two novel algorithms with exponentially better ratios of o log sup 3 n and o log n m sup where m sup is the size of the smallest grammar for that input the latter algorithm highlights a connection between grammar based compression and lz77

minimization of linear objective functions under the constraints expressed by a system of fuzzy relation equations this paper considers the following optimization problem minimization of linear objective functions under the constraints expressed by a system of fuzzy relation equations using max product composition we first show some properties of minimal solutions of a system of fuzzy relation equations with max product composition and present some rules for reducing the original problem then we derive an algorithm which enables us to find all the optimal solutions for solving the optimization problem and provide some numerical examples to illustrate

on the n back and forth types of boolean algebras the objective of this paper is to uncover the structure of the back and forth equivalence classes at the nite levels for the class of boolean algebras as an application we obtain bounds on the computational complexity of determining the back and forth equivalence classes of a boolean algebra for nite levels this result has implications for characterizing the relatively intrinsically 0 relations of boolean algebras as existential formulas over a nite set of relations

an on line universal lossy data compression algorithm via continuous codebook refinement iii redundancy analysis for pt ii see ibid vol 42 p 822 36 1996 the gold washing data compression algorithm is an adaptive vector quantization algorithm with vector dimension n in this paper a redundancy problem of the gold washing data compression algorithm is considered it is demonstrated that for any memoryless source with finite alphabet a and generic distribution p and for any r 0 the redundancy of the gold washing data compression algorithm with dimension n defined as the difference between the average performance of the algorithm and the distortion rate function d p r of p is upper bounded by sub spl delta r sup spl delta d p r a 2 spl xi 4 log n 2n spl sigma logn n where sub spl delta r sup spl delta d p r is the partial derivative of d p r with respect to r a is the cardinality of a and spl xi 0 is a parameter used to control the threshold in the gold washing algorithm in connection with the results of zhang yang and wei see ibid vol 43 no 1 p 71 91 1997 on the redundancy of lossy source coding this shows that the gold washing algorithm has the optimal convergence rate among all adaptive finite state vector quantizers

scott ranks of models of a theory abstract the scott rank of a countable structure is a measure coming from the proof of scott s isomorphism theorem of the complexity of that structure the scott spectrum of a theory by which we mean a sentence of l 1 is the set of scott ranks of countable models of that theory in z f c p d we give a descriptive set theoretic classification of the sets of ordinals which are the scott spectrum of a theory they are particular 1 1 classes of ordinals our investigation of scott spectra leads to the resolution in zfc of a number of open problems about scott ranks we answer a question of montalban by showing for each 1 that there is a 2 in theory with no models of scott rank less than we also answer a question of knight and calvert by showing that there are computable models of high scott rank which are not computably approximable by models of low scott rank finally we answer a question of sacks and marker by showing that 2 1 is the least ordinal such that if the models of a computable theory t have scott rank bounded below 1 then their scott ranks are bounded below

solution algorithms for fuzzy relational equations with max product composition the conditions for the existence of an inverse solution to the max min composition of fuzzy relational equations have been well documented since the original work by sanchez 30 31 these same existence theorems have been extended to the t norm composition of relational equations in which the max product composition is a member 5 13 26 several studies 8 15 24 33 34 38 have shown that the max min operator may not always be the most desirable fuzzy relational composition and in fact the max product operator was superior in these instances this paper reviews the algorithms necessary to determine the complete solution of the inverse for fuzzy relational equations with max product composition

solving fuzzy relation equations with a linear objective function abstract an optimization model with a linear objective function subject to a system of fuzzy relation equations is presented due to the non convexity of its feasible domain defined by fuzzy relation equations designing an efficient solution procedure for solving such problems is not a trivial job in this paper we first characterize the feasible domain and then convert the problem to an equivalent problem involving 0 1 integer programming with a branch and bound solution technique after presenting our solution procedure a concrete example is included for illustration purpose

some properties of sup min fuzzy relational equations on infinite domains abstract this paper deals with a sup min fuzzy relational equation on infinite domains some properties of its solution set will be obtained in particular a necessary and sufficient condition describing the attainable solution resp the unattainable solution is formulated in the end the solution set will be given under some conditions

finitary solvability conditions for systems of fuzzy relation equations in this paper we consider the problem of solving systems of fuzzy relation equations two types of these systems with different compositions are considered and processed simultaneously a new sufficient condition and new solvability criteria are proposed for systems of both types all these conditions are finitary and thus can be easily verified the sufficient condition and criteria of solvability characterize a relationship between a skeleton nxn matrix n is a number of equations and a vector of the right hand side

boolean algebra approximations knight and stob proved that every low4 boolean algebra is 0 6 isomorphic to a computable one furthermore for n 1 2 3 4 every lown boolean algebra is 0 n 2 isomorphic to a computable one we show that this is not true for n 5 there is a low5 boolean algebra that is not 0 7 isomorphic to any computable boolean algebra it is worth remarking that because of the machinery developed the proof uses at most a 0 00 priority argument the technique used to construct this boolean algebra is new and might be useful in other applications such as to solve the lown boolean algebra problem either positively or negatively

optimization of linear problems subjected to the intersection of two fuzzy relational inequalities defined by dubois prade family of t norms abstract in this paper optimization of a linear objective function with fuzzy relational inequality constraints is investigated in doing so the feasible region is formed by the intersection of two inequality fuzzy systems and dubois prade family of t norms which are considered as fuzzy composition the most well known continuous t norms are archimedean such as frank yager hamacher sugeno weber and schweizer sklar family an interesting family of t norms that is not archimedean has been introduced by dubois and prade in this paper the resolution of the feasible region of the problem is initialy investigated when it is defined with max duboise prade composition a necessary and sufficient condition along with three other necessary conditions are derived for determining the feasibility of the problem moreover two procedures have also been presented with the aim of simplifying the current linear problems a method is proposed to generate random feasible max dubois prade fuzzy relational inequalities and an algorithm is accordingly presented to solve the problem finally an example is described to illustrate this algorithm

optimization of fuzzy relation equations with max product composition abstract an optimization problem with a linear objective function subject to a system of fuzzy relation equations using max product composition is considered since the feasible domain is non convex traditional linear programming methods cannot be applied we study this problem and capture some special characteristics of its feasible domain and the optimal solutions some procedures for reducing the original problem are presented the problem is transformed into a 0 1 integer program which is then solved by the branch and bound method for illustration purpose an example of the procedures is provided

approximation of grammar based compression via recompression in this paper we present a simple linear time algorithm constructing a context free grammar of size o g log n g for the input string where n is the size of the input string and g the size of the optimal grammar generating this string the algorithm works for arbitrary size alphabets but the running time is linear assuming that the alphabet of the input string can be identified with numbers from 1 n c for some constant c otherwise additional cost of o n log is needed algorithms with such an approximation guarantee and running time are known the novelty of this paper is a particular simplicity of the algorithm as well as the analysis of the algorithm which uses a general technique of recompression recently introduced by the author furthermore contrary to the previous results this work does not use the lz representation of the input string in the construction nor in the analysis

an algorithm for optimizing the linear function with fuzzy relation equation constraints regarding max prod composition abstract fuzzy sets as the feasible region for optimization problems is an interesting and on going research topic s c fang g li solving fuzzy relations equations with a linear objective function fuzzy sets syst 103 1999 107 113 7 j lu s c fang solving nonlinear optimization problems with fuzzy relation constraints fuzzy sets syst 119 2001 1 20 16 e khorram a ghodpusian linear objective function optimization with fuzzy relation constraints regarding max av composition appl math comput in press doi 10 1016 j amc 2005 04 021 in this paper we focus on these kind problems in which the solutions region is the fuzzy relation equation with max prod composition and the objective function is linear whereas one of the major difficulties in such problems is non convexity of the feasible region it is preferable to study these regions in the first step hence we have primarily investigated two methods and their relationship and then we have determined the feasible region via them after determining the feasible set we have given an algorithm to optimize the linear objective function on such these regions finally we have presented two examples to illustrate the methods and algorithms

some properties of minimal solutions for a fuzzy relation equation abstract some properties of the solution set and minimal solutions of a fuzzy relation equation are considered in this paper we show the necessary and sufficient condition for existence of a minimal solution of a finite fuzzy relation equation defined on infinite index sets

approximation of smallest linear tree grammar a simple linear time algorithm for constructing a linear context free tree grammar of size o r g r g log n r g for a given input tree t of size n is presented where g is the size of a minimal linear context free tree grammar for t and r is the maximal rank of symbols in t which is a constant in many applications this is the first example of a grammar based tree compression algorithm with a good i e logarithmic in terms of the size of the input tree approximation ratio the analysis of the algorithm uses an extension of the recompression technique from strings to trees

ranked structures and arithmetic transfinite recursion atr0 is the natural subsystem of second order arithmetic in which one can develop a decent theory of ordinals sim99 we investigate classes of structures which are in a sense the well founded part of a larger sim pler class for example superatomic boolean algebras within the class of all boolean algebras the other classes we study are well founded trees reduced abelian p groups and countable compact topological spaces using com putable reductions between these classes we show that arithmetic transfinite recursion is the natural system for working with them natural statements such as comparability of structures in the class are equivalent to atr0 the reductions themselves are also objects of interest

sequential codes lossless compression of individual sequences and kolmogorov complexity a general class of sequential codes for lossless compression of individual sequences on a finite alphabet is defined including many types of codes that one would want to implement the principal requirement for membership in the class is that the encoding and decoding operations be performable on a computer the opta function for the class of codes is then considered which is the function that assigns to each individual sequence the infimum of the rates at which the sequence can be compressed over this class of sequential codes two results about the opta function are obtained 1 it is shown that any sequential code in the class compresses some individual sequence at a rate strictly greater than the rate for that sequence given by the opta function and 2 it is shown that the opta function takes a value strictly greater than that of the kolmogorov 1965 complexity rate function for some individual sequences

resolution of finite fuzzy relation equations abstract general schemes for solving fuzzy relation equations with finite sets are derived in this paper the relationship between the results derived here and those of previous publications regarding this subject is also discussed

sample converses in source coding theory the rate and distortion performance of a sequence of codes along a sample sequence of symbols generated by a stationary ergodic information source are studied two results are obtained 1 the source sample sequence is encoded by an arbitrary sequence of block codes which operate at a fixed rate level r and a sample converse is obtained which states that with probability one the lower limit of the code sample distortions is lower bounded by d r the value of the distortion rate function at r 2 the source sample sequence is encoded by an arbitrary sequence of variable rate codes which operate at a fixed distortion level d and a sample converse is obtained which states that with probability one the lower limit of the code sample rates is lower bounded by r d the value of the rate distortion function at d a novel ergodic theorem is used to obtain both sample converses

a computability theoretic equivalent to vaught s conjecture we prove that for every theory t which is given by an l 1 sentence t has less than 2 0 many countable models if and only if we have that for every x 2 on a cone of turing degrees every x hyperarithmetic model of t has an x computable copy we also find a concrete description relative to some oracle of the turing degree spectra of all the models of a counterexample to vaught s conjecture

on the relation between equations with max product composition and the covering problem systems of equations with max product composition are considered it is shown that solving these equations is closely related with the covering problem which belongs to the category of np hard problems it is proved that minimal solutions of equations correspond to irredundant coverings in terms of the covering problem the conditions of compatibility of equations of redundancy of equations of uniqueness of solution of uniqueness of minimal solution are determined concepts of essential non essential semi essential and super essential variables are suggested ways of simplification of a covering problem and methods of its solving are considered

a modified algorithm for solving the proposed models by ghodousian and khorram and khorram and ghodousian abstract in this paper we focus on the proposed algorithm for optimizing the linear function with fuzzy relation equation constraints regarding max prod composition that it has been proposed by ghodousian and khorram a ghodousian e khorram an algorithm for optimizing the linear function with fuzzy relation equation constraints regarding max prod composition appl math comput 178 2006 502 509 firstly we show that the algorithm may not lead to the optimal solution in some cases secondly we propose a new algorithm for solving the presented model by ghodousian and khorram 2006 as mentioned above in fact it modifies the presented algorithm in the ghodousian and khorram s paper also this algorithm is extended to the presented model by khorram and ghodousian e khorram a ghodousian linear objective function optimization with fuzzy relation equation constraints regarding max av composition appl math comput 173 2006 872 886 with max av composition finally some numerical examples are given for illustrating the purposes

an efficient solution procedure for fuzzy relation equations with max product composition we study a system of fuzzy relation equations with max product composition and present an efficient solution procedure to characterize the whole solution set by finding the maximum solution as well as the complete set of minimal solutions instead of solving the problem combinatorially the procedure identifies the nonminimal solutions and eliminates them from the set of minimal solutions

fuzzy relation equations on a finite set abstract the family of all the solutions of a fuzzy relation equation on a finite set is considered it is characterized by the set of all the lower solutions which can be obtained by a combinatorial algorithm