determined theories and limit laws we provide a new model theoretic technique for proving 0 1 and convergence laws as an application we obtain a new slightly less computational proof of convergence laws due to spencer and thoma for the probability functions pnl in n n l in in n n c n

randomness and recursive enumerability one recursively enumerable real alpha dominates another one beta if there are nondecreasing recursive sequences of rational numbers a n n in omega approximating alpha and b n n in omega approximating beta and a positive constant c such that for all n c alpha a n geq beta b n see r m solovay draft of a paper or series of papers on chaitin s work manuscript ibm thomas j watson research center yorktown heights ny 1974 p 215 and g j chaitin ibm j res develop 21 1977 pp 350 359 we show that every recursively enumerable random real dominates all other recursively enumerable reals we conclude that the recursively enumerable random reals are exactly the omega numbers g j chaitin ibm j res develop 21 1977 pp 350 359 second we show that the sets in a universal martin lof test for randomness have random measure and every recursively enumerable random number is the sum of the measures represented in a universal martin lof test

vapnik chervonenkis density in some theories without the independence property ii we study the vapnik chervonenkis vc density of denable families in certain stable rst order theories in particular we obtain uniform bounds on vc density of denable families in nite u rank theories without the nite cover property and we characterize those abelian groups for which there exist uniform bounds on the vc density of denable families

closed asymptotic couples the derivation of a hardy field induces on its value group a certain function if a hardy field extends the real field and is closed under powers then its value group is also a vector space over such ordered vector spaces with function are called h couples we define closed h couples and show that every h couple can be embedded into a closed one the key fact is that closed h couples have an elimination theory solvability of an arbitrary system of equations and inequalities built up from vector space operations the function parameters and the unknowns to be solved for is equivalent to an effective condition on the parameters of the system the h couple of a maximal hardy field is closed and this is also the case for the h couple of the field of logarithmic exponential series over we analyze in detail finitely generated extensions of a given h couple

conditional probability logic lifted bayesian networks and almost sure quantifier elimination abstract we introduce a formal logical language called conditional probability logic cpl which extends first order logic and which can express probabilities conditional probabilities and which can compare conditional probabilities intuitively speaking although formal details are different cpl can express the same kind of statements as some languages which have been considered in the artificial intelligence community we also consider a way of making precise the notion of lifted bayesian network where this notion is a type of lifted probabilistic graphical model used in machine learning data mining and artificial intelligence a lifted bayesian network in the sense defined here determines in a natural way a probability distribution on the set of all structures in the sense of first order logic with a common finite domain d our main result theorem 3 14 is that for every noncritical cpl formula x there is a quantifier free formula x which is almost surely equivalent to x as the cardinality of d tends towards infinity this is relevant for the problem of making probabilistic inferences on large domains d because a the problem of evaluating by brute force the probability of x being true for some sequence d of elements from d has in general highly exponential time complexity in the cardinality of d and b the corresponding probability for the quantifier free x depends only on the lifted bayesian network and not on d some conclusions regarding the computational complexity of finding are given in remark 3 17 the main result has two corollaries one of which is a convergence law and zero one law for noncritial cpl formulas

towards human consistent multistage decision making and control models using fuzzy sets and fuzzy logic abstract fuzzy multistage decision making control models are presented two generations of them are distinguished the first consists of conventional models derived from bellman and zadeh s general framework fuzzy constraints goals decision etc the second consists of newer approaches which additionally allow for a more general aggregation of control stage scores outcomes e g at most of the control stages instead of at all conventionally used both generations have roots in papers co authored by bellman they are further steps towards human consistent soft models whose implementation should be much easier

linguistic quantifiers modeled by sugeno integrals since quantifiers have the ability of summarizing the properties of a class of objects without enumerating them linguistic quantification is a very important topic in the field of high level knowledge representation and reasoning this paper introduces a new framework for modeling quantifiers in natural languages in which each linguistic quantifier is represented by a family of fuzzy measures and the truth value of a quantified proposition is evaluated by using sugeno s integral this framework allows us to have some elegant logical properties of linguistic quantifiers we compare carefully our new model of quantification and other approaches to linguistic quantifiers a set of criteria for linguistic quantification was proposed in the previous literature the relationship between these criteria and the results obtained in the present paper is clarified some simple applications of the sugeno s integral semantics of quantifiers are presented

randomness and semigenericity let l contain only the equality symbol and let l be an arbitrary finite symmetric relational language containing l suppose probabilities are defined on finite l structures with edge probability n t by t the almost sure theory of random l structures we mean the collection of l sentences which have limit probability 1 t denotes the theory of the generic structures for ka the collection of finite graphs g with 6 a g igi a a edges of g hereditarily nonnegative 0 1 theorem t the almost sure theory of random l structures is the same as the theory tc of the ka generic model this theory is complete stable and nearly model complete moreover it has the finite model property and has only infinite models so is not finitely axiomatizable this paper unites two apparently disparate lines of research in 8 shelah and spencer proved a 0 1 law for first order sentences about random graphs with edge probability na where a is an irrational number between 0 and 1 answering a question raised by lynch 5 we extend this result from graphs to hypergraphs i e to arbitrary finite symmetric relational languages let t denote the set of sentences with limit probability 1 the spencer shelah proof proceeded by a process of quantifier elimination which implicitly showed the theories t were nearly model complete see below and complete hrushovski in 3 refuted a conjecture of lachlan by constructing an no categorical strictly stable pseudoplane baldwin and shi 1 considered a variant on his methods to construct strictly stable but not to categorical theories ta indexed by irrational a in this paper we show that for each irrational a t ta and thus deduce that t is not finitely axiomatizable and that ta is stable each ta is the theory of a generic model m of an amalgamation class ka of finite structures although the hrushovski examples are easily seen to be nearly model complete this is less clear for the t since they are not no categorical we show that each ta is nearly model complete in the first purely model theoretic section of the paper we describe our basic framework and prove a sufficient condition for certain theories including the ta to be nearly model complete these conditions depend upon a generalization of the notion of genericity of a structure semigenericity which is introduced in this paper in the second section we consider the addition of random relations and received by the editors september 7 1994 1991 mathematics subject classification primary 03c10 05c80

group decision making with a fuzzy linguistic majority abstract the determination of solutions in group decision making is considered the point of departure is a collection of individual fuzzy preference relations a solution is derived either directly from the individual fuzzy preference relations or by constructing first a social fuzzy preference relation as opposed to conventional approaches in which a crisp threshold type majority rule is used we employ a fuzzy majority rule specified by a fuzzy linguistic quantifier e g most much more than 50 etc a calculus of linguistically quantified propositions is applied using the fuzzy majority various solution concepts are derived mainly of the type of core minimax opposition set and consensus winner

quantifiers in the formulation of multiple objective decision functions abstract we discuss the interpretation of linguistically quantified statements using fuzzy sets we indicate that there are two methods of interpreting these statements the algebraic approach and the substitution approach we then use quantified statements to express overall decision functions involving multiple objectives these statements are of the type a solution should satisfy many of the objectives using the algebraic and substitution approach we develop a whole family of forms for combining multiple objectives we show that algebraic approach corresponds to situations in which we are unconcerned about the allocation of satisfaction amongst the objectives but just interested in getting much satisfaction the substitution approach on the other hand leads to forms which are dependent upon the allocation of satisfaction between the objectives

a soft measure of consensus in the setting of partial fuzzy preferences abstract consensus as traditionally meant to be a full and unanimous agreement is often not reachable in practice a degree of consensus for indicating how far a particular group of individuals is from consensus may be therefore very useful we propose a new measure degree of consensus which is more human consistent in the sense that it better reflects a real human perception of the essence of consensus in practice basically our consensus measure expresses the degree to which say most of the important individuals agree as to their testimonies concerning almost all of the relevant options the point of departure is the set of individual testimonies which are here the individual fuzzy preference relations as a formal tool we use a fuzzy logic based calculus of linguistically quantified propositions

on sufficient conditions for unsatisfiability of random formulas a descriptive complexity approach to random 3 sat is initiated we show that unsatisfiability of any significant fraction of random 3 cnf formulas cannot be certified by any property that is expressible in datalog combined with the known relationship between the complexity of constraint satisfaction problems and expressibility in datalog our result implies that any constraint propagation algorithm working with small constraints will fail to certify unsatisfiability almost always our result is a consequence of designing a winning strategy for one of the players in the existential pebble game the winning strategy makes use of certain extension axioms that we introduce and hold almost surely on a random 3 cnf formula the second contribution of our work is the connection between finite model theory and propositional proof complexity to make this connection explicit we establish a tight relationship between the number of pebbles needed to win the game and the width of the resolution refutations as a consequence to our result and the known size width relationship in resolution we obtain new proofs of the exponential lower bounds for resolution refutations of random 3 cnf formulas and the pigeonhole principle

analytic p adic cell decomposition and integrals we prove a conjecture of denef on parameterized p adic analytic integrals using an analytic cell decomposition theorem which we also prove in this paper this cell decomposition theorem describes piecewise the valuation of analytic functions and more generally of subanalytic functions the pieces being geometrically simple sets called cells we also classify silbanalytic sets up to subanalytic bijection

linguistic summaries of locally periodic time series this paper proposes a method to linguistically summarise the local periodic components of a time series it identifies subparts of the data which are periodic together with their periodicity degree and period and provides a linguistic description thereof the generated sentences can be illustrated by the example approximately from march to june the series is highly periodic with a period of exactly 2 weeks the method proposed to identify local periodic zones relies on the determination of relevant auto adaptive windows based on an analytical expression of the probability distribution of the considered periodicity criterion the linguistic description generation in the protoform approach framework expresses three core aspects of the identified periodic intervals namely their time context or localisation in time their periodicity and their period intensive experiments performed on both artificial and real data validate the proposed method

phase transitions in number theory from the birthday problem to sidon sets in this work we show how number theoretical problems can be fruitfully approached with the tools of statistical physics we focus on g sidon sets which describe sequences of integers whose pairwise sums are different and propose a random decision problem which addresses the probability of a random set of k integers to be g sidon first we provide numerical evidence showing that there is a crossover between satisfiable and unsatisfiable phases which converts to an abrupt phase transition in a properly defined thermodynamic limit initially assuming independence we then develop a mean field theory for the g sidon decision problem we further improve the mean field theory which is only qualitatively correct by incorporating deviations from independence yielding results in good quantitative agreement with the numerics for both finite systems and in the thermodynamic limit connections between the generalized birthday problem in probability theory the number theory of sidon sets and the properties of q potts models in condensed matter physics are briefly discussed

generically stable and smooth measures in nip theories we formulate the measure analogue of generically stable types in first order theories with nip without the independence property giving several characterizations answering some questions from 9 and giving another treatment of uniqueness results from 9 we introduce a notion of generic compact domination relating it to stationarity of keisler measures and also giving group versions we also prove the approximate definability of arbitrary borel probability measures on definable sets in the real and p adic fields

softer optimization and control models via fuzzy linguistic quantifiers a softening of a basic formulation of multicriterion optimization and control multistage decision making is presented for optimization instead of seeking an optimal solution that best satisfies all the fuzzy objectives as has been done so far we seek an optimal solution that best satisfies most much more than 50 etc a linguistic quantifier in general of the fuzzy objectives for control we seek in turn an optimal sequence of controls that best satisfies the fuzzy constraints and fuzzy goals at most much more than 50 etc of the control stages a calculus of linguistically quantified statements based upon fuzzy sets and possibility theory is used some applications to softer evidence aggregation in expert systems are also indicated

on finding compromise solutions in multicriteria problems using the fuzzy min operator abstract this paper presents an application of fuzzy approaches to the linear vectormaximum problem it shows that using the fuzzy min operator together with linear as well as special nonlinear membership functions the obtained solutions are always compromise solutions of the original multicriteria problem

random graphs and the parity quantifier the classical zero one law for first order logic on random graphs says that for every first order property p in the theory of graphs and every p 0 1 the probability that the random graph g n p satisfies p approaches either 0 or 1 as n approaches infinity it is well known that this law fails to hold for any formalism that can express the parity quantifier for certain properties the probability that g n p satisfies the property need not converge and for others the limit may be strictly between 0 and 1 in this work we capture the limiting behavior of properties definable in first order logic augmented with the parity quantifier foltr over g n p thus eluding the above hurdles specifically we establish the following modular convergence law for every foltr sentence p there are two explicitly computable rational numbers a0 a1 such that for i 0 1 as n approaches infinity the probability that the random graph g 2npi p satisfies p approaches ai our results also extend appropriately to fo equipped with modq quantifiers for prime q in the process of deriving this theorem we explore a new question that may be of interest in its own right specifically we study the joint distribution of the subgraph statistics modulo 2 of g n p namely the number of copies mod 2 of a fixed number of graphs f1 fe of bounded size in g n p we first show that every foltr property p is almost surely determined by subgraph statistics modulo 2 of the above type next we show that the limiting joint distribution of the subgraph statistics modulo 2 depends only on n mod 2 and we determine this limiting distribution completely interestingly both these steps are based on a common technique using multivariate polynomials over finite fields and in particular on a new generalization of the gowers norm the first step is analogous to the razborov smolensky method for lower bounds for ac0 with parity gates yet stronger in certain ways for instance it allows us to obtain examples of simple graph properties that are exponentially uncorrelated with every foltr sentence which is something that is not known for ac0ltr

on the number of sets definable by polynomials abstract we show that the known algorithms used to re write any first order quantifier free formula over an algebraically closed field into its normal disjunctive form are essentially optimal this result follows from an estimate of the number of sets definable by equalities and inequalities of fixed polynomials finally we apply our results to obtain similar estimates in the real case

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

a generalization of fuzzy multistage decision making and control via linguistic quantifiers abstract multistage incision making control under fuzziness is considered at each control stage a fuzzy constraint is imposed on the control applied and a fuzzy goal is imposed on the state attained an optimal sequence of controls is sought which maximizes the membership function of the fuzzy decision which is assumed to be the intersection of fuzzy constraints and fuzzy goals thin is basically equivalent to the determination of an optimal sequence of controls which best satisfies the fuzzy constraints and fuzzy goals at nil the control stages the generalization proposed in the paper replaces the strict all the something milder for example most thus we seek an optimal sequence of controls which best satisfies the fuzzy constraints and fuzzy goals at for example most of the control stages a calculus of linguistically quantified statements based upon fuzzy sets and possibility theory is used first the algebraic method is employed the resulting control problem is solved by dy

fuzzy quantification a state of the art abstract quantified sentences are a very powerful notion for modelling statements in natural language nl but in practice they have been used to solve several problems this paper is intended to offer a global view of the development on this branch until now focusing in the different approaches dealing with quantification specially those involving imprecision called fuzzy quantification we put attention to the different mechanisms for defining them the evaluation methods for measuring their fulfilment as well as the properties they should satisfy

a generalization of discounted multistage decision making andcontrol through fuzzy linguistic quantifiers an attempt to introduce commonsense knowledge abstract this paper is an extension of work published by kacprzyk in 1983 in which by using a fuzzy logic based calculus of linguistically quantified propositions some generalizations of multistage decision making and control have been proposed namely a new class of models has been developed that make it possible roughly speaking to find an optimal sequence of controls best satisfying fuzzy constraints and goals at say most or any other suitable linguistic quantifier for example almost all at least a few of the control stages in this paper we further extend the above mentioned approach by introducing discounting a variable importance of particular control stages in principle the earlier the more important we propose a new model that can find an optimal sequence of controls best satisfying the fuzzy constraints and goals at say most of the earlier control stages we employ a fuzzy logic based calculus of linguistically quantified propositions and develop a branch and bound algo

recursively enumerable reals and chaitin ω numbers a real is called recursively enumerable if it is the limit of a recursive increasing converging sequence of rationals following solovay unpublished manuscript ibm thomas j watson research center yorktown heights new york may 1975 215 pp and chaitin ibm j res develop 21 1977 350 359 496 we say that an r e real dominates an r e real if from a good approximation of from below one can compute a good approximation of from below we shall study this relation and characterize it in terms of relations between r e sets solovay s unpublished manuscript ibm thomas j watson research center yorktown heights new york may 1975 215 pp like numbers are the maximal r e real numbers with respect to this order they are random r e real numbers the halting probability of a universal self delimiting turing machine chaitin s number j assoc comput mach 22 1975 329 340 is also a random r e real solovay showed that any chaitin number is like in this paper we show that the converse implication is true as well any like real in the unit interval is the halting probability of a universal self delimiting turing machine

quasi random set systems there are many properties of mathematical objects that satisfy what is sometimes called a 0 1 law in the following sense under some natural probability measure on the set of objects the measure of the subset of objects having the given property is either 0 or 1 in the latter case we can say that almost all the objects have the property familiar examples of this phenomenon are the following almost all real numbers are transcendental or normal to every base almost all integers are composite almost all continuous real functions are nondifferentiable etc it is often the case that the objects under consideration can be partitioned into a countable number of finite classes cn with the probability assigned to an object in cn being just 1 icn i in this case we say that a property pn satisfies a 0 1 law if the fraction of the number of objects in cn that satisfy pn either tends to 0 or tends to 1 as n x 0o for example almost all graphs on n vertices have maximum cliques and maximum independent sets of size at most 2 log n almost all boolean functions with n variables have circuit complexity 1 o 1 2n and almost all binary codes of length n with at most 2nr codewords with r less than the binary symmetric channel capacity c have arbitrarily small error probability a special case of shannon s coding theorem see s48 one of the first general results of this type was the theorem of fagin f76 and glebskii et al gklt69 which asserts that every property of graphs that can be expressed in first order logic satisfies a 0 1 law see ss88 for recent striking developments in this topic one obvious method for finding explicit objects having some property pn shared by almost all objects in cn is simply to select one at random with overwhelming probability tending to 1 as n x oc the selected object will have property pn unfortunately it may be and often is extremely difficult to prove that any particular object does indeed satisfy pn it is our purpose in this paper to describe a method that can to a certain extent circumvent this difficulty we will show that for a variety of families it is possible to identify a natural hierarchy of equivalence classes of properties all of which are shared by almost all objects in the family any object satisfying

linear equations in variables which lie in a multiplicative group let k be a field of characteristic 0 and let n be a natural number let r be a subgroup of the multiplicative group k n of finite rank r given a 1 a n e k write a a 1 a n for the number of solutions x x 1 x n of the equation a 1 x 1 a n x n 1 such that no proper subsum of a 1 x 1 a n x n vanishes we derive an explicit upper bound for a a 1 a n r which depends only on the dimension n and on the rank r

when does the zero one law hold in 1960 paul erdbs and alfred renyi er began the subject of random graphs the 1985 book of bela bollobas b provides the standard reference for this field in modem terminology the random graph g n p is a graph on vertex set n 1 n where each pair i j of vertices are adjacent with independent probability p more accurately g n p is a probability space over the space of graphs on vertex set n for any property a of graphs there is a probability denoted pr g n p l a that g n p satisfies a in their very title on the evolution of random graphs erd6s and renyi envisioned a dynamic process g n p changing character as p moved from zero to one they discovered as did their many successors that for many natural properties a pr g n p t a was usually near zero or near one and made the jump from near zero to near one or back again in a very narrow range the placement of this critical range of p depended on n for example let a be the property of containing a triangle there are n n3 6 3 potential triangles each is a triangle in g n p with probability p 3 and so the expected number of triangles in g n p is asymptotically n3p3 6 this suggests the critical range p 8 1 n indeed erdos and renyi proved that if p p n l n then limn gc pr g n p l a 1 notation f n g n means limn oo f n g n ox they called p n l n a threshold function for this property a as other examples connectivity has threshold function logn n containing a clique on four points has threshold function n 2 3 containing an edge has easily threshold function n 2 and every vertex lying in a triangle has threshold function log n 13n 23 it was the observation that threshold functions seemed to be of the form log n n fl with a f rational that motivated our current line of research what can we say about the possible threshold functions of properties a not much if we place no restrictions on a for example the property that the number of edges is even shows no threshold function behavior if we restrict

rings of finite representation type and modules of finite morley rank this paper deals with rings whose modules have very pleasant decomposition properties the right pure semisimple rings the category of right modules over such a ring is in many ways similar to the category of modules over a semisimple artinian ring yet it is still an open question whether or not pure semisimplicity actually is a two sided property and hence 30 coincides with being of finite representation type most of the arguments in this paper are model theoretic algebraic in nature it seems that ideas from model theory and from the model theory of modules find natural application in the context considered here the paper is organized into three sections the first is introductory and with it i attempt to make the other sections accessible to a reasonably wide readership in the second section are gathered together various equivalents to right pure semisimplicity their equivalence is given a comparatively short unified essentially model theoretic algebraic proof the main theorem in the third section is that a ring is of finite representation type if and only if all its right modules have finite morley rank a number of related results are developed most of these involve some model theory in their statement but many are in essense algebraic since a good deal of information has been compressed into the introductory section this section should perhaps first be read through quickly and then referred back to as the need arises this section also is meant to serve as background to a sequel in which i consider pp formulas and types in terms of the corresponding matrices and in which the main aim is more purely algebraic the global conventions are r denotes a ring with identity module

groups measures and the nip we discuss measures invariant measures on definable groups and genericity often in an nip failure of the independence property environment we complete the proof of the third author s conjectures relating definably compact groups g in saturated o minimal structures to compact lie groups we also prove some other structural results about such g for example the existence of a left invariant finitely additive probability measure on definable subsets of g we finally introduce a new notion compact domination domination of a definable set by a compact space and raise some new conjectures in the o minimal case

computing halting probabilities from other halting probabilities the halting probability of a turing machine is the probability that the machine will halt if it starts with a random stream written on its one way input tape when the machine is universal this probability is referred to as chaitin s omega number and is the most well known example of a real which is random in the sense of martin lof although omega numbers depend on the underlying universal turing machine they are robust in the sense that they all have the same turing degree namely the degree of the halting problem this means that given two universal prefix free machines u v the halting probability u of u computes the halting probability v of v if this computation uses at most the first n g n bits of u for the computation of the first n bits of v we say that u computes v with redundancy g in this paper we give precise bounds on the redundancy growth rate that is generally required for the computation of an omega number from another omega number we show that for each 1 any pair of omega numbers compute each other with redundancy log n on the other hand this is not true for 1 in fact we show that for each omega number u there exists another omega number which is not computable from u with redundancy log n this latter result improves an older result of frank stephan

interpretability of fuzzy linguistic summaries this paper investigates the question of the interpretability of fuzzy linguistic summaries both at the sentence level and at the summary level seen as a set of sentences the individual sentence interpretability is examined as depending both on its representativity measured by a quality degree and on its linguistic expression different properties at the summary level are also discussed namely their consistency their non redundancy and the information they convey

working with strong reducibilities above totally omega c e and array computable degrees we investigate the connections between the complexity of a c e set as calibrated by its strength as an oracle for turing computations of functions in the ershov hierarchy and how strong reducibilities allow us to compute such sets for example we prove that a c e degree is totally w c e iff every set in it is weak truth table reducible to a hypersimple or ranked set we also show that a c e degree is array computable iff every left c e real of that degree is reducible in a computable lipschitz way to a random left c e real an number

externally definable sets and dependent pairs ii 2015 american mathematical society we continue investigating the structure of externally definable sets in nip theories and preservation of nip after expanding by new predicates most importantly types over finite sets are uniformly definable over a model a family of non forking instances of a formula with parameters ranging over a type definable set can be covered with finitely many invariant types we give some criteria for the boundedness of an expansion by a new predicate in a distal theory naming an arbitrary small indiscernible sequence preserves nip while naming a large one doesn t there are models of nip theories over which all 1 types are definable but not all n types

linear programming with multiple fuzzy goals abstract this paper describes the use of fuzzy set theory in goal programming gp problems in particular it is demonstrated how fuzzy or imprecise aspirations of the decision maker dm can be quantified through the use of piecewise linear and continuous functions models are then presented for the use of fuzzy goal programming with preemptive priorities with archimedean weights and with the maximization of the membership function corresponding to the minimum goal examples are also provided

zero one laws for sparse random graphs for random graph theorists see e g bollobas 1 for general reference p any constant is not the only not even the most interesting case rather they consider p p n a function approaching zero in their seminal paper erd6s and renyi 5 showed that for many interesting a there is a function p n which they called a threshold function so that if r n p n then f n r n a 0 while if p n r n then f n r n a 1 notation p r means lim p r 0 all limits are as n approaches infinity let us say p p n satisfies the zero one law if for all a in gra limf n p a 0 or 1 we shall partially characterize those p p n which satisfy the zero one law