algorithmic and logical characterizations of bisimulations for non deterministic fuzzy transition systems abstract bisimulation is a well known behavioral equivalence for discrete event systems and has been developed in fuzzy systems quickly in this paper we adopt an approach of the relational lifting that is one of the most used methods in the field of bisimulation research to define it for a non deterministic fuzzy transition system an o s 4 2 algorithm is given for testing bisimulation where s is the number of states and the number of transitions in the underlying transition systems two different modal logics are presented one is two valued and indicates whether a state satisfies a formula which is an extension of hennessy milner logic the other is real valued and shows to what extent a state satisfies a formula they both characterize bisimilarity soundly and completely interestingly the second characterization holds under a class of fuzzy logics in addition this real valued logic allows us to conveniently define a logical metric that captures the similarity between states or systems that is the smaller distance the more states alike although the work is inspired by the corresponding work in probabilistic systems it is obviously different in particular the real valued logic in this paper remains unexplored in fuzzy systems even in probabilistic systems

a note on modal logic and possibility theory abstract there are two theories in which the concept of possibility plays an important role modal logic and possibility theory the roles are different and so are the agendas of modal logic and possibility theory to gain an insight into the differences a very simple model of modal logic is constructed the model has the structure of a finite state system referred to as the fs model the fs model may be viewed as a simple interpretation of kripke model an interpretation which is easy to understand the fs model is in the spirit of graph models of modal logic the fs model readily lends itself to generalization concrete versions of the fs model serve as examples

on fuzzy modal logics s5 c the fuzzy variant s5 c of the well known modal logic s5 is studied c being a recursively axiomatized fuzzy propositional logic extending the basic fuzzy logic bl three kinds of kripke models are introduced and corresponding deductive systems are found

fuzzy logics with an additional involutive negation this paper surveys the present state of knowledge on propositional fuzzy logics extending sbl with an additional involutive negation the involutive negation is added as a new propositional connective in order to improve the expressive power of the standard mathematical fuzzy logics based on continuous triangular norms

analytical solutions of the fokker planck equation for generalized morse and hulthen potentials in the present contribution we analytically calculate solutions of the transition probability of the fokker planck equation fpe for both the generalized morse potential and the hulthen potential the method is based on the formal analogy of the fpe with the schrodinger equation using techniques from supersymmetric quantum mechanics

wythoff polytopes and low dimensional homology of mathieu groups abstract we describe two methods for computing the low dimensional integral homology of the mathieu simple groups and use them to make computations such as h 5 m 23 z z 7 and h 3 m 24 z z 12 one method works via sylow subgroups the other method uses a wythoff polytope and perturbation techniques to produce an explicit free z m n resolution both methods apply in principle to arbitrary finite groups

so 3 1 versus sp 4 r as dynamical potential algebra of the symmetrical poschl teller potentials the results of a previous paper ibid vol 21 p 4487 1988 concerned with the first family of two parameter poschl teller potentials are used to obtain potential and dynamical potential algebras for the subfamily of one parameter symmetrical poschl teller potentials they are respectively identified with so 3 and so 3 1 algebras and shown to be subalgebras of the so 4 potential and sl 4 r dynamical potential algebras of the two parameter potentials all the hamiltonian eigenstates corresponding to the subfamily of potentials with quantised potential strengths mu differing by an integer are proved to belong to a single degenerate continuous unitary irreducible representation of so 3 1 which may be labelled by generalised young pattern labels omega 0 where omega 1 1 4i eta and eta is some real parameter an sp 4 r approximately so 3 2 algebra is also constructed and shown to provide an alternative choice for the dynamical potential algebra of the symmetrical potentials all the above mentioned eigenstates belong to a single sp 4 r unitary irreducible representation of the positive discrete series which may be labelled by its lowest weight 1 2 1 2 such an alternative choice for the dynamical potential algebra however does not lead to a unified treatment of the one and two parameter potentials as does the first one

an sl 4 r lie algebraic approach to the bargmann functions and its application to the second poschl teller equation the sl 4 r lie algebraic treatment of the winger su 2 matrices 1959 is extended by analytical continuation to the bargmann su 1 1 matrices 1947 corresponding to the positive discrete series irreducible representations it is then used to obtain an sl 4 r dynamical potential algebra for the negative energy solutions of the second poschl teller equations 1933

explicit wavefunctions for shape invariant potentials by operator techniques the authors obtain explicit expressions for the wavefunctions of all the known shape invariant potentials by using the operator method

algebraic treatment of second poschl teller morse rosen and eckart equations the method of an earlier paper see ibid vol 20 p 4075 1987 is applied to the non compact case to solve a family of second poschl teller morse rosen and eckart equations with quantised coupling constants both discrete and continuous spectra bound state and scattering wavefunctions transmission coefficients are found from the matrix elements of group representations

lie algebras for systems with mixed spectra i the scattering poschl teller potential starting from an n body quantum space we consider the lie algebraic framework where the poschl teller hamiltonian 1 2 2 c sech2 s csch2 is the single sp 2 r casimir operator the spectrum of this system is mixed it contains a finite number of negative energy bound states and a positive energy continuum of free states it is identified with the clebsch gordan series of the d d representation coupling the wave functions are the sp 2 r clebsch gordan coefficients of that coupling in the parabolic basis using only lie algebraic techniques we find the asymptotic behavior of these wave functions for the special pure trough potential s 0 we derive thus the transmission and reflection amplitudes of the scattering matrix

some homotopy equivalences for sporadic geometries a previous work rsy90 established the projectivity of the reduced lefschetz modules of certain sporadic group geometries and the present paper continues that work in a wider context recent developments in sporadic group cohomology include some applications of rsy90 which in turn suggested treatment of a broader class of geometries recurring similarities in the proofs also led to a more unified treatment establishing the stronger result of homotopy equivalence of thep local geometry with the usual elementary poset ap g one equivalence method proceeds by means of a new closed set in a standard technique of quillen it was further observed that the larger list of simple groups now treated essentially coincides with those of characteristicp type suggesting another equivalence method via the poset bp g of radical or stubborn p subgroups in particular one finds that these sporadic groups satisfy an analogue of the borel tits theorem that normalizers ofp groups lie in simplex stabilizers still further intriguing coincidences remain to be explained

an approximate κ state solutions of the dirac equation for the generalized morse potential under spin and pseudospin symmetry by using an improved approximation scheme to deal with the centrifugal pseudo centrifugal term we solve the dirac equation for the generalized morse potential with arbitrary spin orbit quantum number in the presence of spin and pseudospin symmetry the analytic bound state energy eigenvalues and the associated upper and lower spinor components of two dirac particles are found by using the basic concepts of the nikiforov uvarov method we study the special cases when 1 l l 0 s wave the non relativistic limit and the limit when becomes zero kratzer potential model the present solutions are compared with those obtained by other methods

the geometry and cohomology of the mathieu group m12 it is usually very complex to calculate the cohomology of a finite group g to date the most significant results have been on the symmetric and alternating groups n m m ma mu a m m the general linear groups over a finite field q1 and the extra special p groups q2 in the list above only the alternating groups and the gln f2 are simple indeed among the simple groups only a very few aside from those above have been understood it is probable that away from the characteristic p h psln fpn z q is available but even this is not obvious especially for q 2 and certainly we currently have no information on most of the families of groups of lie type for example g2 fq e6 fq e7 fq e8 fq or any of the sporadic groups but m11 and j1 about 15 years ago quillen introduced very powerful techniques that allowed for a much deeper understanding of group cohomology however it is also apparent that successful calculations over z p have only occurred when the group has a well behaved lattice of p elementary abelian subgroups formidable combinatorial problems arise in the general situation as well as necessarily complicated multiplicative relations for example the general calculation of h gln fp z p remains quite inaccessible and there has been only marginal progress since quillen s landmark results

the potential group approach and hypergeometric differential equations this paper proposes a generalized realization of the potential groups so 2 1 and so 2 2 to describe the confluent hypergeometric and the hypergeometric equations respectively it implies that the classes of schrodinger equations with solvable potentials whose analytical solutions are related to the confluent hypergeometric and the hypergeometric functions can be realized in terms of the above group structure

an sl 4 r lie algebraic treatment of the first family of poschl teller potentials an sl 4 r dynamical potential algebra containing the so 4 potential algebra is constructed for the two parameter poschl teller potentials of the first kind for this purpose the relation between the wigner rotation matrices and the solutions of the first poschl teller equation is used explicit expressions are given for the sl 4 r generators as well as for their action on the normalised solutions all the hamiltonian eigenstates corresponding to the family of potentials with quantised potential strengths m m differing by integers are proved to belong to a single sl 4 r unitary irreducible representation of the ladder series for integral values of m and m the latter is dladd 0 0 eta while for half integral values it is dladd 1 2 1 2 eta where eta is some real parameter both irreducible representations are also shown to be characterised by generalised young pattern labels pqr0 where p 2 1 2i eta and q r 0

bisimulations for fuzzy transition systems there has been a long history of using fuzzy language equivalence to compare the behavior of fuzzy systems however the comparison at this level is too coarse recently a finer behavioral measure i e bisimulation has been introduced to fuzzy finite automata however the results obtained are applicable only to finite state systems in this paper we consider bisimulation for general fuzzy systems which may be infinite state or infinite event by modeling them as fuzzy transition systems ftss to help understand and check bisimulation we characterize it in three ways by enumerating whole transitions comparing individual transitions and using a monotonic function in addition we address composition operations subsystems quotients and homomorphisms of ftss and discuss their properties connected with bisimulation the results presented here are useful to compare the behavior of general fuzzy systems in particular this makes it possible to relate an infinite fuzzy system to a finite one which is easier to analyze with the same behavior

symmetric invariants and cohomology of groups some time ago nakaoka in determined the additive structure of h z f2 around the same time the ring structure of h 274 f2 was calculated and it was assumed that the same type of simple multiplicative relations would appear for n 4 since then most work has concentrated on the embedding 2 o using the polynomial ring structure of h xo f2 in particular most of the recent work in hi was explicitly or implicitly described in m m more than a decade ago and provides no new information on the multiplicative structure of the mod 2 cohomology of finite symmetric groups in this paper we use invariant theory to detect unexpected multiplicative relations in h z f2 the complicated nature of the numerical evidence which we present explains the absence of complete calculations for n 16 the rings of invariants which we study are at the core of any computation of h 2 f2 they build up sucessively yielding relations rich in symmetry but of a highly convoluted type the following problem lies at the heart of all this let p be a polynomial ring k over f2 on n variables and let p k p then ek acts by permuting blocks of generators 1

generalized morse potential symmetry and satellite potentials we study in detail the bound state spectrum of the generalized morse potential gmp which was proposed by deng and fan as a potential function for diatomic molecules by connecting the corresponding schrodinger equation with the laplace equation on the hyperboloid and the schrodinger equation for the poschl teller potential we explain the exact solvability of the problem by an symmetry algebra and obtain an explicit realization of the latter as we prove that some of the generators connect among themselves wavefunctions belonging to different gmps called satellite potentials the conserved quantity is some combination of the potential parameters instead of the level energy as for potential algebras hence belongs to a new class of symmetry algebras we also stress the usefulness of our algebraic results for simplifying the calculation of frank condon factors for electromagnetic transitions between rovibrational levels based on different electronic states

a behavioral distance for fuzzy transition systems in contrast with the existing approaches to exact bisimulation for fuzzy systems we introduce a robust notion of behavioral distance to measure the behavioral similarity of nondeterministic fuzzy transition systems which are a generalization of fuzzy automata this behavioral distance provides a quantitative analogue of bisimilarity and is defined as the greatest fixed point of a suitable monotonic function the behavioral distance has the important property that two systems are at zero distance if and only if they are bisimilar moreover for any given threshold we find that systems with behavioral distances bounded by the threshold are equivalent in addition we show that two system combinators parallel composition and product are nonexpansive with respect to our behavioral distance which makes compositional verification possible the theory developed here is applicable to the quantitative verification approximate reduction and reliability analysis of fuzzy transition systems

a new realisation of dynamical groups and factorisation method a new method of algebraisation of quantum mechanical eigenvalue equations is presented in this method the dynamical algebra is represented on the space of group matrix elements the ladder operators of the dynamical algebra are obtained from infeld hull miller factorisations the method is used to study the first poschl teller equation even in the non symmetric case the energy spectrum and the exact normalised solutions are obtained in agreement with the results of non algebraic methods

bisimulations for fuzzy automata bisimulations have been widely used in many areas of computer science to model equivalence between various systems and to reduce the number of states of these systems whereas uniform fuzzy relations have recently been introduced as a means to model the fuzzy equivalence between elements of two possible different sets here we use the conjunction of these two concepts as a powerful tool in the study of equivalence between fuzzy automata we prove that a uniform fuzzy relation between fuzzy automata a and b is a forward bisimulation if and only if its kernel and co kernel are forward bisimulation fuzzy equivalence relations on a and b and there is a special isomorphism between factor fuzzy automata with respect to these fuzzy equivalence relations as a consequence we get that fuzzy automata a and b are ufb equivalent i e there is a uniform forward bisimulation between them if and only if there is a special isomorphism between the factor fuzzy automata of a and b with respect to their greatest forward bisimulation fuzzy equivalence relations this result reduces the problem of testing ufb equivalence to the problem of testing isomorphism of fuzzy automata which is closely related to the well known graph isomorphism problem we prove some similar results for backward forward bisimulations and we point to fundamental differences because of the duality with the studied concepts backward and forward backward bisimulations are not considered separately finally we give a comprehensive overview of various concepts on deterministic nondeterministic fuzzy and weighted automata which are related to bisimulations

logical characterizations of regular equivalence in weighted social networks abstract social network analysis is a methodology used extensively in social science classical social networks can only represent the qualitative relationships between actors but weighted social networks can describe the degrees of connection between actors in a classical social network regular equivalence is used to capture the similarity between actors based on their links to other actors specifically two actors are deemed regularly equivalent if they are equally related to equivalent others the definition of regular equivalence has been extended to weighted social networks in two ways the first definition called regular similarity considers regular equivalence as an equivalence relation that commutes with the underlying graph edges while the second definition called generalized regular equivalence is based on the notion of role assignment or coloring a role assignment resp coloring is a mapping from the set of actors to a set of roles resp colors the mapping is regular if actors assigned to the same role have the same roles in their neighborhoods recently it was shown that social positions based on regular equivalence can be syntactically expressed as well formed formulas in a kind of modal logic thus actors occupying the same social position based on regular equivalence will satisfy the same set of modal formulas in this paper we present analogous results for regular similarity and generalized regular equivalence based on many valued modal logics

dynamical potential algebras for gendenshtein and morse potentials a differential realization of so 2 1 is shown to be the potential algebra for the one dimensional systems with the morse or gendenshtein potentials this shows that two classes of gendenshtein potentials will support the same eigenvalues as the morse potential and that the three sets of eigenfunctions may be derived in a common formalism the potential algebra is then extended to a dynamical potential algebra with operators connecting states both in different potentials and with different energies giving new dynamical algebras for the gendenshtein problem the matrix elements of certain corresponding operators in the three types of system may then be given by a single formula involving so 2 1 wigner coefficients the authors also give ladder operators connecting the gendenshtein potential eigenstates

making fuzzy description logic more general a version of fuzzy description logic based on the basic continuous t norm based fuzzy predicate logic bl is presented problems of satisfiability validity and subsumption of concepts are discussed and reduced to problems of fuzzy propositional logic known to be decidable for any continuous t norm for lukasiewicz t norm some stronger results are obtained

computation of the greatest simulations and bisimulations between fuzzy automata recently two types of simulations forward and backward simulations and four types of bisimulations forward backward forward backward and backward forward bisimulations between fuzzy automata have been introduced if there is at least one simulation bisimulation of some of these types between the given fuzzy automata it has been proved that there is the greatest simulation bisimulation of this kind in the present paper for any of the above mentioned types of simulations bisimulations we provide an efficient algorithm for deciding whether there is a simulation bisimulation of this type between the given fuzzy automata and for computing the greatest one whenever it exists the algorithms are based on the method developed in ignjatovic et al on the greatest solutions to weakly linear systems of fuzzy relation inequalities and equations fuzzy sets syst 161 2010 3081 3113 which comes down to the computing of the greatest post fixed point contained in a given fuzzy relation of an isotone function on the lattice of fuzzy relations

bisimulation and bisimilarity for fuzzy description logics under the godel semantics abstract description logics dls are a suitable formalism for representing knowledge about domains in which objects are described not only by attributes but also by binary relations between objects fuzzy extensions of dls can be used for such domains when data and knowledge about them are vague and imprecise one of the possible ways to specify classes of objects in such domains is to use concepts in fuzzy dls as dls are variants of modal logics indiscernibility in dls is characterized by bisimilarity the bisimilarity relation of an interpretation is the largest auto bisimulation of that interpretation in dls and their fuzzy extensions such equivalence relations can be used for concept learning in this paper we define and study fuzzy bisimulation and bisimilarity for fuzzy dls under the godel semantics as well as crisp bisimulation and strong bisimilarity for such logics extended with involutive negation the considered logics are fuzzy extensions of the dl alc r e g a variant of pdl with additional features among inverse roles nominals qualified or unqualified number restrictions the universal role local reflexivity of a role and involutive negation we formulate and prove results on invariance of concepts under fuzzy resp crisp bisimulation conditional invariance of fuzzy tboxes aboxes under bisimilarity resp strong bisimilarity and the hennessy milner property of fuzzy resp crisp bisimulation for fuzzy dls without resp with involutive negation under the godel semantics apart from these fundamental results we also provide results on using fuzzy bisimulation to separate the expressive powers of fuzzy dls as well as results on using strong bisimilarity to minimize fuzzy interpretations

cohomology of the double cover of the mathieu group m12 abstract in this paper we calculate the mod two cohomology of the double cover of the mathieu group m 12 the starting point is the calculation by adem maginnis and milgram of the mod two cohomology of m 12 we use a hypercohomology spectral sequence to determine a differential in the lyndon hochschild serre spectral sequence of the central extension and this gets us as far as the e page ungrading requires restriction to the sylow 2 subgroup and here some computer calculations come to our rescue

fuzzy bisimulation for godel modal logic bisimulation is a central concept in the model theory of modal logic with extensive computational applications it is a relation between two models in which related states have identical atomic properties and matching transition possibilities bisimulation captures the expressive power of propositional modal logic in the sense that bisimilar states are not distinguishable by any propositional modal formula in recent years fuzzy modal logic has received much attention because of the connection between uncertainty measures and fuzzy modalities in this paper we define the notion of fuzzy bisimulation and prove that in a fuzzified version it bears the same relationship to fuzzy modal logic that bisimulation bears to propositional modal logic