a complete axiom system for propositional projection temporal logic with cylinder computation model to specify and verify multi core parallel programs in a uniform framework this paper proposes an axiom system for ccm pptl which extends that of pptl by including transformation rules for sequence expressions and axioms as well as inference rules on the ccm construct further the soundness and completeness of the extended axiom system are proved

on consistency and ranking of alternatives in fuzzy ahp abstract in fuzzy ahp where pairwise comparisons are described by membership values the treatment of consistency has largely been sketchy this paper proposes a fuzzy consistency definition with consideration of a tolerance deviation essentially the fuzzy ratios of relative importance allowing certain tolerance deviation are formulated as constraints on the membership values of the local priorities the fuzzy local and global weights are determined via the extension principle the alternatives are ranked on the basis of the global weights by application of a maximum minimum set ranking method our analysis uses only the data which are considered as consistent the proposed approach is computationally efficient as it consists of solving only linear programs an example is included for illustrative purpose

min transitivity of fuzzy leftness relationship and its application to decision making abstract we show that the fuzzy leftness relationship left a b between two intervals a and b on a line which we had recently defined for modeling spatial relationships satisfies the min transitivity property the previous definitions of leftness given by krishnapuram et al and miyajima et al do not have this property the min transitivity property plays an important role in fuzzy decision making based on orlovsky s result we give several general schemes for constructing concrete min transitive relations based on our left a b for use in fuzzy decision making

stabilization of the lattice boltzmann method using the ehrenfests coarse graining idea the lattice boltzmann method lbm and its variants have emerged as promising computationally efficient and increasingly popular numerical methods for modeling complex fluid flow however it is acknowledged that the method can demonstrate numerical instabilities e g in the vicinity of shocks we propose a simple technique to stabilize the lbm by monitoring the difference between microscopic and macroscopic entropy populations are returned to their equilibrium states if a threshold value is exceeded we coin the name ehrenfests steps for this procedure in homage to the vehicle that we use to introduce the procedure namely the ehrenfests coarse graining idea

fuzzy hierarchical analysis the lambda max method abstract we present a new method of finding the fuzzy weights in fuzzy hierarchical analysis which is the direct fuzzification of the max method used by saaty in the analytical hierarchical process an example is presented where there are five criteria and three alternatives

fuzzy logarithmic least squares ranking method in analytic hierarchy process abstract we xu and zhai 1992 have discussed the extensions of analytic hierarchy process in fuzzy environment and presented a procedure for constructing the fuzzy judgment matrix this paper continues the discussion and goes further into the problem about extracting the fuzzy weights from the fuzzy judgment matrix by logarithmic least squares method which is a main ranking method in analytic hierarchy process first we define the metric of the fuzzy judgment space and develop the expression form of logarithmic least squares method in fuzzy environment next we derive the associated normal equations from the problem of fuzzy logarithmic least squares and prove that the solutions of fuzzy logarithmic least squares model are equivalent to the solutions of its normal equations finally for every t 0 1 we can get its explicit solutions by solving the normal equations the technique proposed in this paper is rather simple to realize and can be used for dealing with actual problems

on comparing interval numbers abstract this paper first presents a brief survey of the existing works on comparing and ranking any two interval numbers on the real line and then on the basis of this gives two approaches to compare any two interval numbers the first one describes a value judgement index along with a discussion on its strength and weakness over the other approaches this index has also been called an optimistic decision maker s preference index the other approach defines strict and fuzzy preference ordering between any two interval numbers from a pessimistic decision maker s point of view choice of different decision makers with different degrees of pessimism has also been considered

model checking of pushdown systems for projection temporal logic abstract in this paper we study the model checking problem of pushdown systems for projection temporal logic ptl an interval based temporal logic that is as expressive as the full regular languages for this we provide an algorithm to decide whether a pushdown system determined by a pushdown automaton satisfies a desired logic property by using relevant automata notations and operations the algorithm terminates in exponential time for normal ptl properties i e properties specified as normal ptl formulas to show the lower bound of complexity of the model checking problem we also construct a polynomial time reduction from the acceptance problem of a linearly bounded alternating turing machine as a result the model checking problem of pushdown systems for normal ptl properties is exptime complete

a fuzzy extension of saaty s priority theory we present a fuzzy method for choosing among a number of alternatives under conflicting decision criteria a fuzzy version of saaty s pairwise comparison method 1980 extended by de graan 1980 and lootsma 1981 each ratio expressing the relative significance of a pair of factors is displayed in a matrix from which suitable weights can be extracted since these ratios are essentially fuzzy they express the opinion of a decision maker on the importance of a pair of factors we have adapted the above mentioned method in such a way that decision makers are asked to express their opinions in fuzzy numbers with triangular membership functions we apply the method at two distinct levels first to find fuzzy weights for the decision criteria and second to find fuzzy weights for the alternatives under each of the decision criteria by a suitable combination of these results we obtain fuzzy scores for the alternatives as well as their sensitivities using these fuzzy scores the decision makers should be able to make a choice for one of the alternatives

preference simulation and preference programming robustness issues in priority derivation abstract decision makers often resist having to make what appears to them as precise numerical judgements in fuzzy situations pairwise verbal comparisons used in the ahp are fuzzy in the sense that decision maker s need not relate verbal judgment to precise numbers because of the redundancy inherent in each set of judgments accurate priorities can be derived from such fuzzy verbal judgments another way of making fuzzy judgments is to express each judgment as a numerical interval this paper explores two new approaches for priority derivation when preferences are expressed as interval judgments one based on a simulation approach and the other based on mathematical programming the first approach assumes that the interval judgments are uniformly distributed and proceeds to derive the priority vectors and their underlying rank order by randomly sampling from these distribution this approach provides in addition to the priority vectors a measure of robustness given by the probability of rank reversal the second approach generates a region if one exists that encloses all priority vectors derived from inequalities representing the original interval judgments the two approaches are described and illustrated through a numerical example

approximate articulation of preference and priority derivation a comment abstract in a recent paper by a arbel 1989 the standard analytic hierarchy process ahp model is extended to the case where preference strength is expressed in terms of a range of scale values rather than a single value while this extension holds and is natural for the consistent case it is shown that the proposed solution for the more common inconsistent case is erroneous

uncertainty and rank order in the analytic hierarchy process abstract the analytic hierarchy process uses paired comparisons to derive a scale of relative importance for alternatives we investigate the effect of uncertainty in judgment on the stability of the rank order of alternatives the uncertainty experienced by decision makers in making comparisons is measured by associating with each judgment an interval of numerical values the approach leads to estimating the probability that an alternative or project exchanges rank with other projects these probabilities are then used to calculate the probability that project would change rank at all we combine the priority of importance of each project with the probability that it does not change rank to obtain the final ranking

fuzzy hierarchical analysis revisited abstract we present a new method of finding the fuzzy weights in fuzzy hierarchical analysis which is the direct fuzzification of the original method used by saaty in the analytic hierarchy process we test our new procedure in two cases where there are formulas for the crisp weights an example is presented where there are five criteria and three alternatives

interval weight generation approaches based on consistency test and interval comparison matrices a simple yet pragmatic method of consistency test is developed to check whether an interval comparison matrix is consistent or not linear programming method is used to derive consistent interval weights from consistent interval comparison matrices and to aggregate local interval weights to generate global interval weights in cases where an interval comparison matrix is inconsistent an eigenvector method based nonlinear programming nlp approach is developed to generate interval weights that can meet pre determined consistency requirements a simple and effective preference ranking method is utilized to compare the interval weights of criteria or rank alternatives three numerical examples including a hierarchical ahp decision problem are provided to illustrate the validity and practicality of the proposed methods

preference programming through approximate ratio comparisons abstract in the context of hierarchical weighting this paper operationalizes interval judgments which allow the decision maker to enter ambiguous preference statements by indicating the relative importance of factors as intervals of values on a ratio scale through such judgments the decision maker can capture the subjective uncertainty in his preferences and thus avoid the often cumbersome elicitation of exact ratio estimates after each new statement the interval judgments are synthesized into dominance relations on the alternatives by solving a series of linear programming problems this leads to an interactive process of preference programming which provides more detailed results as the decision maker gradually enters a more specific preference description moreover the overall effort of preference elicitation is smaller than in the analytic hierarchy process because the most preferred alternative can usually be identified before all possible comparisons between pairs of factors have been completed

index set expressions can represent temporal logic formulas abstract in temporal logic tl a well formed formula is generally formed by applying rules of its syntax finitely many times however under some circumstances although formulas such as ones expressed by index set expressions are constructed via applying rules of the syntax infinitely many times they are possibly still well formed since their equivalent concise syntax formulas can be found with this motivation this paper investigates the relationship between formulas specified by index set expressions and concise syntax by means of fixed point approach firstly we present two kinds of formulas namely i n 0 i q and i n 0 q i and prove they are indeed well formed by proving they are equivalent to formulas q and q respectively further we generalize i n 0 i q to i n 0 p i i q and explore the least and greatest fixed points of an abstract equation x q p x based on these some well formed special instances of i n 0 p i i q are obtained besides with the index set expression technique we equivalently represent u strong until and w weak until constructs of propositional linear temporal logic ltl within propositional projection temporal logic pptl

multicriteria decision analysis with fuzzy pairwise comparisons abstract we present a modification of the fuzzy multi criteria method proposed by van laarhoven and pedrycz 1983 first the weights of the decision criteria are calculated by the minimization of a logarithmic regression function next the weights of the decision alternatives are calculated for each criterion separately lastly the fuzzy final scores of the alternatives are determined by an appropriate aggregationm of the calculated weights we will show that our modified method in contrast with the original one yields weights which are optimal with respect to the logarithmic regression function we also present a version of our method with geometric ratio scales rather than saaty s equi distant scale to quantify the gradations of human comparative judgement this approach also enables us to develop an efficient procedure for investigating the scale sensitivity of the calculated weights and the final scores of the alternatives

fuzzy least squares priority method in the analytic hierarchy process the least squares priority method is an effective priority method in the analytic hierarchy process but it is not appropriate under fuzziness this paper is concerned with the question of estimation of weights of factors by least squares from a fuzzy judgment matrix first the expression form of least squares under fuzziness is given then the associated system of equation is developed after some properties of the system are investigated the explicit solution is derived finally it is shown that the solution of the system is the solution of the fuzzy least squares in this paper an example is also given to illustrate the procedure

a canonical form based decision procedure and model checking approach for propositional projection temporal logic this paper proposes a canonical form cf for chop formulas of propositional projection temporal logic pptl based on cf an improved algorithm for constructing labeled normal form graph lnfg of a pptl formula is presented this improvement leads to a better decision procedure for pptl with infinite models in addition a transformation from lnfgs to generalized buchi automata gba and then buchi automata ba is formalized thus a spin based model checking approach is generalized for pptl to illustrate how these algorithms work several examples are given

a practical decision procedure for propositional projection temporal logic with infinite models this paper presents a practical decision procedure for propositional projection temporal logic with infinite models first a set prop l of labels l i 0 i n n 0 is used to mark nodes of an lnfg of a formula and a node with l i is treated as an accepting state as in an automaton further the generalized buchi accepting condition for automata is employed to identify a path resulting a word in an lnfg as a model of the formula in addition the implementation details of the decision procedure and relevant algorithms including pre processing lnfg circle finding algorithms are presented as a matter of fact all algorithms are implemented by c programs

a statistical approach to the analytic hierarchy process with interval judgements i distributions on feasible regions abstract this paper addresses the problem of extracting preferences for alternatives from interval judgement matrices in the analytic hierarchy process ahp the method of arbel for extracting such preferences which is based on the assumption that the interval judgements specify a feasible region in the weight space of the alternatives is critically appraised from a statistical perspective and some new ideas emanating from this approach are developed and discussed in particular it is proposed that a distribution for the weights on the feasible region which is both tractable and meaningful be adopted the mean of the distribution can then be used as an assessment of the overall ranking of the alternatives and quantities of interest such as probabilities and marginal distributions can immediately be quantified two specific distributions on the feasible region the uniform distribution and the distribution of random convex combinations with coefficients which are uniform spacings are examined in some detail and the ideas which emerge are illustrated by means of selected examples

a decision procedure and complete axiomatization for projection temporal logic abstract to specify and verify the concurrent and reactive systems with the theorem proving approach a complete axiomatization is formalized for first order projection temporal logic ptl with both finite and infinite time to this end ptl is restricted to a finite domain and the syntax semantics as well as the axiomatization of ptl are presented further the techniques of labeled normal form and labeled normal form graph of ptl formulas are introduced respectively with which a decision procedure for quantifier free ptl qfptl formulas is given moreover a generalized labeled normal form graph is defined and employed to transform a quantified ptl formula into its equivalent qfptl formula finally a decision procedure for ptl is formalized and the completeness of the axiomatic system is proved based on the decidability of ptl formulas

deriving priorities from fuzzy pairwise comparison judgements a new approach for deriving priorities from fuzzy pairwise comparison judgements is proposed based on cuts decomposition of the fuzzy judgements into a series of interval comparisons the assessment of the priorities from the pairwise comparison intervals is formulated as an optimisation problem maximising the decision maker s satisfaction with a specific crisp priority vector a fuzzy preference programming method which transforms the interval prioritisation task into a fuzzy linear programming problem is applied to derive optimal crisp priorities aggregating the optimal priorities which correspond to different cut levels enables overall crisp scores of the prioritisation elements to be obtained a modification of the linear fuzzy preference programming method is also proposed to derive priorities directly from fuzzy judgements without applying cut transformations the formulation of the prioritisation problem as an optimisation task is similar to the previous approach but it requires the solution of a non linear optimisation program the second approach also derives crisp priorities and has the advantage that it does not need additional aggregation and ranking procedures both proposed methods are illustrated by numerical examples and compared to some of the existing fuzzy prioritisation methods

multiobjective programming in optimization of the interval objective function abstract in this paper the mathematical programming problem whose objective function has interval coefficients is investigated to maximize the interval objective function the order relations which represent the decision maker s preference between interval profits are defined by the right limit the left limit the center and the width of an interval the maximization problem with the interval objective function is converted into a multiobjective problem using the order relations the minimization problem is also converted into a multiobjective problem using the order relations which represent the decision maker s preference between interval costs