This is a survey on the relationship between logic and games. What do games have to save about logic, and conversely what does logic have to say about games? Johan Van Benthem in his lengthy manuscript Logic In Games sets forth an axiomatization of game equivalence and asks two main questions. The first is whether the axioms are complete for the semantic notion of game equivalence. Van Benthem also cautions that it is important to distinguish that games are “dynamic” activities, and that the meaning of a game is not fully captured by the assertion player has a winning strategy in it, and hence the second question is what constitutes this “dynamic aspect”. In this survey project, I will briefly discuss the difference between using games to determine results about logic and using logic to determine results about games. I then will discuss two responses in the affirmative to the first question by Van Benthem about the axiomatization of game logic with regard to logic in games. One is by Goranko which employs translations into modal logic to obtain the completeness result; the second is by Venema which uses a more general approach to show that game algebras and board algebras are isomorphic. I will also offer what seems to be a novel approach in responding to Van Benthem’s second question by suggesting that games are not fully captured by understanding whether a player has a winning strategy or not because games involve a dynamic action between intelligent agents who are trying to out think each other. In order to represent this dynamic process mathematically I propose that one must classify strategies themselves, and I will suggest ways of classifying strategies in the context of modal logic.

The approximations of activity mean, project duration belong to the most important activities in managing a construction project. Several researchers made an attempt to provide better Program Evaluation Review Technique (PERT) approximations using diverse probability distribution functions, for instance beta, normal, lognormal, triangular, weibull. Several researchers approximated the time estimates with three parameters optimistic time (a), most likely time(m), pessimistic time(b) and a few with two parameters (a,m) or (b,m). A usual supposition in project management is that the distribution for most activities is right skewed. Mohan et al.[4] suggested lognormal PERT approximations for a right skewed project and proved that the approximations are better than Traditional PERT, normal approximation, when the distributions are highly right skewed. The prime objective of this paper is to find effective distribution in conjunction with effective PERT approximations for right skewed projects. Our PERT approximations[14] are compared with normal, lognormal approximations and also with beta approximations with three parameters. The comparison reveals that PERT [14] performs better than lognormal and also other approximations suggested.

It is the purpose of this article to introduce a new weighted FGP technique by using only under deviation variables to fuzzy goals of fuzzy multi objective linear fractional goal programming problem (FMOLFGPP) to achieve highest degree of each of the membership goals by minimizing their under deviational variables. To assess the relative importance of the fuzzy goals properly, the weighting scheme by the decision of decision maker (DM) and Mohamed’s technique (Fuzzy Sets and Systems, 89 (1997) 215-222) has been used. Illustrative numerical examples of FMOLFGPP are provided to demonstrate the feasibility of the proposed method, which clearly shows the proposed approach yields better optimal solution of FMOLFGP problem than the conventional min sum FGP approach in the sense that it gives the values of the fractional objectives closer to their aspiration level.