As a data structure for representing and manipulating Boolean functions, BDDs (Binary Decision Diagrams) are commonly used in many fields such as model-checking, system verification and so on. For saving space and improving operation speed, all the existing packages limit the number of variables to 216. However, such a limitation also restrains its applicability. In this paper, we present TiniBDD, an efficient implementation of a 232BDD package incorporating sub-allocation of memory and lightweight Garbage Collection as well as a new operator named as Satisfiable Assignment Operator. Compared with the well-known CUDD which is one of the best 216BDD packages that can be attained publicly, the experiments show TiniBDD has comparable performance.

In this paper, we investigate knowledge reasoning within a simple framework called knowledge structure. We use variable forgetting as a basic operation for one agent to reason about its own or other agents\' knowledge. In our framework, two notions namely agents\' observable variables and the weakest sufficient condition play important roles in knowledge reasoning. Given a background knowledge base and a set of observable variables for each agent, we show that the notion of an agent knowing a formula can be defined as a weakest sufficient condition of the formula under background knowledge base. Moreover, we show how to capture the notion of common knowledge by using a generalized notion of weakest sufficient condition. Also, we show that public announcement operator can be conveniently dealt with via our notion of knowledge structure. Further, we explore the computational complexity of the problem whether an epistemic formula is realized in a knowledge structure. In the general case, this problem is PSPACE-hard; however, for some interesting subcases, it can be reduced to co-NP. Finally, we discuss possible applications of our framework in some interesting domains such as the automated analysis of the well-known muddy children puzzle and the verification of the revised Needham-Schroeder protocol. We believe that there are many scenarios where the natural presentation of the available information about knowledge is under the form of a knowledge structure. What makes it valuable compared with the corresponding multi-agent S5 Kripke structure is that it can be much more succinct.

The standard Kripke semantics of epistemic logics has been applied successfully to reasoning communication protocols under the assumption that the network is not hostile. This paper introduces a natural semantics of Kripke semantics called knowledge structure and, by this kind of Kripke semantics, analyzes communication protocols over hostile networks, especially on authentication protocols. Compared with BAN-like logics, the method is automatically implementable because it operates on the actual definitions of the protocols, not on some difficult-to-establish justifications of them. What is more, the corresponding tool called SPV (Security Protocol Verifier) has been developed. Another salient point of this approach is that it is justification-oriented instead of falsificationoriented, i.e. finding bugs in protocols.